a great range of techniques and tools have been developed and studied for the prediction of program behaviours without actually running the program .one of them , originating from type inference in functional programming languages , is the type and effect discipline . as a refinement of type systems in programming languages ,types are annotated with information characterizing dynamic behaviours of programs _ effects_. as a result , a well - typed program satisfies some properties regarding its side - effects as well .this type - based technique has been used for all kinds of static analysis of programs , e.g. flow analysis , dependency analysis , resource allocation analysis , and amortised analysis , etc .in particular , a type and effect system was developed by grabowski et al . to verify that a particular programming guideline for secure web - programming has been adhered to .generalizing from this , one could model a programming guideline as a property of traces that a program might have , where traces are sequences of events that are issued by a certain instrumentation of the program with special event - issuing operations. this instrumentation would be part of the formalised guideline .a finite state machine would then be used to specify the set of acceptable traces .most policies involve safety properties which can be assessed by examining finite portions of traces . in some cases, however , properties pertaining to liveness and fairness can become relevant .for instance , a guideline could be that calls to appropriate logging functions must be made again and again or that event ( sic ! ) handlers should not become stuck , e.g. in the java swing framework .this motivated us to investigate the possibility of using type systems in this situation as well .our aim is not to offer new algorithms for deciding certain temporal properties or indeed to compete with the existing methods which are numerous , but to extend the reach of type systems .our solution goes , however , beyond a simple reformulation of an existing algorithm ; the abstract domain based bchi automata may well be useful in its on right and is an original contribution of this work .  for the sake of simplicity , we introduce and study a small language consisting of recursive first - order procedures and non - deterministic choices .the language explicitly allows infinite recursions . in this language , except for primitive procedures which have events as arguments , other procedures have no inputs .we define _ trace semantics _ which formalize finite and infinite traces generated by programs in this language .we also remark that in essence our language is the same as the _ pushdown systems _ that have been studied in detail by a number of authors .similar trace semantics were also studied by the cousots as a specific case of abstract interpretation .once a satisfactory type system for such a simple language has been found , it can be combined with known techniques to scale to a type system for a large fragment of java or similar languages .alternatively , one can use our simple language as a target of a preliminary abstraction step .then , we develop the bchi type and effect system to capture correctly finite and infinite traces .since branching is non - deterministic in our language , we can even establish a completeness result .completeness , of course , will be lost , once we re - introduce data - dependent branching .  as a demonstration, we extend the bchi type and effect system for this small language to a bchi type and effect system for featherweight java with field update .the main technical contribution of this paper is the design of an abstract domain in the sense of abstract interpretation based on bchi automata or rather a mild extension of those allowing infinite as well as finite words .the proofs of soundness and completeness of the type system are based on clear - cut lattice - theoretic properties of this abstraction .  as in the finitary case ,this _ bchi abstraction _ is based on equivalence relations on finite words generated by the policy bchi automaton .abstracted effects are no longer sets of such equivalence classes , but rather sets of pairs of the form with , classes and representing the infinitary language .while such pairs appear in bchi s original complementation construction for bchi automata and have subsequently been used by a number of authors , they have never been used in the context of type systems and abstract interpretation .as already mentioned , our language is equivalent to pushdown systems for which model - checking of temporal properties has been extensively studied .pushdown systems , on the other hand , are special cases of higher - order recursion systems introduced by knapik et al . andextensively studied by ong and his collaborators , e.g. .the latter work also casts model checking into the form of a type system and is thus quite closely related to our result .more precisely , from an alternating parity automaton a type system for higher - order recursion schemes is derived such that a scheme is typable iff its evaluation tree would be accepted by the automaton . in this way , in particular all mu - calculus definable properties of the evaluation tree become expressible .regarding trace languages as opposed to tree properties alternating parity automata are equivalent to bchi automata since both capture the ( )regular languages .thus for the trace language of interest here the system from loc.cit .is equal in expressive power to our type system .the difference is that kobayashi and ong s system has a much more semantic flavour not unlike the intersection type systems used to characterise strong normalisation .more concretely , the well - formedness condition for recursions in that system requires the solution of a parity game whose size is proportional to the size of the program ( number of function symbols to be precise ) which is known to be equivalent to model checking trees against mu - calculus formulas .our type system , on the other hand , deviates from the standard type systems used in programming and program analysis only very slightly ; instead of the usual recursion rule ( which is clearly unsound in the context of liveness ) we use a rule involving a type variable .no further semantic conditions need to be checked once of course the given bchi automaton has been analysed and preprocessed .we can also mention that our method and approach are rather different . while_ loc.cit ._ uses games and automata we rely on the recently re - popularised ramseyian approach to the study of -regular language and automata .another recent work on the use of types for properties of infinite traces is which embeds formulas of linear temporal logic into types in the context of functional reactive programming .this work , however , relies on an encoding of linear temporal logic in first - order logic with integers , e.g. , one models `` the event occurs infinitely often '' as a formula like where refers to that the event issues at the time .dependent types are being used to turn this into a type system , but questions of inference and decidability are not considered .the discussed works are to our knowledge the only attempts at extending the range of typing beyond safety properties .   in the next sectionwe define a simple first - order language with parameterless recursive procedures and non - deterministic branching ( meant , of course , to abstract ordinary conditionals ) .an alphabet of events is assumed and for each event a primitive procedure is available that outputs and has no effect on control flow or state .programs are toplevel mutually recursive definitions of parameterless procedures comparable to the c - language .given a program , any expression then admits a set of finite and infinite words over traces of terminating and nonterminating computations of the program .we distinguish finite traces stemming from terminating execution from finite traces stemming from nonterminating but `` unproductive '' executions .thus , for every expression ( relative to a well - formed program ) and trace we define a judgement meaning that admits a terminating execution with trace ( necessarily then ) and another judgement meaning that admits a nonterminating computation with trace . in this case both and are possible .formally , this is defined by introducing events that are repeatedly issued so that any nonterminating computation will have an infinite trace _ with _ events .the official trace semantics ( and ) is then defined by discarding these events .we then discuss alternative ways for defining the trace semantics and emphasize that it is merely meant to formalize the intuitively clear notion of event trace occurring during a computation .  in section [ sec - e ]we then define a type - and - effect system whose effects are pairs with and .semantically , an expression has effect when implies and implies .the typing rules are given in figure [ fig - e ] .we notice here that for the -part ( terminating computation ) the typing rules are as usual ; one `` guesses '' a type for a recursively defined procedure and justifies it for its body .the rule for the nonterminating `` -part '' is different .one assumes a type ( and effect ) variable for the recursive calls , typechecks the body and then takes the greatest fixpoint of the resulting type - and - effect equation .  with an ordinary recursive typing rule it would be possible to infer an effect like ( `` infinitely often '' ) for the program which is unsound .we then establish soundness ( theorem [ thm - a ] ) and completeness ( theorem [ thm - ca ] ) for this type - and - effect system .in particular , this shows that the proposed handling of recursive definitions does indeed work .the type system is at this level , however , of limited use since the effects are infinitary objects .therefore , in section [ sec - q ] we introduce an abstraction of this type - and - effect system where effects are taken from a fixed finite set .this finite set is calculated from an a priori given bchi automaton and effects still denote pairs of finite and possibly infinite languages , but no longer is any such pair denotable.  our main result theorem [ thm - cb ] then asserts that if the set of all traces of an expression is accepted by the given bchi automaton then this is provable in the abstracted type - and - effect system .so , no precision is lost in this sense .of course , we also have an accompanying soundness theorem ( theorem [ thm - c ] ) for the abstract type - and - effect system .these results can be modularly deduced from soundness and completeness for the infinitary type - and - effect system ( thms [ thm - a ] and [ thm - ca ] ) with the help of lattice - theoretic properties of the abstraction that are established in section [ ba ] . in particular, we have a galois connection between the lattice of all languages and the lattice of language denotations in the abstract type system and all operations needed in the typing rules , in particular least and greatest fixpoints can be correctly rendered on the level of the abstractions .the crucial building block is the ability to compute abstractions of greatest fixpoints needed for recursive definitions entirely on the level of the abstracted types .this requires the combination of a combinatorial lemma ( lemma [ krit ] ) with known covering properties ( lemma [ lem - d ] ) of the abstractions which follow from ramsey s theorem .section [ xx ] and section [ ba ] contain these lattice - theoretic results .we consider the discovery of this abstract lattice obtained from a bchi automaton an important result of independent interest .section [ bet ] then contains the actual definition of the abstracted type - and - effect system and its soundness and completeness theorems which , as already mentioned , then are direct consequences of earlier results .section [ compl ] discusses automatic type inference and its complexity .an appendix contains several worked out examples that did not fit into the main text and omitted proofs .we also sketch there , as a demonstration , a combination of an existing region - based type and effect system for featherweight java with field update with bchi types .the syntax of expressions is given by where is the only primitive procedure which generates an event taken from a fixed alphabet of events and ranges over procedures defined by expressions .parentheses are used to eliminate ambiguity .we assume that the operator is right - associative and has higher priority than the operator . as an example, we can define procedures and as : and .formally , thus a _ program _ consists of a finite set of procedure identifiers and for each an expression defining where calls to procedures from are allowed and in particular , may occur recursively in .  