linear temporal logic ( ltl ) has been proposed by pnueli  as a formalism to specify properties of parallel programs and concurrent systems , as well as to reason about their behaviour . since then, it has been widely used for these purposes .recent developments require reasoning tasks such as deciding satisfiability , validity , or model checking to be performed automatically . therefore , decidability and computational complexity of the corresponding decision problems are of great interest .the earliest and fundamental source of complexity results for the satisfiability problem ( sat ) and the model - checking problem ( mc ) of ltl is certainly sistla and clarke s paper .they have established -completeness of sat and mc for ltl with the temporal operators ( eventually ) , ( invariantly ) , ( next - time ) , ( until ) , and  ( since ) .they have also shown that these problems are -complete for certain restrictions of the set of temporal operators .this work was continued by markey  .the results of sistla , clarke , and markey imply that sat and mc for ltl and a multitude of its fragments are intractable .in fact , they do not exhibit any tractable fragment .the fragments they consider are obtained by restricting the set of temporal operators and the use of negations .what they do not consider are arbitrary fragments of temporal _ and _ boolean operators . for propositional logic ,a complete analysis has been achieved by lewis .he divides all infinitely many sets of boolean operators into those with tractable ( polynomial - time solvable ) and intractable ( -complete ) sat problems .a similar systematic classification has been obtained by bauland et al . in for ltl .they divide fragments of ltl determined by arbitrary combinations of temporal and boolean operators into those with polynomial - time solvable , -complete , and -complete sat problems .this paper continues the work on the mc problem for ltl .similarly as in , the considered fragments are arbitrary combinations of temporal and boolean operators. we will separate the mc problem for almost all ltl fragments into tractable ( , polynomial - time solvable ) and intractable ( , -hard ) cases .this extends the work of sistla and clarke , and markey , but in contrast to their results , we will exhibit many tractable fragments and exactly determine their computational complexity .surprisingly , we will see that tractable cases for model checking are even very easy that is , -complete or even -solvable .there is only one set of boolean operators , consisting of the binary -operator , that we will have to leave open .this constellation has already proved difficult to handle in , the latter being a paper where sat for basic modal logics has been classified in a similar way .  while the borderline between tractable and intractable fragments in is quite easily recognisable ( sat for fragments containing the boolean function is intractable , almost all others are tractable ) , our results for mc will exhibit a rather diffuse borderline .this will become visible in the following overview and is addressed in the conclusion .our most surprising intractability result is the -hardness of the fragment that only allows the temporal operator and no propositional operator at all .our most surprising tractability result is the -completeness of mc for the fragment that only allows the temporal operators , , and the binary -operator .taking into account that mc for the fragment with only plus is already -hard ( which is a consequence from ) , we would have expected the same lower bound for the `` dual '' fragment with only plus , but in fact we show that even the fragment with and and is tractable . in the presence of the -operator ,the expected duality occurs : the fragment with , plus and the one with , plus are both -hard .table [ tab : overview ] gives an overview of our results .the top row refers to the sets of boolean operators given in definition [ def : clones ] .these seven sets of boolean operators are all relevant cases , which is due to post s fundamental paper and lemma [ lemma : const ] .entries in bold - face type denote completeness for the given complexity class under logspace reductions .( all reductions in this paper are logspace reductions . )the entry stands for logspace solvability .all other entries denote hardness results .superscripts refer to the source of the corresponding result as explained in the legend .this paper is organised as follows .section [ sec : prelims ] contains all necessary definitions and notation . in section [ sec : np ] , we show -hardness of all intractable cases , followed by section [ sec : nl ] with the -completeness of almost all remaining cases .we conclude in section [ sec : concl ] . due to the limitations of space ,we have left out a number of proofs .these can be found in the technical report .a _ boolean function _ is a function . we can identify an -ary propositional function symbol with the -ary boolean function defined by : if and only if the formula becomes true when assigning to for all .an _ operator _ is either a function or a function symbol , which becomes clear from the context .additionally to propositional operators we use the unary temporal operators ( next - time ) , ( eventually ) , ( invariantly ) and the binary temporal operators ( until ) , and ( since ) .let be a finite set of boolean operators and be a set of temporal operators .temporal -formula over _ is a formula that is built from variables , propositional operators from , and temporal operators from .more formally , a temporal -formula over is either a propositional variable or of the form or , where are temporal -formulae over , is an -ary propositional operator from and is an -ary temporal operator from . in , complexity results for formulae using the temporal operators , , ( unary ) , and ,  ( binary ) were presented .we extend these results to temporal -formulae over subsets of those temporal operators .the set of variables appearing in is denoted by if we call a _temporal -formula _ , and if we call a _ propositional -formula _ or simply a _ -formula_. the set of all temporal -formulae over is denoted with   a _ kripke structure _ is a triple , where is a finite set of states , is a total binary relation ( meaning that , for each , there is some such that ) , and for a set of variables .a model in linear temporal logic is a linear structure of states , which intuitively can be seen as different points of time , with propositional assignments .formally , a _ path _ in is an infinite sequence denoted as , where , for all , and .  for a temporal -formula over with variables from , a kripke structure , and a path in , we define what it means that _ satisfies in _ ( ) : let and be temporal -formulae over and let be a variable .lll + & iff & + & iff & and + & iff & + & iff & there is a such that + & iff & for all , + & iff & + & iff & there is an such that , + & & and for every ,  + & iff & there is an such that , + & & and for every ,    since every boolean operator can be composed from and , the above definition generalises to temporal -formulae for arbitrary sets of boolean operators .this paper examines the model - checking problems for finite sets of boolean functions and sets of temporal operators .sistla and clarke have established the computational complexity of the model - checking problem for temporal -formulae over some sets of temporal operators .[ theorem : sicl85 ]   [ part : mc(f;andorneg ) np - c ] is -complete .[ part : mc(f , x|u|u , s , x;andorneg ) pspace - c ] , , and are -complete .since there are infinitely many finite sets of boolean functions , we introduce some algebraic tools to classify the complexity of the infinitely many arising satisfiability problems .we denote with the -ary projection to the -th variable , where , , , and with the -ary constant function defined by . for and simply write 1 and 0 . a set of boolean functionsis called a _ clone _ if it is closed under superposition , which means contains all projections and is closed under arbitrary composition . for a set of boolean functionswe denote with the smallest clone containing and call a _ base _ for . in postclassified the lattice of all clones and found a finite base for each clone .the definitions of all clones as well as the full inclusion graph can be found , for example , in  .the following lemma the following lemma , which we prove in , implies that only clones with both constants are relevant for the model - checking problem ; hence we will only define those clones .note , however , that our results will carry over to all clones .