from now on ,we fix such a program and call an expression _ well - formed _ if it uses calls to procedures from only .since the operator is non - deterministic and non - primitive procedures have no arguments , stacks and heaps are not needed at this level of abstraction .let be the set of all finite and infinite sequences generated from the set of primitive events .we call an element in a _ trace_. given traces and , we define the concatenation as : if and if where and are respectively sets of all finite and infinite sequences over .so , and . as usual , we may write instead of . we are concerned with finite prefixes of the trace generated by a given expression .we call them _ observed traces_. notice that all observed traces are in .let be the definition ( a well - formed expression ) of .the observed trace semantics is given in figure [ fig - a ] . { } { o(a ) \downarrow a } \quad \inferrule*%[left={prim - b } ] { } { o(a ) \uparrow a } \quad \inferrule*%[left={epsilon } ] { } { e \uparrow \epsilon } \quad \inferrule*%[left={call - a } ] { e_f \downarrow w } { f \downarrow w } \quad \inferrule*%[left={call - b } ] { e_f \uparrow w } { f \uparrow w } \inferrule*%[left={seq - a } ] { e_1 \downarrow w\qquad e_2 \downarrow u } { e_1\,;\ , e_2 \downarrow w \cdot u } \quad \inferrule*%[left={seq - b } ] { e_1 \downarrow w\qquad e_2 \uparrow u } { e_1\,;\,e_2 \uparrow w \cdot u}\]] { e_1 \uparrow w } { e_1\,;\,e_2 \uparrow w } \inferrule*%[left={if - a } ] { e_1 \downarrow w } { e_1\,?\,e_2 \downarrow w } \quad \inferrule*%[left={if - b } ] { e_2 \downarrow w } { e_1\,?\,e_2 \downarrow w } \quad \inferrule*%[left={if - c } ] { e_1 \uparrow w } { e_1\,?\,e_2 \uparrow w } \quad \inferrule*%[left={if - d } ] { e_2 \uparrow w } { e_1\,?\,e_2 \uparrow w}\ ] ]  we write to mean that the finite trace generated by is .in particular , terminates .we write to mean that is a finite prefix of the trace generated by .let the notation denote that is a finite prefix of .we have : if or , then for all , .we now turn to define infinite traces of non - terminating programs .unfortunately , the observed trace semantics does not contain enough information for this .let us consider the following definitions : , , and .notice that the observed traces of and are exactly the same .however , the procedure has a path leading to an unproductive infinite recursion while is non - recursive . in order to fix this problem ,let us introduce the extended set of events and use for the set of all _ extended traces_. the _ observed extended trace semantics _ is same as the observed trace semantics except for the rule for function application in which a -event is automatically generated .that is , { e_f \uparrow w } { f \uparrow \checkmark\cdot w } ] of such that implies for all traces .if we have the derivation : , then .[ thm - a ] given an environment and an assignment function satisfying that , for all derivations : of an expression , we have : implies and implies .the only interesting case is that for the last rule in figure [ fig - e ] which relies on lemma [ lem - b ] . for more details , see appendix [ a - p ] .[ cor - a ] for all derivations of an expression , we have : implies and implies .fix for each non - primitive procedure a unique variable .if is a family of languages with define the corresponding environment as to contain the bindings .for each function body we can now derive using the rules except the last one a unique typing .the passage from to defines a monotone operator on the lattice .if is the least fixpoint of this operator then is justified and we get the judgements .successive application of the last rule then gives judgements and a direct induction shows that in fact and .we have thus shown :  [ thm - ca ] the judgements are derivable for each .we have kept the proof of this theorem in the running text since the monotone operator is still needed later .based on equivalence relations on finite words defined by the policy bchi automata , we introduce an abstraction of languages of finite and infinite words : the bchi abstraction .we place this abstraction into the framework of abstract interpretation and show that crucial operations , namely concatenation , least fixpoint , and infinite iteration ( ) can be computed on the level of the abstraction .we also show that the abstraction does not lose any information as far as acceptance by the fixed policy automaton is concerned .this then allows us to replace the infinitary effects in the previous type system by their finite abstraction and thus to obtain a type - and - effect system which is decidable with low complexity ( in the program size ) and yet complete .the soundness and completeness of this system follow directly from lattice - theoretic properties of this bchi abstraction ( lemma [ lem - d ] and theorem [ prese ] ) .given an expression , our goal is to verify that the set of all traces generated by satisfies some property .we use a mild extension of the standard bchi automata which we call _ extended bchi automata _ :  an extended bchi automaton is a quadruple where is a finite set of states , is an alphabet ; hereafter always required to be equal to the fixed alphabet of events ; the transition function , the initial state , and the set of final states .the language of is defined as : the set of all finite words by which a final state can be reached from the initial state and all infinite words for which there is a path which starts from the initial state and goes through final states infinitely often .thus , is the union of s language when understood as a traditional nfa and its language when understood as a traditional bchi automaton .following bchi s original works we use equivalence relations defined by extended bchi automata themselves to obtain finite representations of and .we write to mean that the state is reachable from the state by using the finite word .let denote that by using the finite word , the state can be reached from the state in such a way that a final state is visited on the way . in particular , with or implies .formally , we have iff there exists and such that and and .  for nonempty words we define we write  ] stand for . we write for \} ] ) for and we can choose and  ] and , such that , and and so on .it follows that and for and thus as required .it only remains to prove preservation of .so , fix .we want to show that .note that .the direction `` '' is obvious from monotonicity ; towards proving `` '' assume .since is closed , we may without loss of generality assume that .pick and decompose where .define \mathfrak{a}\mathfrak{a} ] ) , the set of non - empty words consists of ( = a^+ ] ) , the set of words ending with ( = ( a+b)^*b ] is a bchi effect of .further , \}))^\omega ) & = & \{([b ] , [ b ] ) , ( [ ba ] , [ ba])\}\ .\end{aligned}\ ] ] so , . that is ,all traces generated by satisfy the target property .this coincides with our observation that does not generate any finite traces and the only infinite trace generated by is which contains infinite many .consider the following c - like program :  .... 0  # define timeout 65536  1  while ( true ) {  2   i = 0 ; 3   while ( i++ < timeout & & s ! = 0 ) { 4    unsigned int s = auth ( ) ; / * o(a ) ; * / 5   }  / * o(c ) ; * / 6   work ( ) ; / * o(b ) ; * / 7  } ....  we would like to verify that line 6 is executed infinitely often under the fairness assumption that the while loop 3 always terminates .to this end , we can annotate the above program by uncommenting the event - issuing commands and abstract the so annotated program as the definition : we are then interested in the property `` infinitely many '' assuming that `` infinitely often '' ( fairness ) or equivalently : `` infinitely many or finitely many . ''this property can be readily expressed as the following bchi automaton  \\\mathfrak{a } : & * + + [ o][f=]{2}\ar@(ur , ul)_{a , b}&*++[o][f]{0}\ar[l]_{a , b , c}\ar@(dr , dl)^{a , b , c}\ar@/^/[r]^b & * + + [ o][f=]{1 } \ar@/^/[l]^{a , b , c}&}\ ] ] by the definition of , we have that the set consists of the following equivalence classes : = \{\epsilon\}\qquad [ a ] = \{a\}\qquad [ b ] = \{b\}\qquad [ c ] = \{c\ } \\ & & [ aa ] = a^+a \qquad\qquad \qquad \quad [ ba ] = ( a+b)^+a - [ aa]\\ & & [ ab ] = a^+b \qquad\qquad \qquad \quad\,[bb ] = ( a+b)^+b - [ ab]\\ & & [ cb ] = ( a+c)^*c(a+c)^*b\\ & & [ bcb ] = ( a+b+c)^*c(a+b+c)^*b - [ cb]\\ & & [ cca ] = ( a+c)^+c\ \cup\ ( a+c)^ * c ( a+c)^*a\\ & & [ bca ] = ( a+b+c)^+c\ \cup\ \\  & & \qquad\quad\ \ ( a+b+c)^* c ( a+b+c)^*a - [ cca]\ .\end{aligned}\ ] ] further , the set consists of the following pairs : ,[\epsilon])\ \  \quad\quad([a],[\epsilon])\ \  o\quad\quad([b],[\epsilon])\ \\quad\ \ \ \ , ( [ c],[\epsilon ] ) \\ & & ( [ aa],[\epsilon])\ \  \quad \ \ , ( [ ba],[\epsilon])\ \  \quad \ \ , ( [ ab],[\epsilon])\\  \quad \ \ ( [ bb],[\epsilon ] ) \\ & & ( [ cb],[\epsilon])\quad\ \ \,\ \  ( [ bcb],[\epsilon])\quad\ \  ( [ cca],[\epsilon])\quad\ , \ \  ( [ bca],[\epsilon ] ) \\ & & ( [ aa],[aa])\ \ \ \ \  ( [ ba],[aa])\ \ \,\ \  ( [ ba],[ba])\ \ \ \ \,\ \  ( [ bb],[bb ] )  \\ & & ( [ bcb],[bb])\ \ \ \ \  ( [ bcb],[bcb])\,\ \  ( [ cca],[aa])\ \ , \ \( [ cca],[cca])\ ,  \\ & & ( [ bca],[aa])\ \ \ \  ( [ bca],[ba])\ \ \  ( [ bca],[cca])\  \ \  ( [ bca],[bca])\end{aligned}\ ] ] then , the abstractions of are as follows : \}\in \mathcal{m}_*\\ & & \mathcal{v}_l = \mathcal{c}- \ { ( [ \epsilon],[\epsilon ] ) , ( [ cca],[cca ] ) , ( [ bca],[cca ] ) \}\in \mathcal{m}_{\leq\omega}\ .\end{aligned}\ ] ] by using the bchi type and effect system , we get the effect of as the pair : , [ cca]\ } , \ { ( [ aa ] , [ aa])\ } ) \ .\ ] ] then , the effect of is the pair given as follows : ,[aa ] ) , ( [ bca],[aa ] ) , ( [ bcb],[bcb ] ) , ( [ bca],[bca ] ) \})\ .\ ] ] since and , we have that the program satisfies the property .( of theorem [ thm - a ] ) we proceed by induction on the structure of the type and effect system .the only interesting case is that for the last rule in figure [ fig - e ] .let us show that if then is in .we introduce the assignment functions : & \mbox{\quad if\ \ }\ ; \\ \eta[x\mapsto a \cdot x_{\eta_{n-1 } } \cup\ v(\eta ) ] & \mbox{\quad if\ \ }\ .\end{array } \right.\end{aligned}\ ] ] from lemma [ lem - b ] , .assume that . by inductive hypothesis, we have that for all , is in further , by lemma [ lem - b ] , . by mathematical induction, we have that for all natural numbers .define as \ .\end{aligned}\ ] ] we get .notice that is equal to . by the definition of , is in .( of lemma [ lem - d ] ) a ) and b ) are obvious from the definition .property c ) coincides with b ) when .otherwise , let .decompose and so that and for .we have for all so any accepting run for yields an accepting run for by the definition of .property d ) is again trivial when ( choose  ] for all .the claim follows with  ] for otherwise the claim is trivial .decomposing the witnessing word , we get finite words where and .note that .thus , with the class of . as a result , ,so by closedness and finally , since .in order to make the bchi type and effect system given in section [ bet ] more functional and effective in programming practices , by integrating with region types , we extend it to a region - based bchi type and effect system for featherweight java with field update . based on this , the future goal is to develop and implement a powerful type system for java - like languages in which ( )-regular properties of traces can be properly characterized for verification purposes .the syntax of an expression in featherweight java with field update is given as follows : -0.5 cm for the sake of simplicity , we omit primitive types and casting and assume that every expression is in let normal form . in the definition of the * if*-*then*-*else * expression , the expression denotes an unusual judgement between objects which is independent on booleans .the notation denotes a sequence of variables .additionally , the expression is used to produce appealing events .it is a global primitive procedure which is not part of featherweight java with filed update .it is added as annotations to programs for the purposes of property characterization .let be the subclass relation between classes .let and be mappings from a class to its fields and methods respectively .we use to denote the method table which assigns to each method its definition , i.e. , its formal parameters ( a sequence of variables ) and its body ( an expression ) . with these definitions , a program is given as follows : usual well - formedness conditions on methodsare assumed to ensure the inheritance relation between same methods from different classes .we use to denote that the evaluation of terminates and the value is .the notation means that the evaluation of does nt terminate .a value of an expression is a location or .a state is consisted of a stack which is a partial function assigning to each variable a value and a heap which is a partial function assigning to each location a pair of a class and values assigned to fields of this class .in addition , as side effects , an infinite or a finite trace which is generated from the set of events by ordinary concatenations is attached to each evaluation .the whole operational semantics is given as follows : { } { ( s , h)\vdash { \bfo}(a ) \downarrow null , h\ \&\ a}\]] { } { ( s , h)\vdash null \downarrow null , h\ \&\ \epsilon } \qquad\qquad\qquad  \inferrule*[left={var } ] { } { ( s , h)\vdash x \downarrow s(x ) , h\ \&\ \epsilon}\]] { l \not\in dom(h)\qquad f = [ f\mapsto null]_{f\in fields(c ) } } { ( s , h)\vdash { \bf new}\ c \downarrow l , h[l\mapsto ( c , f)]\ \&\ \epsilon}\]] { s(x ) = l\qquad h(l ) = ( c , f ) } { ( s , h)\vdash x.f \downarrow f(f ) , h\ \&\ \epsilon}\]] { s(x ) = l\qquad h(l ) = ( c , f)\\\\  h ' = h[l\mapsto ( c , f[f\mapsto s(y ) ] ) ] } { ( s , h)\vdash x.f : = y \downarrow s(y ) , h'\ \&\ \epsilon}\]] { s(x ) = l\qquad h(l ) = ( c , f)\\\\ mtable(c , m ) = ( \overline{x } , e ) \qquad |\overline{x}| = |\overline{y}| = n\\\\ s ' = [ this\mapsto l]\cup[x_i\mapsto s(y_i)]_{i\in\{1,2,\dots , n\}}\\\\ ( s',h)\vdash e \downarrow v , h'\ \&\ w } { ( s , h)\vdash x.m(\overline{y } ) \downarrow v , h'\ \&\w}\]] { ( s , h)\vdash e_1 \downarrow v_1 , h_1\ \&\w_1\\\\ ( s[x\mapsto v_1],h_1)\vdash e_2 \downarrow v_2 , h_2\ \&\w_2 } { ( s , h)\vdash { \bf let}\ x = e_1\ { \bf in}\ e_2 \downarrow v_2 , h_2\ \&\ w_1\cdot w_2}\]] { s(x ) = s(y)\qquad ( s , h)\vdash e_1 \downarrow v , h'\ \&\ w } { ( s , h)\vdash { \bf if}\ x = y\ { \bf then}\ e_1\ { \bf else}\ e_2 \downarrow v , h'\ \&\ w}\]] { s(x ) \neq s(y)\qquad ( s , h)\vdash e_2 \downarrow v , h'\ \&\ w } { ( s , h)\vdash { \bf if}\ x = y\ { \bf then}\ e_1\ { \bf else}\ e_2 \downarrow v , h'\ \&\ w}\]] { } { ( s , h)\vdash e \uparrow \ \&\ \epsilon } \qquad\qquad\qquad \inferrule*[left={prim - b } ] { } { ( s , h)\vdash { \bfo}(a ) \uparrow \ \&\ a}\]] { s(x ) = l\qquad h(l ) = ( c , f)\\\\ mtable(c , m ) = ( \overline{x } , e ) \qquad |\overline{x}| = |\overline{y}| = n\\\\ s ' = [ this\mapsto l]\cup[x_i\mapsto s(y_i)]_{i\in\{1,2,\dots , n\}}\\\\ ( s',h)\vdash e \uparrow \ \&\ w } { ( s , h)\vdash x.m(\overline{y } ) \uparrow\ \&\ w}\]] { ( s , h)\vdash e_1 \downarrow v_1 , h_1\ \&\ w_1\\\\ ( s[x\mapsto v_1],h_1)\vdash e_2 \uparrow \ \&\ w_2 } { ( s , h)\vdash { \bf let}\ x = e_1\ { \bf in}\ e_2 \uparrow \ \&\ w_1\cdot w_2}\]] { ( s , h)\vdash e_1 \uparrow \ \&\ w } { ( s , h)\vdash { \bf let}\ x = e_1\ { \bf in}\ e_2 \uparrow \ \&\ w}\]] { s(x ) = s(y)\qquad ( s , h)\vdash e_1 \uparrow \ \&\ w } { ( s , h)\vdash { \bf if}\ x = y\ { \bf then}\ e_1\ { \bf else}\ e_2 \uparrow \ \&\ w}\]] { s(x ) \neq s(y)\qquad ( s , h)\vdash e_2 \uparrow\ \&\ w } { ( s , h)\vdash { \bf if}\ x = y\ { \bf then}\ e_1\ { \bf else}\ e_2 \uparrow \ \&\ w}\ ] ]    we now sketch an integration of the bchi type and effect system with the region type system for java given by beringer et al .let us first explain why this integration is interesting and useful by an example .considering the following fragment of java - like code :     then , an object ` o ` typed ` c ` expects a ` string ` as argument to ` f ` and ` o.f ( ) ` will exhibit a ` ( u , v ) ` effect .an object ` o1 ` typed ` c ` expects a ` string ` as argument to ` f ` and ` o1.f ( ) ` will exhibit a ` ( u,v ) ` effect . in this particular case , regions denote locations at which effects are produced .  generally , a region is a static abstraction of concrete locations which can be considered as a set of concrete locations .a class type can then be equipped with a set of regions , yielding a refined type that places the constraint that its members belong to one of the regions in .we summarize these definitions and introduce new varaiabls as follows : here , the type is introduced for typing the expression .it is easy to define the subtype relation between region - based types : if and only if . and to extend this definition to sequences of types as follows :   with respect to the subtype relation , the following field typings : assign to each field in each region - annotated class respectively a set - type which is a contravariant type for data written to the field and a get - type which is a covariant type for data read from the field .the following well - formedness conditions on and are imposed : and if , then   given a bchi automaton , let and be the bchi abstractions for and respectively , as defined in section [ ba ] .we define effects as : . then , the following typing rule : assigns to each method in each region - annotated class a functional type with an effect .the subtype relation between effect annotated functional types is defined as : that is , the input type is contravariant and the output type is covariant .the well - formedness condition on is : for all classes , regions , and methods .  with the above definitions , combinined with the type and effect system given in section [ ba ] , we arrive at the region - based bchi type and effect system as follows : { \gamma \vdash_\mathfrak{a } e : \tau \ \&\ ( \mathcal{u } , \mathcal{v}(\mathfrak{x } ) ) \\\\: \tau'\qquad\qquad \mathcal{u}\subseteq \mathcal{u'}\qquad\qquad \mathcal{v}(\mathfrak{x } ) \sqsubseteq \mathcal{v'}(\mathfrak{x } ) } { \gamma \vdash_\mathfrak{a } e:\tau ' \ \&\ ( \mathcal{u ' } , \mathcal{v'}(\mathfrak{x}))}\]] { } { \gamma \vdash_\mathfrak{a } { \bf o}(a ) : unit \ \&\ ( \alpha_*(\{a\ } ) , \emptyset)}\]] { } { \gamma \vdash_\mathfrak{a } null : c_\emptyset \ \&\ ( \alpha_*(\{\epsilon\ } ) , \emptyset)}\]] { } { \gamma , x:\tau \vdash_\mathfrak{a } x:\tau \ \&\ ( \alpha_*(\{\epsilon\ } ) , \emptyset)}\]] { } { \gamma \vdash_\mathfrak{a } { \bf new}\ c : c_{\{r\ } } \ \&\ ( \alpha_*(\{\epsilon\ } ) ,\emptyset)}\]] { \forall r\in r\,.\,a^{get}(c , r , f ) < : \tau } { \gamma , x : c_r \vdash_\mathfrak{a } x.f : \tau \ \&\ ( \alpha_*(\{\epsilon\ } ) , \emptyset)}\]] { \forall r\in r\,.\,\tau <: a^{set}(c , r , f ) } { \gamma , x : c_r , y : \tau \vdash_\mathfrak{a } x.f : = y : \tau \ \&\ ( \alpha_*(\{\epsilon\ } ) , \emptyset)}\]] { \overline{\sigma}_r\stackrel{(\mathcal{u}_r , \mathcal{v}_r)}{\longrightarrow } \tau_r = m(c , r , m ) \\\\\forall r\in r\,.\ , \overline{\sigma}_r\stackrel{(\mathcal{u}_r , \mathcal{v}_r)}{\longrightarrow } \tau_r\ \ < : \ \ \overline{\sigma}\stackrel{(\mathcal{u } , \mathcal{v})}{\longrightarrow } \tau } { \gamma ,x : c_r , \overline{y}:\overline{\sigma } \vdash_\mathfrak{a } x.m(\overline{y } ) : \tau \ \&\ ( \mathcal{u } , \cup_{r\in r}\{x_r\})}\]] % { mtable(c , m)=(\overline{x},e)\\\\ \gamma , x : c_r , this : c_r , \overline{x}:\overline{\sigma } , [ c_r.m : ( \overline{\sigma}_r,\tau_r)]_{r\in r},\delta , [ c_r.m\ \&\ ( u_r , x_r)]_{r\in r } \vdash e : \tau\ \&\ ( u , \cup_{r\in r}(a_r\cdot x_r ) \cup v(\mathfrak{x}-\cup_{r\in r}x_r ) ) % } % { \gamma , x : c_r , \overline{y}:\overline{\sigma } , \delta \vdash x.m(\overline{y } ) : \tau \ \&\ ( u , ( \cup_{r\in r}a_r)^*\cdot v(\mathfrak{x}-\cup_{r\in r}x_r)\cup ( \cup_{r\in r}a)^\omega ) } % \]] { \gamma \vdash_\mathfrak{a } e_1 : \tau_1\ \&\ ( \mathcal{u}_1 , \mathcal{v}_1(\mathfrak{x}))\\\\ \gamma , x:\tau_1 \vdash_\mathfrak{a } e_2 : \tau_2\ \&\ ( \mathcal{u}_2 , \mathcal{v}_2(\mathfrak{x } ) ) } { \gamma \vdash_\mathfrak{a } { \bf let}\ x = e_1\ { \bf in}\ e_2 : \tau_2 \ \&\ ( \mathcal{u}_1\cdot \mathcal{u}_2 , \mathcal{v}_1(\mathfrak{x})\cup \mathcal{u}_1\cdot \mathcal{v}_2(\mathfrak{x}))}\]] { \gamma , x : c_{r\cap s } , y : d_{r\cap s } \vdash_\mathfrak{a } e_1 : \tau\ \&\( \mathcal{u}_1 , \mathcal{v}_1(\mathfrak{x}))\\\\ \gamma , x : c_r , y : d_s \vdash_\mathfrak{a } e_2 : \tau\ \&\ ( \mathcal{u}_2 , \mathcal{v}_2(\mathfrak{x } ) ) } { \gamma , x : c_r , y : d_s \vdash_\mathfrak{a } \\ { \bf if}\ x = y\ { \bf then}\ e_1\ { \bf else}\  e_2 : \tau \ \&\( \mathcal{u}_1\cup \mathcal{u}_2 , \mathcal{v}_1(\mathfrak{x})\cup \mathcal{v}_2(\mathfrak{x}))}\ ] ] the typing judgement for expressions is  with the type environment , the policy bchi automaton , the type of , the set of finite traces , and the expression for infinite traces . among the typing rules , is the following rule :  where notice that in section [ ba ] , there are two typing rules for function calls .that is , one rule is used to directly get the effect if the effect has been assumed in the environment and another rule is used to derive the effect with an effect assumption added into the environment . however , in order to integrate with region - based type systems , in our region - based bchi type and effect system , we only use the following rule :   further , a program is well - typed if and only if for all classes , regions , and methods such that the following typing : is derivable and there is an assignment satisfying : that is , a program is well - typed if and only if the constraints produced by the region - based bchi type and effect system are satisfiable with respect to region - based type and effect declarations for all classes , all regions and all methods defined in this program .