[ lemma : const ] let be a finite set of boolean functions and be a set of temporal operators. then . is trivial . for let be an instance of for a kripke structure andlet and be two fresh variables .we define a new kripke structure where and we define to be a copy of where every appearance of 0 is replaced by and every appearance of 1 by .it holds that is an instance of and that if and only if .  because of lemma [ lemma : const ] it is sufficient to look only at the clones with constants , which are introduced in definition [ def : clones ] .their bases and inclusion structure are given in figure [ figure : clones with constants ] .[ def : clones ] let denote the binary exclusive or .let be an -ary boolean function . is the set of all boolean functions . is the set of all monotone functions , that is , the set of all functions where , , implies . is the set of all linear functions , that is , the set of all functions that satisfy , for constants . is the set of all functions where , for constants . is the set of all functions where , for constants . is the set of all functions that depend on at most one variable . is the set of all projections and constants .there is a strong connection between propositional formulae and post s lattice .if we interpret propositional formulae as boolean functions , it is obvious that  ] .then for every set of temporal operators .let . from lemmas1.4.4 and 1.4.5 in  we directly conclude : let be one of the functions , , and such that .let be the arity of .then there is a -formula representing , such that every variable occurs only once in . hence . from lemma [ lemma : const ]follows .it is essential for this lemma that . for , , ,it is open whether .this is a reason why we can not immediately transform upper bounds proven by sistla and clarke  for example , upper bounds for all finite sets of boolean functions , it is open whether for all finite sets of boolean functions , .sistla and clarke and markey have considered the complexity of model - checking for temporal -formulae restricted to atomic negation and propositional negation , respectively .we define a temporal -formula with _propositional negation _ to be a temporal -formula where additional negations are allowed , but only in such a way that no temporal operator appears in the scope of a negation sign . in the casethat negation is an element of , a temporal -formula with propositional negation is simply a temporal -formula . in , _ atomic negation _ is considered , which restricts the use of negation even further negation is only allowed directly for variables .we will now show that propositional negation does not make any difference for the complexity of the model checking problem .since this obviously implies that atomic negation inherits the same complexity behaviour , we will only speak about propositional negation in the following .the proof of the following lemma is similar to that of lemma [ lemma : const ] .[ lemma : propositional negation ] let be a set of temporal operators , and a finite set of boolean functions .we use to denote the model - checking problem extended to -formulae with propositional negation. then .the reduction is trivial . for ,assume that negation is not an element of , otherwise there is nothing to prove .let be an instance of , where .let be the variables that appear in , and for each formula of the kind appearing in let be a new variable .note that since only propositional negation is allowed in in these cases is purely propositional .we obtain from by extending to the variables in such a way that is _ true _ in a state if and only if is _false_. finally , to obtain from we replace every appearance of with now , is a temporal -formula . by the construction it is straightforward to see that iff .propositional negation does not affect the complexity of the model checking problem .this can be proven similarly to lemma [ lemma : const ] . using lemma [ lemma : subred ] in addition , we can generalise the above mentioned hardness results from for temporal monotone formulae to obtain the following intractability results for model - checking .the proofs of all results in this section are given in .[ theorem : propositional negation ] let be a finite set of boolean functions such that . then  [ part : mc(f , g , x;m ) pspace - h ] is -hard .[ part : mc(f|g|x;m ) np - h ] , , and are -hard .[ part : mc(u|g , x;m ) pspace - h ] and are -hard .  [ part : mc(s , g|s , f;m ) pspace - h ] and are -hard .  in theorem 3.5 in it is shown that is -hard .in fact , sistla and clarke give a reduction from to .the result for arbitrary bases generating a clone above follows from lemma [ lemma : subred ] .[ cor : mc(f;e ) np - h ] let be a finite set of boolean functions such that . then is -hard .  in general , the model - checking problem for the temporal operators and is -complete due to .it is in when we only allow propositional negation , again following .therefore , this problem is in for monotone formulae .combining this with the result that we are -complete above even if we only have the temporal operator ( corollary [ corollary : mc(f;e ) np - c ] ) , immediately gives the following result .[ corollary : mc(x , f;e|m ) np - c ] let be a finite set of boolean functions such that . then is -complete .the model - checking problem for temporal --formulae is -complete ( theorem [ theorem : propositional negation][part : mc(u|g , x;m ) pspace - h ] due to ) .the boolean operators are a basis of , the class of monotone boolean formulae .what happens for fragments of ? in theorem [ theorem : mc(f , x;v ) mc(g , x;e ) nl - c ] we will show that is -complete , , the model - checking problem for temporal -formulae over is very simple .we can prove that switching from to makes the problem intractable .as notation , we use to denote the literals obtained from variables that appear in .  [theorem : mc(g , x;v ) np - h ] let be a finite set of boolean functions such that . then is -hard .it suffices to give a reduction from  to ( due to lemma [ lemma : subred ] ) .a formula in 3cnf is mapped to an instance of as follows .let consist of clauses , and variables .the kripke structure has states containing one state for every clause , a sequence of states for every literal , and a final sink state .that is , the set of states is .the variables of are .variable is assigned _ true _ in a state iff literal is contained in clause . in all other states , every is _false_. variable is assigned _ true _ in all states in .the relation between the states is as follows .it starts with the path : . has an edge to and an edge to : .each is the starting point of a path : .each endpoint of these paths has both the literals with the next index resp .the final sink state as neighbours : .the final sink state has an edge to itself , .   for .]  figure [ figure : structure for gxv ] shows an example for a formula and the kripke structure .notice that every path in such a kripke structure corresponds to an assignment to the variables in .a path corresponds to a satisfying assignment iff for every the path contains a state that is assigned to .we are now going to construct a formula to express this property .if we were allowed to use the in , this would be easy .but , the formula consists only of operators , , , and of variables . in order to define , we use formulae and defined as follows . for  intuitively , says that is satisfied in a state in distance , where . the state is the only state in where can hold .every path in has the form .every state except for appears at most once in . for the sake of simplicity , we use the notation for ( for ) , and for .we use for a path in and the notation for .[ claim : np1 ] for every path in and holds : if , then .assume , where . by the definition of , it follows that for some with .consider any path in .after the initial part of follows a sequence of states , where ( for ) . therefore , for some ( that does not matter here ) . but implies , by the definition of , and therefore . since , it follows that .the formulae are defined inductively for as follows ( as before , we can use in our construction ) : finally , .it is clear that the reduction function can be computed in logarithmic space .it remains to prove the correctness of the reduction .using claim [ claim : np1 ] , we make the following observation .  [claim : np2 ] for every path in and holds :     if and only if  for holds .the direction from right to left is straightforward . to prove the other direction , we use induction .as base case we consider .assume .by construction of holds , and therefore holds .for the inductive step , assume .claim [ claim : np1 ] proves for , and with we obtain .this implies and . by the inductive hypothesis, the claim follows .  for a path in ,let be the corresponding assignment for .it is clear that if and only if satisfies clause of . using claim [ claim : np2 ] , it follows that if and only if satisfies all clauses of , , satisfies . using the one - to - one correspondence between paths in and assignments to the variables of we get if and only if .  from follows that is in .it remains open whether or have an upper bound below .  if we allow only one of the temporal operators and in monotone formulae , the complexity of model - checking drops to -complete .[ lemma : mc(g|x;m ) np ] and are in .-membership of is known from  .it remains to show that . given a formula , a kripke structure , and a state ,let denote the maximal nesting depth of -operators in .clearly .notice that for any -ary boolean operator from , the formula is equivalent to .hence can be transformed in polynomial time into a formula , where the -operators occur in only in front of atomic propositions or constants ( possibly in sequences ) .the truth of at in can now be determined as follows : guess a path in starting at . for each subformula of , , if , then mark with .finally , evaluate that boolean formula at which is obtained from if all subformulae are considered as atomic propositions .it is clear that all three steps can be performed in polynomial time .combining lemma [ lemma : mc(g|x;m ) np ] and theorem [ theorem : propositional negation][theorem : propositional negation : mc(f|g|x;m ) np - c ] we obtain the following result .[ theorem : mc(g|x;m ) np - c ] let be a finite set of boolean functions such that . then and are -complete .next , we consider formulae with the until - operator or the since - operator .we first show that using the until - operator makes model - checking intractable .[ theorem : mc(u;. ) np - h ] let be a finite set of boolean functions . then is -hard .we give a reduction from to .this means , that we do not need any boolean operators in the temporal formula over to which a instance is mapped .let be a 3cnf formula consisting of clauses and variables .the structure has states , with initial state .the assignment for state is ( for ) , and for state it is . in state , no variable is assigned true .the relation between the states is as follows .each ( ) has an edge to , has edges to and to , each ( ) has edges to and to , and and have an edge to . has an edge to only .figure [ figure : structure for until ] gives an example .   for following facts are easy to verify for any path in . for the sake of simplicity , we use for a path in and the notation for .fact 1 : :  for holds :   .fact 2 : :  for holds :   iff   .  the formulae are defined inductively as follows . reduction from to is the mapping , where is a 3cnf - formula with clauses .this reduction can evidently be performed in logarithmic space . to prove its correctness, we use the following claim .  [ claim : np3 ] let be constructed from a formula with clauses , and let be a path in . for , it holds that   we prove the claim by induction .the base case is straightforward : is equivalent to which by fact 2 is equivalent to .the inductive step is split into two cases .first , assume .since , it follows that .using fact 2 , we conclude .by fact 1 , . by the initial assumption, this leads to .second , assume and . using the induction hypothesis, we obtain for . by the construction of immediately get .we have a one - to - one correspondence between paths in and assignments to variables of . for a path we will denote the corresponding assignment by . using claim [ claim : np3 ] , it is easy to see that the following properties are equivalent .  1 . is a satisfying assignment for .path in contains for every a state with assignment .3 .  for .4 .  .this concludes the proof that if and only if .although the until - operator and the since - operator appear to be similar , model - checking for formulae that use the since - operator as only operator is as simple as for formulae without temporal operators see theorem [ theorem : mc(s;bf ) in l ] .the reason is that the since - operator has no use at the beginning of a path of states , where no past exists .it needs other temporal operators that are able to enforce to visit a state on a path that has a past .[ theorem : mc(s , g;. ) np - h][theorem : mc(s , x;. ) np - h ] let be a finite set of boolean functions . then and are -hard .we give a reduction from to that is similar to that in the proof of theorem [ theorem : mc(u;. ) np - h ] for .let be an instance of , and let be the structure as in the proof of theorem [ theorem : mc(u;. ) np - h ] . from obtain the structure as follows .first , we add a new state , , .second , replace by its inverse from which the loop at state is removed .the state has in - degree and will be seen as initial state of .the new state will be used as sink state .therefore , we add the arcs and .this results in . finally , we add a new variable that is _ true _ only in state , and a variable that is _ true _ in states . for all other variables , is the same as .   for   ( figure [ figure : structure for since ] shows an example . )the formulae are defined inductively as follows . the reduction from to is the mapping , where is a 3cnf - formula with clauses .this reduction can evidently be performed in logarithmic space . to prove its correctness, we use the following claim .every path in that begins in state corresponds to an assignment to the variables in , that sets all literals to _ true _ that appear on . for the sake of simplicity, we use the notation for .[ claim : np4 ] let be constructed from a formula with clauses , and let be a path in .for it holds that    if and only if  the assignment satisfies clauses .notice that satisfies clause if and only if contains a state with .we prove the claim by induction .since the variable holds in all predecessors of in but not in , it follows that holds in iff holds in . since , it follows that holds in iff holds in a predecessor of iff satisfies .this completes the base case .for the inductive step , notice that iff and . by the construction of follows that iff satisfies , and the rest follows from the induction hypothesis .finally , let be a 3cnf formula , and let be a path in .on the first states of , the variable holds . therefore , and henceforth is satisfied in all these states . on the following states , neither nor holds .notice that iff ( for ) . by claim [ claim : np4 ] , andhenceforth is satisfied in all these states iff satisfies . on the remaining states , only the variable holds .hence , is satisfied in all the latter states iff satisfies . concluding, it follows that iff satisfies . since for every assignment to the structure contains a corresponding path , the correctness of the reduction is proven .the future - operator alone is not powerful enough to make the since - operator -hard : we will show in theorem [ theorem : mc(s , f;v ) nl - c ] that for is -complete .but with the help of or , the model - checking problem for and becomes intractable .[ theorem : mc(s , f;e|n ) np - h ] let be a finite set of boolean functions such that . then is -hard .  by lemma [ lemma : subred ] it suffices to give a reduction from  to .for a 3cnf formula , let be the instance of as described in the proof of theorem [ theorem : mc(s , g;. ) np - h ] . using , it follows that , where the latter is a -formula over .the correctness of the reduction the same line as the proof of theorem [ theorem : mc(s , g;. ) np - h ] .  [ theorem : mc(s , x;. ) np - c ] let be a finite set of boolean functions .then is -hard .to prove -hardness , we give a reduction from to . for a 3cnf formula ,let be the structure as described in the proof of theorem [ theorem : mc(s , g;. ) np - h ] .the reduction function maps to .the `` moves '' to state on any path in .the correctness proof follows the same line as the proof of theorem [ theorem : mc(s , g;. ) np - h ] .an upper bound better than for the intractable cases with the until - operator or the since - operator remains open .we will now show that one canonical way to prove an upper bound fails , in showing that these problems do not have the `` short path property '' , which claims that a path in the structure that fulfills the formula has length polynomial in the length of the structure and the formula .hence , it will most likely be nontrivial to obtain a better upper bound.  