Abstract: we develop a new type and effect system based on bchi automata to capture finite and infinite traces produced by programs in a small language which allows non - deterministic choices and infinite recursions . there are two key technical contributions : ( a ) an abstraction based on equivalence relations defined by the policy bchi automata , the bchi abstraction ; ( b ) a novel type and effect system to correctly capture infinite traces . we show how the bchi abstraction fits into the abstract interpretation framework and show soundness and completeness .


an important question for any theory of dense matter is whether that theory ( or an approximate calculation which uses it ) permits the propagation of information faster than the speed of light  .the requirement of poincar invariance is not sufficient to guarantee causality , and in any event practical calculations involve approximations which may also lead to non - causal properties .the uncertainty principle leads to additional problems of interpretation .the simplest criterion can be obtained by examining sound propagation . in the long - wavelength limit, small density fluctuations can cause isentropic pressure - wave propagation whose velocity is given by where is the pressure and the energy per unit volume .the wave equation follows from considering energy momentum conservation for a linear isentropic perturbation of the pressure in a perfect fluid .the sound speed follows from the equation of state .the presence of is shown explicitly : and may or may not depend explicitly upon .most of the discussion in the literature concentrates on determining whether specific theories satisfy . for a system of non - interacting relativistic fermions , , in a mean - field approximation , zeldovich  showed that can approach from below for a system of fermions interacting via repulsive vector meson exchange .bludman and ruderman  also studied questions of causality within the context of meson - nucleon field theory . among other things , they found that the group velocity can exceed the speed of light in vacuum ( ) , but that real information propagation does not violate causality as long as the index of refraction satisfies the kramers - kronig dispersion relation.  it would be useful to gain some insight as to the behavior of dense systems which employ particle degrees of freedom .indeed , dense - matter calculations which start from a field theory typically end up with models in which particles interact directly .in addition , it turns out that many calculations of the equation of state of nuclear or neutron matter still satisfy the condition , even though they are based on nonrelativistic quantum mechanics .the primary motivation for this paper is therefore twofold :  1 .examine causal and non - causal behavior in theories involving directly interacting particles .2 .  understand why nonrelativistic calculations work as well as they do with respect to causality , in spite of their only approximate connection to the principles of special relativity .  to do this, it will be necessary to discuss various definitions of causality itself , as well as what is meant by spacetime position in a given theory .this discussion is presented first , followed by an examination of specific examples .in relativistic quantum mechanics there are two distinct types of causality .the first is the requirement that the there is a well posed initial value problem .we call this cauchy causality .the second is the requirement that no signal can propagate faster than the speed of light .we call this einstein causality .  a violation of cauchy causality is a serious problem . in a relativistic quantum theorythe poincar invariance implies the existence of a unitary representation of the poincar group , that acts on the hilbert space .the time evolution subgroup relates a state at time to an initial state at time : this shows explicitly that cauchy causality is a consequence of poincar invariance .the group properties ensure that this condition holds in all inertial coordinate systems : which shows that the transformed final state is also uniquely determined by time evolving the transformed initial state .this shows that cauchy causality in all inertial coordinates systems is a consequence of poincar invariance .cauchy causality is independent of any considerations concerning einstein causality .this is distinct from the situation in relativistic classical physics , where cauchy causality and poincar covariance imply einstein causality . to see this ,assume that events can be precisely identified .the poincar group is defined as the group of transformations that preserve the proper time between space - time events .the poincar group provides an invariant classification of events that are past , future , and spacelike , relative to a given event .the temporal order of spacelike separated events is coordinate system dependent .assume that the theory satisfies cauchy causality in all coordinate systems related by poincar transformations .by contradiction , assume that the theory also violates einstein causality .violating einstein causality means that it is possible to send a signal from event to event where and are events with a relative spacelike separation . since these events are spacelike separated it is possible to find a coordinate systems where event occurs before event .this leads to a violation of cauchy causality in the new coordinate system , contradicting the assumption that the theory violates einstein causality .thus , we conclude that if we can precisely measure all spacetime events and the theory satisfies cauchy causality in all coordinate systems related by poincar transformation , then the theory must also satisfy einstein causality .  in a classical _ dynamical _ theory ,the situation is not as clear as suggested by the previous paragraph .for instance , it is known that is impossible to construct an interacting canonical hamiltonian theory with a non - trivial representation of the poincar group ( with the lie algebra realized in terms of classical poisson brackets ) that has spacetime events that transform covariantly  .this gives some clues concerning the difficulties that occur in the quantum case .in the quantum case , the poincar symmetry is preserved , but the concept of an event that occurs at a fixed spacetime point can not be precisely formulated in terms of particle degrees of freedom . specifically , one can not use the complete set of commuting operators that define the state of a single particle to construct a set of commuting hermitian operators whose eigenvalues transform like the spacetime coordinates of an event . more specifically , the uncertainty principle in relativistic quantum mechanics does not permit one to localize a particle with arbitrary precision . its position can only be determined up to an uncertainty on the scale of the particle s compton wavelength .a local field operator is needed for a precise formulatation of einstein causality .all of these issues can be easily illustrated for the case of a free spinless quantum particle of mass . for a free particle ,the momentum eigenstates form a complete set of states . the unitary representation of the poincar group , is determined up to an overall phase by unitarity , the mass and spin of the particle , and the requirement that transforms like a four vector .this representation of the poincar group contains the time evolution subgroup corresponding to a free particle of mass .it leads to a well posed initial - value problem that is consistent in any frame of reference .the single - particle dynamics thus satisfies cauchy causality and poincar invariance .classically , an event transforms like a four - vector under poincar transformations . in order to investigate einstein causality ,assume by contradiction that it is possible to find a single particle state corresponding to the particle being found at the point at a given time .the requirement that transform like a classical event is where it follows from ( [ eq : ac ] ) and ( [ eq : ad ] ) using spacetime translations that and using lorentz transformation that where is a normalization constant . combining ( [ eq : af ] ) and ( [ eq : ag ] ) implies the state necessarily has the form of a plane wave state to check einstein causality we calculate the overlap of states corresponding to different space - time points : where is the negative frequency part of the pauli - jordan commutator function , which has the form  +   { m \theta ( -\tau^2 ) \over 4 \pi^2 \sqrt{-\tau^2 } } \theta  k_1 ( m  \sqrt{-\tau^2 } ) , \end{aligned}\ ] ] and is the proper time between the events .this does not vanish for spacelike separation ( ) .the bessel function has the asymptotic form which decays exponentially for large .this suggests an apparent violation of einstein causality over a range given by the compton wavelength of the particle .note that what appears in the exponent is , which involves the invariant spacelike separation. the actual spatial separation can be much larger than the compton wavelength , provided the time difference between the events is also large .the problem is not with einstein causality ; it is related to the non - existence of a position operator that transforms as the space components of a four vector , or , equivalently , to the non - existence of a state vector representing a point particle .the problem is that if one interprets the fourier transform of the momentum space wave function as a position probability amplitude , then the position operator is related to the generator of lorentz boosts by equation ( [ eq : ac ] ) , which also must be expressible in terms of the derivatives with respect to the momentum .for spinless particles , this relation is with slightly more complicated expressions for particles with spin .the transformation properties of the hamiltonian ( zeroeth component of a four vector ) and the boost generator ( components of a rank two antisymmetric tensor ) determine the transformation properties of under lorentz transformation .to see that the problems with defining a position operator have nothing to do with einstein causality , consider the free local scalar field .this operator satisfies the local commutation relations -=[\phi^{\dagger } ( x ) ,  \phi^{\dagger } ( y ) ] _ -=[\phi^{\dagger } ( x ) , \phi ( y ) ] _-=0 \label{eq : ak}\ ] ] when is spacelike , and the covariance conditions  the field operator can be used to construct observables corresponding to a space - time region where for .the underlying quantum theory satisfies einstein causality ( [ eq : ak ] ) .however , if we define the state it follows from ( [ eq : al ] ) that satisfies ( [ eq : ad ] ) .by direct computation , where .this expression is identical ( up to a constant multiplicative factor ) to ( [ eq : aha ] ) . in this casethe result is derived in local field theory , which clearly satisfies einstein causality .the key observation is that is a single particle state , but the argument does not correspond to a precise space - time position of the particle .what is relevant in the case of the field is that antiparticle degrees of freedom are needed ensure the commutation relations ( [ eq : ak ] ) .the point is that the relevant local observables are the fields - not the particle excitation of the field .although it is possible to localize the field , the single particle eigenstates are not localizable , even for the structureless particles of a free field theory . what distinguishes a particle theory from a field theoryis that it is a theory of an infinite number of degrees of freedom .particle theories do not contain enough operators to separate points in arbitrarily small spacetime regions .thus , we see that in a theory of particles or fields , the particle degrees of freedom can not be used as a reliable test of einstein causality when the separation between particles is on