we will now sketch such families of structures and formulae using an inductive definition .let be the family of graphs presented in figures [ fig : exponential size graph 1 ] and [ fig : exponential size graph 2 ] .notice that is inserted into using the obvious lead - in and lead - out arrows .  ]  ]  the truth assignments for these graphs are as follows : now the formulae are defined as follows :   the rough idea behind the construction is as follows : to satisfy the formula in the path has to repeat the circle once . in the inductive construction , this leads to an exponential number of repetitions .this subsection is concerned with fragments of ltl that have a tractable model - checking problem .we will provide a complete analysis for these fragments by proving that model checking for all of them is -complete or even solvable in logarithmic space .this exhibits a surprisingly large gap in complexity between easy and hard fragments .the following lemma establishes -hardness for all tractable fragments .it is proven in .[ lemma : mc(f|g|x;. ) nl - h ] let be a finite set of boolean functions . then , , and are -hard .first consider .we reduce the accessibility problem for digraphs , gap , to .the reduction is via the following logspace computable function .given an instance of gap , where is a digraph and , map it to the instance of with , where denotes the reflexive closure of , and is given by and , for all .it is immediately clear that there is a path from to in if and only if there is a path in starting from such that .  for, we use an analogous reduction from gap to .given an instance of , where , transform it into the instance of with the kripke structure from above .now it is clear that there is a path from to in if and only if there is a path of length from to in the reflexive structure , if and only if there is a path in starting from such that .now consider .we reduce the following problem to . given a directed graph and a vertex , is there an infinite path in starting at ?it is folklore that this is an -hard problem ( see lemma [ lem : existence_of_infinite_path_is_nl - hard ] in the appendix ) .given an instance of this problem , transform it into the instance of , where .here , , for all , and , for all .it is immediately clear that there is an infinite path in starting at if and only if there is a path in starting from such that .it now remains to establish upper complexity bounds .let be one of the clones , , , and , and let be a finite set of boolean functions such that .whenever we want to establish -membership for some problem , it will suffice to assume that formulae are given over one of the bases , , , or , respectively .this follows since these clones only contain constants , projections , and multi - ary versions of , , , and , respectively .[ theorem : mc(f , g , x;n ) nl - c ] let be a finite set of boolean functions such that then is -complete .the lower bound follows from lemma [ lemma : mc(f|g|x;. ) nl - h ] . for the upper bound , first note that for an ltl formula the following equivalenceshold : , , , , , and . furthermore , it is possible to interchange and adjacent - , - , or -operators without affecting satisfiability . under these considerations ,each formula can be transformed without changing satisfiability into a normal form , where is a prefix ranging over the values `` empty string '' , , , , and ; is the number of occurrences of in ; is either the empty string or ; and is a variable or a constant .this normal form has two important properties .first , it can be represented in logarithmic space using two binary counters and .the counter stores , and takes on values to represent each possible combination of and . note that takes on values less than , and has a constant range .hence both counters require at most logarithmic space . it is not necessary to store any information about , because it can be taken from the representation of .second , can be _ computed _ from in logarithmic space .the value of is obtained by counting the occurrences of in , and is obtained by linearly parsing with the automaton that is given in figure [ fig : automaton_for_gfneg_prefix ] , and which ignores all occurrences of .  ]the state of this automaton at the end of the passage through determines the values of and in .now let be an \{,,},n_--formula , a kripke structure and .if is constant , the problem is trivial , therefore it remains to consider the case where is a variable . according to the possible values of and in ,there are ten cases to consider .we only present the argumentation for those five in which is empty .( for the dual cases , kindly replace each occurrence of `` '' by `` '' . ) in the following list , we assume that . as per explanation below , this is not a significant restriction . is empty : :  then if  and only if there is a state in   accessible from via such that   . : :  in this case we have to check whether there is a state   that can be reached from via   , and . : :  we define and   .it holds that   if and  only if there is some such that   is accessible from via and   belongs to a cycle in . : :  we can reduce this case to the previous one :   if and  only if there is some that can be reached from   via , and   . : :  we have to check whether there exists some that  can be reached from via such that   and belongs to a cycle .since the questions whether there is a path from any vertex to another and whether any vertex belongs to a cycle in a directed graph can be answered in , all previously given procedures are -algorithms .the restriction is removed by the observation that if and only if there exists some state in that is accessible from in -steps such that .this reduces the case to .hence we have found an -algorithm deciding : given , compute , guess a state accessible from in -steps , apply the procedure of one of the above five cases to , and accept if the last step was successful .we show in how to obtain an algorithm using the normal form .[ theorem : mc(f , x;v ) mc(g , x;e ) nl - c ]  [ part : fxv ] let be a finite set of boolean functions such that . then is -complete .[ part : gxe ] let be a finite set of boolean functions such that .+ then is -complete .the lower bounds follow from lemma [ lemma : mc(f|g|x;. ) nl - h ] .first consider the case it holds that as well as and .therefore , every formula can be rewritten as where are variables or constants ( note that this representation of can be constructed in ) .now let be an instance of , where , and let be of the above form .thus , and only if for some , there is a state such that and is accessible from in exactly -steps or if , for some , there is a state such that and is accessible from in at least -steps . this can be tested in .  as for the case we take advantage of the duality of and , and and , respectively .analogous considerations as above lead to the logspace computable normal form let .for each , we define and .furthermore , let be the union of for ( ! ) , and let .now if and only if there is some state satisfying the following conditions .* there is an -path of length at least from to , where the first states on are , , , . *the state lies on a cycle in .* for each , each state of from to is from . * for each , the state is from .these conditions can be tested in as follows .successively guess and verify their membership in the appropriate sets .then guess , verify whether , whether lies on some -cycle , and whether there is an -path from to .  in the proof of theorem [theorem : mc(f , x;v ) mc(g , x;e ) nl - c ] , we have exploited the duality of and , and and , respectively .furthermore , the proof relied on the fact that and ( and and ) are interchangeable .this is not the case for and , or and , respectively .hence it is not surprising that is -hard ( corollary [ cor : mc(f;e ) np - h ] ) .however , the -membership of _ is _ surprising .before we formulate this result , we try to provide an intuition for the tractability of this problem .the main reason is that an inductive view on -formulae allows us to subsequently guess parts of a satisfying path without keeping the previously guessed parts in memory .this is possible because each -formula can be rewritten as where the are variables ( or constants ) , and each is an -formula of the same form with a strictly smaller nesting depth of -operators .now , is _ true _ at the begin of some path iff one of its disjuncts is _ true _ there . in case none of the or _ true _ , we must guess one of the ( or ) and check whether ( or ) is _ true _ on the entire path ( or on minus some finite number of initial states ) .now is again of the above form .so we must either find an infinite path on which is _ true _ everywhere ( a cycle containing at least states satisfying some or suffices , where is the set of states of the kripke structure ) , or we must find a _path satisfying the same conditions and followed by an infinite path satisfying one of the ( or ) at its initial point .hence we can recursively solve a problem of the same kind with reduced problem size .note that it is neither necessary to explicitly compute the normal form for or one of the , nor need previously visited states be stored in memory .[ theorem : mc(f , g;v ) nl - c ] let be a finite set of boolean functions such that then and are -complete .the lower bound follows from lemma [ lemma : mc(f|g|x;. ) nl - h ] .it remains to show -membership of .for this purpose , we devise the recursive algorithm as given in table [ tab : alg_ltl - f , g , v - mc ] .note that we have deliberately left out constants .this is no restriction , since we have observed in lemma [ lemma : const ] that each constant can be regarded as a variable that is set to _ true _ or _ false _ throughout the whole kripke structure .* algorithm *  .the algorithm [ cols= " < , < " , ]   ' '' ''  ' '' ''  ' '' ''  ' '' ''  ' '' ''  [ tab : alg_ltl - f , g , v - mc ]  the parameter _ mode _ indicates the current `` mode '' of the computation .the idea is as follows . in order to determine whether is satisfiable at the _ initial _ point of some structure starting at in , the algorithm has to be in mode ` now ` .this , hence , is the default setting for the first call of .as soon as the algorithm chooses to process a -subformula of , it has to determine whether is satisfiable at _ every _ point in some structure starting at the currently visited state in .it therefore changes into ` always ` mode and calls itself recursively with the first parameter set to , see line 17 .hence , given an instance of the problem , we have to invoke in order to determine whether there is a satisfying path for in starting at .it is easy to see that this call always terminates : first , whenever the algorithm calls itself recursively , the first argument of the new call is a strict subformula of the original first argument .therefore there can be at most recursive calls .second , within each call , each passage through the _ while _ loop ( lines 232 ) either decreases or increases .hence , there can be at most passages through the _ while _ loop until the algorithm accepts or rejects .is an algorithm : the values of all parameters and programme variables are either subformulae of the original formula , states of the given kripke structure , counters of range , or booleans .they can all be represented using , , or constantly many bits .furthermore , since the algorithm uses no _ return _command , the recursive calls may re - use the space provided for all parameters and programme variables , and no return addresses need be stored .it remains to show the correctness of , which we will do in two steps . in ` always ` mode , which will be shown by induction on the nesting depth of the -operator in .we denote this value by .claim [ claim : fgv2 ] will then ensure the correct behaviour in ` now ` mode .both claims are proven in .[ claim : fgv1 ] for each , each , and each :     + .  _ for the base case of the induction , _let .because of the equivalences and , we may assume w.l.o.g .that any occurrence of the -operator is in front of some variable in .if we think of as a tree , this means that -operators can only occur in direct predecessors of leaves .note that the algorithm computes this normal form implicitly : whenever it guesses a path from the root ( ) to some leaf ( a variable ) in the tree and encounters an -operator in line 6 , the flag _ ffound _ is set . only after processing all -operators on the remaining part of the path ,the -operator is processed in lines 1015 .now let be all variables that occur in the scope of an -operator in , and let be all other variables in ._ for the `` '' direction , _suppose .then there exists a path in such that , and for all , .this means that , for each , either there exists some such that , or there is some such that .now it can be seen that there is a non - rejecting sequence of runs through the _ while _ loop in lines 232 after which has value , which then leads to the _ accept _ in line 33 :  consider the begin of an arbitrary single run through the _ while _ loop in line 2 .let be the current value of .if , then the algorithm can `` guess its way through the tree of '' in lines 35 and finally reaches line 19 with .it does not reject in line 20 , increases in line 23 , guesses in line 24 , and resets _ ffound _ and appropriately in lines 25 , 26 .otherwise , if , then there is some such that satisfies .it is safe to assume that because otherwise the path from to would describe a cycle within which could be replaced by a shorter , more direct , path without affecting satisfiability of the relevant subformulae in the states .now the algorithm can proceed as in the previous case , but , in addition , it has to guess the correct value of and the sequence in lines 1015 ._ for the `` '' direction , _let there be an accepting run of . since the algorithm is in `always ` mode , and is -free , the acceptance can only take place in line 33 , without a recursive call in line 17 .hence the counter reaches value in the _ while _ loop in lines 232 .let be the sequence of states guessed in this run in lines 13 and 24 , where .furthermore , let be an index sequence that determines a subsequence of such that  * ,and * for each , is the value assigned to in line 24 after having set to value in line 23 .now it is clear that for all , there must be a variable such that . if , then , and each structure extending beyond satisfies ( and hence ) at .otherwise , and the accepting run of the algorithm has guessed the states in line 13 . in this case , each structure extending beyond satisfies ( and hence ) at .from these two cases , we conclude that each such satisfies in all states .we now restrict attention to the states . among these states ,some of the states of has to occur twice .assume and represent the same state from , where .then we can create an ( infinite ) structure from that consists of states , followed by an infinite repetition of the sequence .it is now obvious that satisfies in every state , hence , that is , ._ for the induction step , _let .for the same reasons as above , we can assume that any -operator only occurs in front of variables or in front of some -operator in .this `` normal form '' is taken care of by setting _ ffound _ to ` true ` when is found ( line 7 ) and processing this occurrence of only when a variable or some -operator is found ( lines 1015 ) ._ for the `` '' direction , _suppose .then there exists a path in such that , and for all , .we describe an accepting run of .consider a single passage through the _ while _ loop with the following configuration .the programme counter has value , has value at most , has value , and has value . since , there are four possible cases .the argumentation for the first two of them is the same as in the base case .blub   , for some . , for some . , for some maximal -subformula of that is _ not _ in the scope of some -operator .this means that is _ true _ everywhere on the path .hence , due to the induction hypothesis , has an accepting run . by appropriate guesses in line 4, the current call of the algorithm can reach that accepting recursive call in line 17 . , for some maximal -subformula of that _ is _ in the scope of some -operator .  