the order of a compton wavelength of the particle .nevertheless , it is possible to establish a violation of einstein causality if it occurs on a scale significantly larger than a compton wavelength of the particle . certainly spacetime positions of particles are determined experimentally to within uncertainties , although large compared to the particles compton wavelength , that are sufficiently small to localize the particle within experimental uncertaintites .if we start with a localized disturbance , and evolve it in time with a non - local hamiltonian , then we can test to see the extent to which the disturbance propagates in a manner that violates einstein causality . specifically ,if non - causal effects exist on scales beyond the uncertainties in position , then the theory can ultimately lead to non - causal effects .we now illustrate how the addition of an interaction affects einstein causality .the simplest illustration is to consider the first order effect due to the interaction in a bakamjian - thomas  two - body model  .the effect of the interaction can be treated using perturbation theory .let , where is the difference between the full bakamjian - thomas hamiltonian and the kinematic hamiltonian .the time evolution operator can be expanded in powers of the interaction using the formula where the first order correction is if we let and be the fourier transforms of the single particle momenta and let then we can write in this form , it is easy to examine the causality properties .the operators are free - particle time evolution operators .the coordinates describe the position of the particle up to an uncertainty that is about the size of the compton wavelength of the particle .the matrix element can be computed : which again vanishes exponentially when is much larger than the square of a compton wavelength .the interaction in ( [ eq : at ] ) is instantaneous .thus propagation from to in a time is causal up to uncertainties in position .the propagation from to is instantaneous over the range of the interaction , and finally the propagation from to in time is causal up to uncertainties in position .it follows that the range of the interaction sets the scale on which non - causal effects occur in perturbation theory .if the range of the interaction is much larger than the compton wavelength of one of the particles , then it should in principle be possible to detect such non - causal effects experimentally .this analysis is based on perturbation theory and only gives an indication of what can happen .exact calculations can be easily performed .many of these issues arise in a many - body system .questions of testing causality are intertwined with questions related to the interpretation of position variables .these issues are more complicated if the particles are detected in the nuclear medium .although it is worthwhile to understand these issues in a many - body problem , for the purpose of testing for violations of einstein causality in particle theories , it is sensible to utilize an idealized model where all of the questions of interpretation become irrelevant . consider a dilute gas of particles and imagine a long chain of successive scattering events where the mean separation of the particles is much larger than either the range of the interaction or the compton wavelength of the particles .if each successive interaction leads to a small violation of causality , then it follows that this violation can be amplified to any desired amount by a sufficiently long chain of such scattering events . in this picture , many - body effects are unimportant .the amplification allows one to obtain violations of causality that occur on macroscopic scales . for a sufficiently dilute gas , we detect free particles with no medium effects .the spacetime position of the detected particle can be determine up to experimental uncertainty .with sufficient amplification , the causality violations can be made large on a scale that is large compared to experimental uncertainties .  in thissetting , einstein causality can be reduced to the study of the two - body problem , which can be dealt with in a practical manner .this type of study can be used to establish the possibility of being able to establish violations of einstein causality in particle models . in dense matter , there are addidtional effects that require additional attention .most notably among are the effects of the pauli principle and cluster properties .the pauli principle restricts the allowable final states in a scattering reaction , while cluster properties requires the existance of a number of many - body interactions .the role of these effects is difficult to estimate without a specific model .as noted in the introduction , the velocity of so - called zero - sound propagation is governed by the pressure - energy relation , where is the equation of state .( we assume for this discussion that the system has zero temperature . ) the nonrelativistic expression , where is the mass density , differs from the relativistic expression by the replacement .the usual procedure is to obtain a relation between the energy density and the number density in terms of a dynamical theory . the pressure is then related to the energy density and the number density via a test of causality is to see whether the resulting equation of state satisfies ( at least for physically reasonable values of ) .nonrelativistic calculations of the nuclear equation of state have very minimal dependence upon the velocity of light .the energy density can be written as where is the mass of the constituent particle and includes the kinetic and interaction energy resulting from a calculation which does not depend explicitly upon . from eq .[ evsn ] , it can be seen that the rest energy does not contribute to the pressure , and thus the pressure does not depend upon . the quantity does depend upon via the energy density : ^{-1 }   n^2 { d^2\delta\epsilon_{nr}\over dn^2}.\ ] ] the appearance of in eq .[ eq : nrdpde ] is trivial unless is comparable to , but in that case a nonrelativistic approximation is no longer valid .thus , the test whether may be less severe than examining the internal consistency of the kinematics leading to the equation of state for nonrelativistic theories of dense matterlocal quantum field theories are not necessary for describing an interacting system of particles consistent with poincar invariance and quantum mechanics .one can also formulate models using a hamiltonian , in which particles are the designated degrees of freedom .the dynamics can be specified in terms of a series of direct interactions among particles .cluster properties requires that widely separated clusters of particles behave as they would in complete isolation , the presence of momentum dependent many - body interactions is a consequence .a strongly interacting relativistic many - body system at high density would therefore satisfy poincar invariance and macroscopic causality , but in general it would not satisfy the microscopic causality that is characterized by local interacting fields .nevertheless , the physical test of such a system is whether it permits signal propagation faster than the velocity of light .bludman and ruderman suggest that this can be tested by examining the poles of the green function of the interacting system ( the sound modes ) . however , such a green function is difficult to calculate for a many - body system .our purpose in this study is to understand the conditions where a breakdown in causality ( in the sense of signal propagation ) might occur . to get a preliminary picture, we consider a system of interacting particles in one dimension .one can imagine an incoming projectile striking a constituent particle , which in turn recoils forward , striking a second constituent , which in turn strikes a third , _ etc ._ classically , if the particles interact via a rigid barrier ( _ e.g. , _ a string of croquet balls ) , then causality is violated in the sense that each particle responds instantly to the near approach of a projectile , that is , the time of light propagation across the barrier is not taken into account . for a quantum mechanical system , information propagates via a wave packet .we will consider two properties of causality with respect to wave packets : the spacetime structure of packet propagation , and the overall time advance of a packet .  in order to study the spacetime structure of packet propagation ,it is useful to examine the negative - frequency portion of pauli - jordan function in one dimension : after a variable change , we have for the case ( timelike interval ) , the integral can be reduced as follows :  where is the zeroeth - order hankel function of the second kind . for the case ( spacelike interval ) , the integral can be reduced as follows :  where is the zeroeth - order modified bessel function .for spacelike separation , does not vanish identically , but rather falls off with a range given by the particle compton wavelength .in the examples which follow , we will encounter integrals similar to that for , of the form even if the integral can not be evaluated analytically , we can gain some insight by considering the integrand in the complex plane . for spacelike separation ( ), the phase factor vanishes exponentially as in the upper half plane .if is analytic in the upper half plane , then the integration contour can be distorted away from the real axis , with the contour `` pinned '' around the branch cut beginning at .the falloff of the result will therefore be limited by the factor , _ i.e. , _ the compton wavelength scale .this qualitative argument applies to functions which do not have additional exponential behavior in .if has poles or branch points in the upper half plane , then there may be additional scales determining the rate of falloff . in any event , the rate of falloff will be no sharper than the compton wavelength .the time evolution of a wave packet is given by this expression can be evaluated by inserting a complete set of states . in the center of momentum , assuming no bound states , the sum is saturated by scattering states : the key element in the structure of wave packet propagation is the spectral integral in eq .[ completeness ] : for non - interacting particles , it has the form of the modified pauli - jordan function , as defined in eq . [ paulijordanmodified ] . for interacting systems , we will collect exponential factors in order to use the analysis following eq .[ paulijordanmodified ] .we can learn about information propagation through a dense system by studying the scattering of two of its constituents .the relevant question for the series of packet backscatterings described above is whether the backscattered packet arrives at a given spatial point `` too early '' in terms of light propagation time across the range of a direct interaction .this _ time advance _ of the backscattered packet is in principle an observable effect : each pairwise scattering contributes a time advance , which can accumulate over many scatterings .whether one can observe a violation of causality on a macroscopic scale depends upon whether information travel time over a macroscopic distance is less than when , where is the number of pairwise scatterings .observable consequences would require both close to and close to ( a dense system ) .the time advance is determined from the wigner time delay  , which can be expressed in terms of the matrix  via where is the matrix for scattering of a particle incident from to .a simple example is the potential barrier : the solution is where for the case , , where .  