by combining the arguments of cases 3 and 2, we can find an accepting run for this case .if only cases 1 or 2 occur more than times in a sequence , then will finally take on value , and this call will accept in line 31 .otherwise , whenever one of cases 3 and 4 occurs , than the acceptance of the new call and hence of the current call is due to the induction hypothesis ._ for the `` '' direction , _let there be an accepting run of .since the algorithm is in ` always ` mode , the acceptance can only take place in line 33 or in the recursive call in line 17 . if the run accepts in line 33 , the same arguments as in the base case apply . if the acceptance is via the recursive call , then let be the sequence of states guessed such that , and is the value of when the recursive call with takes place .due to the induction hypothesis , and , hence , there is an infinite structure extending beyond such that .furthermore , we can use the same argumentation as in the base case to show that , for each , .therefore , , which proves .[ claim : fgv2 ] for each , each , and each :   _ for the `` '' direction , _suppose .then there exists a path in such that and .we describe an accepting run of .consider the first passage through the _ while _ loop with the following configuration .the programme counter has value , has value ( this value does not change in _ now _ mode ) , has value , and has value . since , there are four possible cases .the argumentation for them is very similar to that in the proof of claim [ claim : fgv1 ] .blub   , for some .  as in the proof of claim [ claim : fgv1 ], the algorithm can guess the appropriate disjuncts in lines 35 , does not reject in line 20 and accepts ( it is in _ now _ mode ! ) in line 28 . , for some .  as in the proof of claim [ claim : fgv1 ], there exists some with such that satisfies .the algorithm can proceed as in the previous case , but , in addition , it has to guess the correct value of and the sequence in lines 1015 . , for some maximal -subformula of that is _ not _ in the scope of some -operator .this means that is _ true _ everywhere on the path .hence , due to the induction hypothesis , has an accepting run . by appropriate guesses in line 4, the current call of the algorithm can reach that accepting recursive call in line 17 . , for some maximal -subformula of that _ is _ in the scope of some -operator .  by combining the arguments of cases 3 and 2, we can find an accepting run for this case ._ for the `` '' direction , _suppose there exists an accepting run of .since the algorithm is in ` now ` mode , the acceptance can only take place in line 28 or in the recursive call in line 17 . if the run accepts in line 28 , then there is some variable such that either and , or and the run guesses a path with and . in both cases ,each structure extending the sequence of states guessed so far , satisfies at .on the other hand , if the run accepts in the recursive call , we can argue as in the proof of claim [ claim : fgv1 ] .unfortunately , the above argumentation fails for because of the following considerations .the -algorithm in the previous proof relies on the fact that a satisfying path for , where is of the form , can be divided into a `` short '' initial part satisfying the disjunction of the atoms , and the remaining end path satisfying one of the at its initial state . when guessing the initial part , it suffices to separately guess each state and consult .if were in our language , the disjuncts would be of the form and . not only would this make the guessing of the initial part more intricate .it would also require memory for processing each of the previously satisfied disjuncts .an adequate modification of would require more than logarithmic space .we have shown -hardness for in theorem [ theorem : mc(g , x;v ) np - h ] .[ theorem : mc(x;l ) nl - c ] let be a finite set of boolean functions such that then is -complete .* algorithm *  the lower bound follows from lemma [ lemma : mc(f|g|x;. ) nl - h ] .  for the upper bound , let be a formula , a kripke structure , and a state .let denote the maximal nesting depth of -operators in .since for any -ary boolean operator from , the formula is equivalent to , is equivalent to a formula of the form , where for each .it is not necessary to compute all at once , because it will be sufficient to calculate each time the variable is encountered in the algorithm given in table [ tab : xlalgo ] .it is easy to see that returns if and only if is satisfiable . from the used variables ,it is clear that runs in nondeterministic logarithmic space .   in the fragment with  as the only temporal operator ,  is without effect , since we can never leave the initial state .hence , any formula is satisfied at the initial state of any structure if and only if is .this leads to a straightforward logspace reduction from to : given a formula , successively replace every subformula by until all occurrences of are eliminated .the resulting formula is initially satisfied in any structure iff is .now is the formula value problem , which has been shown to be solvable in logarithmic space in .thus we obtain the following result .  [theorem : mc(s;bf ) in l][theorem : mc(s;. ) in l ] let be a finite set of boolean functions. then .  in our classification of complexity , which is based on logspace reductions , a further analysis of -fragments is not possible .however , a more detailed picture emerges if stricter reductions are considered , see ( * ? ? ?* chapter 2 ) .[ theorem : mc(s , f;v ) nl - c ] let be a finite set of boolean functions such that then is -complete .the lower bound follows from lemma [ lemma : mc(f|g|x;. ) nl - h ] . for the upper bound, we will show that can be reduced to by disposing of the -operator as follows .consider an arbitrary kripke structure and a path therein .then the following equivalences hold .statements and are standard properties and follow directly from the definition of satisfaction for and .statement is simply due to the fact that there is no state in the past of . as for, we consider both directions separately .assume that .then there is some such that .this implies that there is some with and .hence , . for the other direction , let .then there is some such that .this implies .hence , .now consider an arbitrary formula .let be the formula obtained from by successively replacing the outermost by until all occurrences of  are eliminated .this procedure can be performed in logarithmic space , and the result is in . due to , for any path in any kripke structure , it holds that hence , the mapping is a logspace reduction from to .we have almost completely separated the model - checking problem for linear temporal logic with respect to arbitrary combinations of temporal and propositional operators into tractable and intractable cases .we have shown that all tractable mc problems are at most -complete or even easier to solve .this exhibits a surprisingly large gap in complexity between tractable and intractable cases .the only fragments that we have not been able to cover by our classification are those where only the binary -operator is allowed .however , it is not for the first time that this constellation has been difficult to handle , see . therefore , these fragments can justifiably be called ugly .the borderline between tractable and intractable fragments is somewhat diffuse among all sets of temporal operators without . on the one hand , this borderline is not determined by a single set of propositional operators ( which is the case for the satisfiability problem , see ) . on the other hand , the columns anddo not , as one might expect , behave dually .for instance , while is tractable , is not although and are dual , and so are and.  further work should find a way to handle the open cases from this paper as well as from .in addition , the precise complexity of all hard fragments not in bold - face type in table [ tab : overview ] could be determined .furthermore , we find it a promising perspective to use our approach for obtaining a fine - grained analysis of the model - checking problem for more expressive logics , such as ctl , ctl * , and hybrid temporal logics .bcrv03  e. bhler , n. creignou , s. reith , and h. vollmer .playing with boolean blocks , part i : post s lattice with applications to complexity theory ., 34(4):3852 , 2003 .m. bauland , e. hemaspaandra , h. schnoor , and i. schnoor .generalized modal satisfiability . in b.durand and w. thomas , editors , _ stacs _ , volume 3884 of _ lecture notes in computer science _ , pages 500511 .springer , 2006 .m. bauland , t. schneider , h. schnoor , i. schnoor , and h. vollmer .the complexity of generalized satisfiability for linear temporal logic . in h. seidl , editor , _ fossacs _ , volume 4423 of _ lecture notes in computer science _ ,pages 4862 .springer , 2007 .v. dalmau . .phd thesis , department de llenguatges i sistemes informtica , universitat politcnica de catalunya , 2000 .h. lewis .satisfiability problems for propositional calculi . , 13:4553 , 1979 .nancy a. lynch .log space recognition and translation of parenthesis languages ., 24(4):583590 , 1977 .nicolas markey .past is for free : on the complexity of verifying linear temporal properties with past . , 40(6 - 7):431458 , 2004 .g. nordh .a trichotomy in the complexity of propositional circumscription . in _ lpar _ ,volume 3452 of _ lecture notes in computer science _ , pages 257269 .springer verlag , 2005 .n. pippenger . .cambridge university press , cambridge , 1997 .a. pnueli .the temporal logic of programs . in _ focs _ , pages 4657 .ieee , 1977 .e. post .the two - valued iterative systems of mathematical logic . , 5:1122 , 1941 .s. reith . .phd thesis , fachbereich mathematik und informatik , universitt wrzburg , 2001 .s. reith and h. vollmer .optimal satisfiability for propositional calculi and constraint satisfaction problems . , 186(1):119 , 2003 .s. reith and k. w. wagner .the complexity of problems defined by boolean circuits . in _mfi 99_. world science publishing , 2005 .w. j. savitch .maze recognizing automata and nondeterministic tape complexity ., 7:389403 , 1973 .a. sistla and e. clarke .the complexity of propositional linear temporal logics ., 32(3):733749 , 1985 .h. schnoor . .phd thesis , university of hannover , 2007 .[ lem : existence_of_infinite_path_is_nl - hard ]   we reduce from the graph accessibility problem ( gap ) , which is defined as follows . given a directed graph andtwo nodes , is there a path in from to ?this problem is known to be -complete  .  for the reduction ,consider an arbitrary instance of gap , where and .let .we transform into a new graph that consists of `` layers '' each of which contains a copy of the nodes from .whenever there is an edge from node to node in , the new graph will have edges from each copy of to the copy of on the next layer .this destroys all cycles from .now we add an edge from each copy of to the first copy of .it is easy to see that this transformation is a logspace reduction .let the size of a graph be determined by the size of its adjacency matrix .hence has size , and is of size . apart from the representation of , the only space required by the described transformationis spent for four counters that take values between and . with their help , each bit of the new adjacency matrix is set according to the definition of , where only a look - up in the old adjacency matrix is required .[ claim : graph_problems ] for each directed graph and each pair of nodes , there exists a path in from to if and only if there exists an infinite path in starting at .`` '' .suppose there is a path in from to .we can assume that no node occurs more than once on this path , and included .hence there exist nodes with such that , , and for each , . due to its construction, has the cycle that contains .hence has an infinite path starting at .`` '' .suppose there is an infinite path in starting at .since is finite , some node must occur infinitely often on .this , together with the layer - wise construction of , implies that there are infinitely many nodes of layer 1 on . among layer-1 nodes, only has ingoing edges .hence must occur infinitely often on .now the path from some occurrence of to the next is a cycle , where the predecessor node of must be some .this implies that there is a path in from to . due to the construction of ,this corresponds to a path in from to .

Abstract: in a seminal paper from 1985 , sistla and clarke showed that the model - checking problem for linear temporal logic ( ltl ) is either -complete or -complete , depending on the set of temporal operators used . if , in contrast , the set of propositional operators is restricted , the complexity may decrease . this paper systematically studies the model - checking problem for ltl formulae over restricted sets of propositional and temporal operators . for almost all combinations of temporal and propositional operators , we determine whether the model - checking problem is tractable ( in ) or intractable ( -hard ) . we then focus on the tractable cases , showing that they all are -complete or even logspace solvable . this leads to a surprising gap in complexity between tractable and intractable cases . it is worth noting that our analysis covers an infinite set of problems , since there are infinitely many sets of propositional operators .


a variety of creatures from many small organisms to social animals including humans show altruistic behavior .altruism occurs even when being selfish is better for an individual , which constitutes the social dilemma .emergence of altruism under social dilemmas can be explained by various mechanisms , such as kin selection , direct reciprocity , indirect reciprocity , and group selection . in the prisoners dilemma , altruism is also promoted by the viscosity of populations . if players are aligned on a spatially structured graph such as the square lattice , cooperators form close - knit clusters of conspecifics to survive the invasion of selfish defectors .maintenance of such clusters is impossible in well - mixed populations modeled by the random graph and the all - to - all connected network .spatial structure affects cooperation in other games as well ( e.g. ) .particularly , spatial structure can be detrimental to cooperation in the snowdrift game .  to be more realistic, players often inhabit networks of social contacts more complex than the square lattice , the random graph , and the all - to - all connected network .first , real social networks are small - world , implying the combination of abundant localized interactions , as in the square lattice , and sufficient shortcuts , as in the random graph .second , players are heterogeneous in terms of the number of contacts with others .an extreme case of this is the scale - free network in which the number of neighbors is distributed according to the power law .in contrast , the number of neighbors of conventional networks is the same for everybody ( regular lattices and the all - to - all connected network ) or distributed with a narrow tail ( so - called erds - rnyi random graph ) .even though not all social networks are scale - free , the number of neighbors is naturally heterogeneous .recently , it was shown that heterogeneous networks promote evolution of cooperation in the prisoner s dilemma , the snowdrift game , and the stag hunt game .particularly , scale - free networks are strong amplifiers of altruism .  to explain the mechanism of enhanced cooperation , let us introduce the notion of the temperature of players . in evolutionary graph theory , hot players are those replaced often by others . by modifying this definition slightly , we denote by hot players those who play often , namely , players with many neighbors .cold players are those with a small number of neighbors , such as leaves in a network .hot players are allowed in more rounds of the game than cold players per generation . if the typical payoff obtained by playing a game is positively biased , which is a common assumption adopted in the previous studies cited above and others ( but see for other setups ) , it is largely worth participation. then hot players earn more than cold players because everybody earns positive ` base ' payoffs proportional to the number of neighbors . as a result ,hot players are more successful in disseminating their strategies .particularly , cooperation once employed by a hub is stable due to cooperative reactions in its neighborhood . in addition, if hubs tend to be connected to each other , as in the model proposed by barabsi and albert ( ba model ) , cooperators on hubs form loose clusters to take advantage of a variant of spatial reciprocity .however , hot players are successful not owing to playing well but owing to the connectivity . in the present paper ,we critically re - examine the effect of heterogeneous networks on emergence of cooperation .in real lives , participation in the game may be costly .a link to a neighbor implies building and maintaining communication , and this cost has actually been modeled for studying network formation .expensive entry fees would dismiss the premium of hot players to change the entire scenario .we study two - person games on networks with participation costs . for simplicity ,networks