the simplest case to consider is the rigid barrier : solution can be written as in the spectral sum of eq .[ spectralsum ] , there will appear exponentials associated with free particles , },\ ] ] as well as exponentials associated with the barrier : }.\ ] ] the spectral sum over this second set of exponentials will yield a decaying function of range under the condition thus , there is a range of spacetime events where , but also ( the actual physical causality condition ) , for which the spectral sum yields oscillating rather than a decaying behavior .this range corresponds precisely to the region of overlap of the barriers .physically , two hard spheres will repel each other at the instant their edges come in contact .the struck sphere starts moving instantly , leaving no time for information about the interaction to travel across the sphere at the speed of light .the non - causal behavior illustrated above has a range characterized by the interaction , and goes beyond the compton wavelength associated with particle masses .how does this non - causal behavior manifest itself in terms of information propagation in a dense system ?one way to see this is to examine the propagation of a wave packet which represents a pulse of information .consider the evolution of a gaussian packet : the prototype for successive scatterings through a medium is the backscattered portion of a packet in a two - particle subsystem in its center of momentum , as shown in fig . 1 .for a causal system , we expect the centroid of a backscattered packet to arrive at a point at a time , where is the velocity in the center - of - momentum frame .note that both and are negative .the time advance given by eq .[ delay ] is , _i.e. , _ the travel time for the packet to go from to and back again , which is eliminated by the presence of the rigid barrier .we have evaluated numerically the packet propagation for various parameters , , and , and find that in all cases considered , backscattered packets reform almost immediately upon collision with the barrier , and reflect the time advance .  for the case of a _ finite _ barrier , the coefficients have a complicated dependence on through or ,so it is more difficult to formulate a simple analytic picture .numerical evaluation of eq .[ delay ] yields a time advance , though the amount is smaller than .the scattering wave function satisfies the equation where the green function reflects the choice of adding interactions to the non - interacting mass squared : and has a separable yamaguchi form . for definiteness , we use using the yamaguchi pole form , we get ,\ ] ] where .\ ] ] equation [ xwave ] is the sum of the incident wave , plus a scattered wave containing two pieces , one obeying `` outgoing '' boundary conditions ( in one dimension ) , the other having an exponential falloff outside the interaction region .we now consider the spectral sum in eq .[ spectralsum ] .all of the dependence appears as explicit exponentials in eq .[ xwave ] .the oscillating exponentials , together with the time dependence , yield exponentials of the form }.\ ] ] while there is additional dependence multiplying this overall exponential , it has the form given by eq .[ paulijordanmodified ] , and we thus expect the usual causality condition associated with the and functions  to arise .however , there will also appear cross terms involving oscillating exponentials and decaying exponentials , as well as a bilinear term of the form the integration variable does not appear in these exponentials , but they do affect the causality condition . for sufficiently small values of , therefore , these terms will contribute a non - vanishing amount in the acausal region .this acausal effect vanishes for values and/or greater than the scale .while one can see explicitly how this works for a separable yamaguchi interaction , it seems clear that the real issue is one of scales .similar effects should occur in more realistic three dimensional quantum mechanical models .how will a wave packet be affected ?while the packet overlaps the interaction region , there will certainly be non - causal contributions . outside the interaction , however , where values of or less than have little support , the non - causal contributions are suppressed . for repulsive potentials ( ) , there is a non - vanishing packet time advance , but its value is smaller than all other scales in the problem .   of course, the examples of two - particle scattering considered here completely ignore the presence of the dense surrounding medium .apart from an exact treatment of many - body correlation effects , an important effect in a strongly interacting many - fermion system such as dense nuclear matter comes from the fact that scattering into final states below the fermi surface is forbidden because the states are already filled  .thus , for an incident plane wave in one dimension with momentum below the fermi surface , forward scattering is permitted but backward scattering is not . in general , backscattered wave packets can only contain components above the fermi surface .such higher momentum components will be less sensitive to the details of the nuclear two - particle potential , and the link to non - causal effects will be commensurately reduced .   while both the local barrier and the yamaguchi interaction violate a causal condition via the spectral sum in eq .[ spectralsum ] , only the local barrier ( finite or infinite ) displays a backscattered wave packet advance . in the latter case ,the consequences are in principle observable , but one must construct a wave packet with properties suitable for such an observation . the distance scale of the violation is the range of the interaction ( _ i.e. , _ size of the barrier ) . in order to observe such a violation for a single scattering , one would need a packet whose overall width is comparable to .thus , a nucleon - nucleon core radius of 0.5 fm would correspond to a relevant packet whose momentum spread is roughly 2  mev , which is quite different from the characteristics of a particle emerging from an accelerator .  to observe non - causal effects in a many - particle system, one could use much wider packets , since the accumulated spacetime advance has a much larger scale .such a packet would then have a very sharp momentum distribution , and a qualitative picture emerges by considering individual plane - wave states . as noted above , only momentum components above the fermi surface contribute to backscattering . furthermore ,if the repulsive core is not singular , only a portion of the incident packet will be backscattered .this means that each successive scattering will reduce the amplitude of the packet , making it much more difficult to observe in a macroscopic system .we have examined a number of issues associated with the nature of causality in theories of dense matter .a strong causality condition is the vanishing of field commutators ( or anticommutators for fermions ) for spacelike separations of spacetime .this condition precludes any observable non - causal behavior over arbitrarily small distance scales . in field theories, this condition can not be achieved without the antiparticle contributions to the field .it should however be noted that the presence of antiparticles does not guarantee locality .the important observation is that the field itself is the appropriate local observable ( in the case of bose fields ) .the particle excitations of the field are not adequate for testing einstein causality because there are problems in localizing particle degrees of freedom within a compton wavelength .this problem , which occurs on the same scale as the violations of local commutation relations when antiparticles are dropped , is related to the inability to define a position operator that transforms as a four - vector .it is a manifestation of the uncertainty principle . in a sense, a relativistic free particle behaves like a composite particle with its size given by its compton wavelength .if a dense medium is described via the direct interaction of particles rather than fields , then there are apparent violations of causality on the compton wavelength scale .this is not a real problem ; it is related to the inability to localize the particle within a compton wavelength .however , without antiparticle degrees of freedom , the theory has no local operators that can be used to precisely test einstein causality .more seriously , there are violations of causality on the scale of the range interaction that are in principle observable .these come from true non - localities in the dynamics .although our discussion and analysis was restricted to models of directly interacting particles , one can anticipate that similar violations of causality can be identified in any truncation of local field theory that leads to a violation of locality .these include the use of cutoffs , effective field approximations , phenomenological quasiparticle models , and fock space truncations .clearly , one does not expect that any experiment can lead to violations of cauchy or einstein causality ; however , our discussion suggests that for any given model , there are in principle some limits to its applicability .empirically , we find that typical models do not seem to lead to large violations of einstein causality .one of the questions which prompted this study is why conventional nonrelativistic calculations of dense matter properties work as well as they do with respect to causal requirements .there appear to be several reasons :  * coincidence : equations of state based upon nonrelativistic dynamics have no _ a priori _ reason either to satisfy or to violate a causality condition .the dependence upon the speed of light of these calculations appears only trivially through the rest energy contribution to the energy density .* the densities of interest ( say , even up to 10 times normal nuclear density ) are still very low compared to the close - packed limit ( closer to 100 times nuclear density ) where any non - causal effects would be most pronounced . *finite range and/or smooth potentials mitigate the time advance characteristic of backscattered wave packets .repeated backscattering greatly reduces a time - advanced packet of information over macroscopic distances . *pauli principle : filled scattering states mean that the backscattered packet is either damped or eliminated .this work was prompted in part by a discussion between bdk and professor j. d. walecka on this subject many years ago .he also thanks professor b. d. serot for a helpful conversation .this work was supported in part by the u.s .national science foundation under grant phy-9319641 .s. l. shapiro and s. a. teukolsky , _ black holes , white dwarfs , and neutron stars _( new york : wiley , 1983 ) .b. zeldovich , _ zh . eksperim .fiz . _ * 41 * , 1609 ( 1961 ) [ _ sov. phys .jetp _ * 14 * , 1143 ( 1962 ) ] .s. a. bludman and m. a. ruderman , _ phys ._  * 170 * , 1176 ( 1968 ) .m. ruderman , _ phys ._ * 172 * , 1286 ( 1968 ) .s. a. bludman and m. a. ruderman , _ phys .d _ * 1 * , 3243 ( 1970 ) . d. g. currie , t. f. jordan and e. c. g. sudarshan , _ rev .phys . _ * 35 * , 350 ( 1963 ) . n. n. bogoliubov and d. v. shirkov , _ quantum fields _ ( reading ,ma : benjamin - cummings , 1983 ) .b. bakamjian and l. h. thomas , _ phys .rev . _  * 92 * , 1300 ( 1953 ) . b. d. keister and w. n. polyzou , _ adv. nucl .* 20 * , 225 ( 1991 ) .e. p. wigner , _ phys . rev ._ * 98 * , 145 ( 1955 ) .r. g. newton , _ scattering theory of waves and particles , second ed ._ ( new york : springer - verlag , 1982 ) . j. m. jauch and j .- p .marchand , _ helv .acta _ * 40 * , 217 ( 1967 ) .v. f. weisskopf , _ helv .acta _ , * 23 * , 187 ( 1950 ) .

Abstract: the possibility of non - causal signal propagation is examined for various theories of dense matter . this investigation requires a discussion of definitions of causality , together with interpretations of spacetime position . specific examples are used to illustrate the satisfaction or violation of causality in realistic calculations .   = 10000


the option pricing formula of black - scholes is arguably one of the most successful modern applications of quantitative mathematical analysis . despite its great simplicityit gives prices of the most actively traded options on leading markets which are typically not off by more than a few percent .this has paved the way for an influx of advanced ideas from the theory of stochastic processes into finance . in the most general formulation of the no - arbitrage argument of black - scholes  and cox et al  it was established by harrison - kreps  that a price system ( under certain restrictions such as no trading costs ) admits no arbitrage opportunities if and only if there exists an equivalent probability measure , with respect to which all discounted price processes of underlying and derivative securities are martingales .this theory is presented in several excellent monographs , see  .nevertheless , these developments do not yield a fully general solution to the derivative pricing problem , since the equivalent martingale measure is not uniquely defined , except if the market is complete . if the market is incompletethere are many possible equivalent martingale measures , and each of these specifies a price system without arbitrage opportunities .  in the lognormal model of black - scholesthe market of one share and one bond is already complete .one can thus test its predictions by computing the volatility of the price process of the underlying security , plugging that into the black - scholes formula , and then comparing with observed market prices of a derivative : such tests were initiated already by black - scholes  and black  .the volatility from historical data is obviously subject to measurement errors and statistical fluctuations .a more robust test is therefore if observed option prices can be fitted by a black - scholes formula with volatility treated as a free parameter .there are several recent reports that this is not so , see rubinstein  , and earlier investigations  .in addition to the lognormal model there exists a wider class of it processes in continuous time for which the market is also complete , and the derivative pricing problem can be solved only by no - arbitrage conditions . to distinguish which model in this classis the best approximation to observed data , and then to use it , is however already not a trivial task , see e.g  .  for the rest of this paper we will consider the general case of an incomplete market , but we do keep the standard assumption of no market friction .we introduce a criterion to price derivatives in this situation .stated differently , we advance an argument as to which out of many possible equivalent martingale measures is the appropriate one .we refer to this criterion as the principle of no almost sure arbitrage  : it is based on the theory of optimal gambling of kelly  .we will consider an investor that gambles on investing in an underlying and a derivative security , and who looks for the growth - optimal investment strategies among such combinations .the paper is organized as follows : in section [ s : kelly ] we summarize and discuss the kelly theory in a version appropriate for our purposes .we discuss also here the papers of samuelson  and merton - samuelson  . in section [s : principle - nasa ] , the technical core of the paper , we apply the kelly theory to derivative pricing , and in section [ s : remarks ] we discuss the relations and differences to other proposed alternative derivative pricing procedures .it is straight - forward to see that in complete markets our procedure agrees with the standard ones , e.g. with the models of cox - ross - rubinstein and black - scholes . in section [ s : lognormal ] we discuss the lognormal limit of our procedure , the limits of its applicability , and that it is not exactly equivalent to the black - scholes formula . in section [ s : error ] we study the expected scatter of the optimal investment fraction and report a small - scale empirical test of pricing options on the swedish omx index . in section [ s : multiasset ] we generalize the method to a market with correlated stocks . in section [ s : conclusion ] we summarize and discuss our results .an earlier version of this paper was circulated in 1998  .a longer sequel was also made available in the electronic archive  , lacking however in the version accepted for publication , material corresponding to sections [ s : error ] and [ s : multiasset ] below .the presentation in the present paper is significally different from  and contains a fuller discussion of the implementation of the model and the economic background .the theory to be exposed in this section is due to kelly  , who was looking for an interpretation of information theory of shannon outside of the context of communication , and to the treatment of the st .petersburg paradox by bernoulli  .for a recent review of growth - optimal investment strategies , see hakanson - ziemba  .consider a price movement of stock or some other security which is described by where time is discrete , is the price at time and the s are independent , identically distributed random variables .the assumptions on the s can be relaxed .consider now an investor which starts at time with a wealth , and who decides to gamble on this stock repeatedly .suppose that the investor chooses to commit at each time a fraction of his capital in stock , and the rest in a risk - less security .we set the risk - less rate to zero .a non - zero risk - less rate corresponds to a discount factor in the definition of the share prices , and can be accounted for by a redefinition of the s .the investor will hence hold at time a number of shares , and his wealth at successive instants of time follows a multiplicative random process in the large time limit we have by the law of large numbers that the exponential growth rate of the wealth tends with probability one to a constant .that is , .\label{eq : lambda - definition}\ ] ]  the optimal gambling strategy in the sense of kelly consists in maximizing in ( [ eq : lambda - definition ] ) by varying .the solution must be unique because the logarithm is a concave function of its argument .which values of are reasonable in our problem ?first , the optimum must be such that is positive on the support of .second , we must decide if the investor is allowed to borrow money . in the original formulation of kelly he is not ,but here it is useful to assume that the investor may do so at a risk - less rate .furthermore we also assume that the investor can go short i.e. , it is possible to borrow stocks .hence we allow to take any finite positive or negative value , and look for the maximum of .the desired strategy is hence the only finite which solves =0 \label{eq : l - star - definition}\ ] ] and the maximum realized growth rate is \label{eq : lambda - star - definition}\ ] ] let us look more closely at ( [ eq : l - star - definition ] ) if can take the values with probabilities .let us write such that ( [ eq : l - star - definition ] ) reads   from being positive it follows that the s are all positive , and from ( [ eq : q - equation ] ) it follows that the sum of the s is one .the s thus form a new set of probabilities , and , again by ( [ eq : q - equation ] ) , with respect to these the stock price process is a martingale .it is worthwhile to discuss this point at some length in light of the well - known papers of samuelson  and merton - samuelson  .these authors demonstrated that if the risk / return preferences of an agent is described by an expected utility , then the growth - optimal strategy does not maximize that utility , except if the utility is logarithmic , and the price process is similar to ( [ eq : u_i - definition ] ) .it is a mathematical fact that the growth rate ( [ eq : lambda - definition ] ) holds with probability one . if a gambler would choose any other strategy in but that which maximizes then he would in the large time limit almost surely end up with an exponentially smaller capital .  in the counter - examples of samuelson - mertonthe dominant contribution to non - logarithmic utility comes from events with asymptotically ( in time ) vanishing probability .when sets of measure zero are involved , the relation between sample and ensemble averages becomes delicate .suppose some agents want to maximize a non - logarithmic utility , and that they choose strategies accordingly .if they do so , and we compare with them using the growth - optimal strategy , they would almost surely end up with less utility according to their own criterion .only if there would be an ensemble of infinitely many agents , all using this same strategy , then some few would actually end up with higher utility . and, in fact , so much higher that the averaged utility over the ensemble would be increased .this is the sense in which other strategies can be ` better ' than the growth - optimal strategies .therefore , merton - samuelson give preference to ensemble averages over sample average .this is a problematic point , which has apparently not received much attention in the economic literature .a simple counter - argument against the growth - optimal criterion is that time in economical problems , in the sense considered here , is perhaps often not very long .however , this is essentially a quantitive question , which has to be decided case - by - case . for recent discussions of characteristic times in the kelly problem with transaction costs ,see  us now consider the stock of section [ s : kelly ] and one derivative security on the same stock .the argument generalizes at every step in an obvious way to an arbitrary number of derivatives , but to keep the notation simple we will write out the formulae for one derivative . to begin , let us write the present share price at the time  , and let us assume that the realized value of the derivate at the next future instant of time is .the unknown parameter is the present price of derivative , which we will write .we will further suppose that this situation is repeated over and over again .that is , we assume that the investor finds himself many times in the situation that if he keeps some money in stock it will be worth , while if he keeps the same money in the derivative it will be worth , with the same random variable entering into both expressions .the simplest example is one time step before expiration time of a call option with a present share price and a strike price ; in this case one has .if at the stage in this gamble the investor commits a fraction of his wealth in shares and a fraction in the derivative , this means shares and derivatives .after the random variable has taken a value , the wealth is changed to if the investor plays this game many times , his wealth will almost surely grow at an exponential rate ( over these games ) which is \label{eq : double - lyapunov - exponent}\ ] ]  let us now assume that we compute ( [ eq : double - lyapunov - exponent ] ) and that we find an optimal growth rate \label{eq : double - lyapunov - exponent - optimal}\ ] ] and that this is realized at fractions and .if is larger than zero , then all operators would like to buy the derivative . hence its present price would tend to rise , and its rate of return would fall . the fraction would thus tend to fall . if , on the other hand , would be less than zero , then all operators would want to go short on the derivative , its present price would fall , its rate of return would rise and would tend to rise .the only value of in which the market can be in equilibrium is therefore the one such that is zero .this statement was referred to as the principle of no almost sure arbitrage in .we remark that from the technical point of view , this procedure , where is taken to vanish , is closely similar to the samuelson s methodology of warrant pricing in the ` incipient case ' , see merton  . and the fractions and thus satisfy two equations .one is \label{eq : l_s - star - definition}\ ] ] which is identical to ( [ eq : l - star - definition ] ) of the kelly model , and therefore determines .the other is \label{eq : c_0-star - definition}\ ] ] if we now compare with ( [ eq : q - definition ] ) and ( [ eq : q - equation ] ) we see that ( [ eq : c_0-star - definition ] ) can be written which states that the derivative price process is a martingale with respect to .we have constructed the equivalent martingale measure over one elementary time - step . by compoundingwe can construct , from convolutions of distribution , the price distributions over many time - steps .the price of , say , a european call option with expiration time at time , can thus be written in the standard manner as \quad c(t , s_t ) = max ( s_t - k,0 ) \label{eq : equivalent - martingale - pricing - t}\ ] ] where the expectation value is taken with respect to the many time - steps distribution over share prices at time , conditioned by the share price having been at time .we have taken the risk - less rate to be zero .a non - zero risk - less rate can be put back in as a discount factor in the share prices , which leads to a different definition of the s , the returns over elementary periods of time , and hence to different s .the pricing procedure proposed in section [ s : principle - nasa ] is based on a particular choice of an equivalent martingale measure .it therefore follows that this procedure is arbitrage - free .it also follows that it must agree with no - arbitrage pricing in complete markets.  it may nevertheless be instructive to carry through the calculations of section [ s : principle - nasa ] explicitly for the cox - ross - rubinstein model .we hence assume that the discounted share price can go up by a fraction with probability , down by with probability , and that , where is the risk - free interest rate .equation [ eq : l_s - star - definition ] is then solved for as and this reproduces the cox - ross - rubinstein measure , independent of .the maximum growth rate however still depends on , and is the black - scholes model can now be derived as the continuous - time limit of the cox - ross - rubinstein model .the growth rate ( [ eq : crr - kullback - contrast ] ) is the kullback contrast of the measure with respect to the measure in the dichotomic case .it is easy to see that this is the general form of the solution , and that the measure can be obtained as follows : consider a security with price process determined by a measure , and consider all measures with respect to which the price process of the security is a martingale. then the particular measure which we compute from ( [ eq : l_s - star - definition ] ) ( see also equations ( [ eq : q - definition ] ) and ( [ eq : q - equation ] ) ) is the one that minimizes the kullback contrast =  -\sum_n  p_n \log\frac{p_n}{q_n }  \label{eq : kullback - contrast}\ ] ] our proposal therefore coincides , over one time step , with one of the two proposals recently put forward by stutzer  .stutzer s formulae differ from ours in that we posit the minimization of ( [ eq : kullback - contrast ] ) under the martingale constraint for the elementary process over each time - step , while stutzer applies ( [ eq : kullback - contrast ] ) directly on the probability distribution over a finite time interval .a similar approach minimizing the relative entropy distance to calibrate a pricing model has been suggested by avellaneda et al .it is also interesting to remark that if we perform the quadratic approximation of the logarithm in ( [ eq : lambda - definition ] ) around the expected return on capital ,  ] is finite .then , for the most probable realizations , is well approximated by a lognormal distribution , i.e.   where   \quad ;  \quad \delta^2= e^q[(\log u)^2]-(e^q[\log u])^2 \label{eq : lambda - delta - definitions}\ ] ]  for large only low - probability events are not distributed according to ( [ eq : lognormal - approximation ] ) .for the treatment of such large , but very rare , events the proper mathematical setting is the theory of large deviations , see varadhan  . in we discuss the use of large deviations theory to price derivatives related to rare events .we note that is by construction a martingale measure , while , for the log - normal approximation , =\int q_t^{(ln)}(s_t|s_0 ) \ , s_t \ , d s_t =   s_0 \exp \ , ( \lambda+{1 \over 2 } \delta^2 ) t \quad .\label{eq : lognormal - martingale - condition}\ ] ] in general .that is , in the lognormal approximation one looses the martingale property .our lognormal approximation does not agree with a straight - forward application of the black - scholes formula .we recall that the black - scholes theory also involves lognormal distribution . in the parameterization of ( [ eq : lognormal - approximation ] ) it is characterized by drift coefficient and volatility , related through -(e^p[\log u])^2\ ] ] it is easy to understand that if has support concentrated around one , then the difference between and is small . in the continuous limitthe differences between , and disappear altogether .  on the other hand, the risk - neutral price distribution can actually be close to lognormal but still significantly different from black - scholes . in an extreme situation of very large  ] .  according to equation ( [ sdl ] ) ,it is evident that in most cases the error of is large .a non - intuitive result is that it is easier to determine with accuracy for distribution with larger variance because of the in the denominator .this fact can be understood as follows .a stock with small fluctuations allows for a bigger fraction be held against the interest rate to make money , but just a small change in the distribution will make a big change in the optimal fraction . on the contrary , for a wildly fluctuating stockthe fraction invested is smaller , and a little change in the distribution will not have much effect on the optimal portfolio .   empirical study is needed to verify the growth optimal pricing procedure here discussed . to verify the model we looked at the european call option on the omx - index , traded at the om exchange in stockholm , during the period 2 january 1995 until 22 september 1995 .we consider closing prices of each day .the time period between trading is one day .the time series contained observations , providing a sample of of size ( data points ) .the empirical distribution of of the omx - index is built by the fraction of the price of the omx - index today and tomorrow , .weekends and other holidays are treated as non - existing .the thirty day interest rate was as used discount factor to compute the discounted daily returns , . during the period the interest rate fluctuate between , of yearly yield .the sample space was divided into cells and the optimal determined using the newton - raphson algorithm  . because of the concave of the logarithm the convergence is very fast , the optimal for the omx - index is equal to .the distribution of is obtained using bootstrap technique with a set of new samples .the standard deviation of is and can be compared to obtained estimating the standard deviation using formula ( [ sdl ] ) . from the empirical distribution the empirical -distribution can be created and from -distribution the empirical -distribution can be derived .the empirical -distribution is used to estimate parameters of the lognormal approximation ( [ eq : lambda - delta - definitions ] ) and .we determine , using the bootstrap technique , the distribution and standard deviation for and , and respectively .these parameters correspond , inside the experimental errors , to the black - scholes model .the consequence is that in this case lognormal approximation of the kelly pricing scheme is practically equivalent to black - scholes model .the comparison between empirical data and theoretical one is shown in figure [ fig : th_exp ] , for the call options with maturity date less or equal then ten working days ( two weeks ) .we observe a coincidence inside the errors .finally we observe that to be able to tell a difference between classical way of pricing derivatives and growth - optimal pricing procedure either the price movement of the underlying has to have ` wilder ' fluctuations , or one has to have access to more high - frequency data .in order to study the effect of correlated stocks , the market in this section is assumed to have stocks and derivatives written on them .to get transparency of the principle we discuss the case of just two stocks with one derivative each and a risk free interest rate equal to . the conclusion of this section will be given in a general notation .the features of the stock price movements are given by : the random variable is assumed to be correlated on other stocks .an investor of thgs market commits a fraction of his capital in stock and a fraction in the derivative .the exponential growth rate of the capital is : \,\ , .\\ \end{array } \label{eq : growthgeneralcase}\end{aligned}\ ] ] according to the general equlibrium argument , one obtains the two equations : =  0 \;\;\;\;\;\ ; k=1,2  \label{eq : optimaleqstock}\ ] ] and   \;\;\;\;\;\ ; j=1,2 \label{eq : optimaleqderivative}\ ] ] where we have defined , as in section [ s : principle - nasa ] , a new probability measure : where the index and stands for some realization of the random variables and with the joint probability .the solution of equations ( [ eq : optimaleqstock ] ) gives the optimal fraction to invest in each of the different stocks and equations ( [ eq : optimaleqderivative ] ) give the option prices as the expected value under the measure ( [ eq : generalq ] ) .the martingale measure may be compounded so that the option is priced by using the pay out function at expire date :    \label{compmartingale}\ ] ] where the denotes the compounded martingale measure of .the case of independent stock price movements is of particular interest from a theoretical point of view .the price of the derivative on an underlying stock and even the hedge position do not depend on the other stocks in the classical models , i.e. complete markets and risk minimization pricing procedure .this result is a consequence of the property where and are the martingale measures of one and two shares respectively .  in the case of a complete marketit is a known fact that the martingale measure is unique  , then it can not be anything else but the measure ( [ eq : q - definition ] ) found by optimizing the growth rate for just that particular stock the property ( [ eq : property ] ) is also preserved in the risk minimization pricing procedure but not in the general case of an incomplete market . in this situation the associated martingale measure ( [ eq : generalq ] ) depend on the number of stocks traded in the market in figure [ fig : multiasset ] we show this difference on the option prices for the trichotomic case .this fact is clear in an incomplete market with large number of stocks and the returns of these stocks are independent identically distributed random processes .the optimal fraction invested in each stock is and the excess rate of return becomes almost sure . to avoid arbitrage opportunities the expected excess rate of return must vanish and the measure pricing the option becomes close to the observed probability i.e , $ ] .let us notice that this corresponds to a generalization to a non gaussian of the bachelier theory  .we have in this paper introduced a new criterion to price derivatives in incomplete markets derived from the theory of optimal gambling strategies in repeated games .the criterion say that one should not be able to construct a portfolio using derivatives that grows almost surely at a faster exponential rate than using only the underlying security .a corner - stone in modern derivative pricing theory is that discounted prices ( in friction - less markets ) can be expressed as expected pay - off , expectations taken with respect to equivalent martingale measures .we use the criterion to decide which out of many possible measures to choose .we therefore do not need to assume a complete market to fix unambiguously a price .  in section [ s :remarks ] we compared our approach to other procedures proposed in the literature , and in section [ s : lognormal ] we discussed the lognormal limit . in our constructionwe determined first the equivalent martingale measure over one elementary time - step . by compositionwe can then get the risk - neutral distribution at finite times . in a large class of modelsthis compounded probability tends to the lognormal form , at the points close to the maximum , where probability is high .one could then conclude that we just rederive the black - scholes model , except for rare events .this is however not true , because , for us , the condition that the risk - neutral distribution is a martingale involves also the distribution of rare and large events .these are not lognormal .in fact , the best lognormal approximation does not by itself have to be a martingale , and in general differs from black - scholes .e.a . thanks marco avellaneda , phil dybvig and svante johansson for email conversations and useful remarks .r.b . thanks duccio fanelli and guglielmo lacorata for interesting discussions about the topic .e. aurell , r. baviera , o. hammarlid , m. serva and a. vulpiani , a general methodology to price and hedge derivatives in incomplete markets , _ international j. of theoretical and applied finance _[ in press ] and _ cond - mat/9810257 _ , ( http://xxx.lanl.gov/ )              j .- p . bouchaud and d. sornette , the black - scholes option pricing problem in mathematical finance ; generalization and extensions for a large class of stochastic processes , _ j. de physique i ( france ) _ * 4 * ( 1994 ) 863881 .e. derman and i. kani stochastic implied trees : arbitrage pricing with stochastic term and strike structure of volatility , _ quantitative strategies technical notes _ ( goldman sachs & co. , april 1997 ) , also in _ international j. of theoretical and applied finance _ * 1 * ( 1998 ) 61110 .m. rubinstein , nonparametric test of alternative option pricing models using all reported trades and quotes on the 30 most active cboe option classes from august 23 , 1976 through august 31 , 1978 , _ j. of finance _ * 40 * ( 1978 ) 454480 .

Abstract: we introduce a criterion how to price derivatives in incomplete markets , based on the theory of growth optimal strategy in repeated multiplicative games . we present reasons why these growth - optimal strategies should be particularly relevant to the problem of pricing derivatives . under the assumptions of no trading costs , and no restrictions on lending , we find an appropriate equivalent martingale measure that prices the underlying and the derivative security . we compare our result with other alternative pricing procedures in the literature , and discuss the limits of validity of the lognormal approximation . we also generalize the pricing method to a market with correlated stocks . the expected estimation error of the optimal investment fraction is derived in a closed form , and its validity is check with a small - scale empirical test .  ,  ,  ,  and  .