are assumed to be fixed in size and topology .we show that there are three regimes depending on how costly participation is .first , when participation is inexpensive , cooperation is enhanced on heterogeneous networks as discovered previously .second , when the participation cost is intermediate , the effect of the local payoff structure , namely , the configuration of the two - person game , and that of the network are comparable. then altruism does not develop .third , when participation is very costly , initial strategies of cold players have long transients and propagate to hot players . with small and large participation costs , the network rather thanthe local payoff structure determines evolutionary dynamics . in the intermediate regime , evolution is most sensitive to the local payoffs .we compare effects of two types of networks on the evolution of cooperation by means of monte carlo simulations .a diluted well - mixed population is modeled by the regular random graph in which each player has 8 neighbors that are chosen randomly from the population .the heterogeneous networks are modeled by scale - free networks generated by the ba model , in which the number of neighbors denoted by follows the power law  .the average number of neighbors in the scale - free networks is set equal to 8 for fair comparison with the regular random graph .both types of networks consist of players .to probe the network effect , we consider only two simple strategies without memory , namely , unconditional cooperation and unconditional defection .the initial fraction of cooperators is set equal to 0.5 . in one generation, everybody participates in the two - person game with all the neighbors .the payoff matrix will be specified in the next section .each player tends to copy successful strategies in its neighborhood .we apply the update rule compatible with the replicator dynamics , following the previous literature .suppose that player with neighbors has obtained generation - payoff . to update the strategy , selects a player among the neighbors with equal probability ( ) .then copies s strategy with probability  [ uppermost payoff in one game - lowermost payoff in one game ] if .the denominator is the normalization constant so that the replacement probability ranges between 0 and 1 . if , the strategy of is unchanged .all the players experience updating according to this rule synchronously .this completes one generation .each evolutionary simulation consists of 5000 generations .the final fraction of cooperators denoted by is calculated as the average fraction of cooperators based on the last 200 generations of 5 runs with different initializations for each network and 5 different realizations of the network . when the participation cost is not very large , corresponds to values close to stochastic stationary values .otherwise , represents transient values .we start with two well - known payoff matrices of the simplified prisoner s dilemma . the first one is given by the entries of eq .( [ eq : pd1-basic ] ) indicate the payoff that the row player gains when playing against the column player .the first ( second ) row and column correspond to cooperation ( defection ) .the prisoner s dilemma arises when , and larger results in more defectors .with participation cost , the payoff matrix becomes note that introducing does not trespass the notion of the prisoner s dilemma as far as .  actually , the game defined by eq .( [ eq : pd1-basic ] ) or eq .( [ eq : pd1-net ] ) lies on the boundary between the snowdrift game analyzed in sec .[ sub : sg ] and the prisoner s dilemma .therefore we also examine another standard payoff matrix of the generic prisoner s dilemma given by    where . with the participation cost , eq .( [ eq : pd2-basic ] ) is transformed to where . in the following ,we refer to numerical results for the payoff matrix given by eq .( [ eq : pd1-net ] ) .the results that are qualitatively the same as the following are obtained for eq .( [ eq : pd2-net ] ) , as shown in supplementary figure [ fig : supp1](b - d ) .the fraction of cooperators is not affected by on the regular random graph ( supplementary figure [ fig : supp1](a ) ) .because each player has the same number of neighbors , participation cost does not differentiate the players .in contrast , drastically affects for the scale - free networks , as shown in fig .[ fig : pd](a ) . in fig .[ fig : pd](b ) , for the scale - free networks relative to for the regular random graph is plotted .we identify three qualitatively different regimes in terms of , which roughly correspond to ( i ) , ( ii ) , and ( iii ) .the transition between ( ii ) and ( iii ) is fairly gradual .   when , participation is inexpensive , and hot players such as hubs are strong competitors regardless of the strategies of their cold neighbors .the payoff of a player increases linearly with the number of contacts , as shown by thin lines with positive slopes in fig .[ fig : pd_flip](a ) .in particular , cooperation spreads from hot cooperators to their cold neighbors , the local density of cooperators increases , and hot cooperators gain more by mutual cooperation .cooperation triggered by hot players is self - promotive .defective hot players may also win for a moment .however , defection then prevails in their neighborhood so that hot defectors can no longer exploit the neighbors because of mutual defection .this results in a null generation - payoff of hot defectors so that they can be outperformed by their cold neighbors .a hot player sticks to cooperation but not to defection .  in sum ,heterogeneous networks enhance altruism , which recovers the previous work corresponding to  .note that this regime extends to , that is , when gifts are given for participation so that everyone wins a positive payoff .  to illuminate on different dynamics of hot and cold players , we measure how often players flip the strategy . the average number of flips throughout the evolutionary run ( including the contribution from both transients and stationary states ) is shown in fig .[ fig : pd_flip](b ) .colder players experience more flips when ( thin lines ) .they myopically follow what hotter players do both in transients and in stationary states .cooperation on hubs is stabilized in an early stage , yielding less flips of hotter players .interestingly , cold players spread their strategies to hot players when ( regimes ( ii ) and ( iii ) ) , which is opposite to what happens in regime ( i ) . as a result ,enhanced cooperation diminishes , even with a relatively small participation cost .  regime ( ii ) is defined by small to intermediate ( ) .now the local payoff structure as well as the network topology is relevant .when , scale - free networks surpass the regular random graph in terms of the number of cooperators only for .when , this range shrinks to .the privilege of scale - free networks is entirely lost when . in regime ( ii ) , the payoff of a player linearly decreases with the number of contacts ( thick lines with negative slopes in fig .[ fig : pd_flip](a ) ) .consistent with this , hot players flip strategies more often than cold players ( thick lines in fig . [fig : pd_flip](b ) ) .hubs no longer conduct the dynamics .   when is roughly greater than 2 , participation is really costly. then cold players with any strategy surpass hot players and govern the dynamics .this is not because cold players are tactical but because they play less often and lose less than hot players do .strategies of cold players are not often replaced owing to the small number of neighbors .consequently , cold players persist in their initial strategies , actually up to generations . in the meantime ,strategies of cold players tend to spread to hot neighbors . as a result , , which is measured at 5000 generations , is almost the same as the initial fraction of cooperators ( ) regardless of ( fig .[ fig : pd](a ) ) .figure [ fig : pd](b ) indicates that is larger for the scale - free networks than for the random graph , but this is due to long transients . after a very long period, the fraction of cooperators will be smaller than shown in fig .[ fig : pd](a ) to eradicate the spurious advantage of scale - free networks .the snowdrift game , which is also known as the chicken game or the hawk - dove game , is illustrated as a situation of two drivers caught in a snowdrift .for the two cars to get out , which is equivalent to payoff  for each driver , the snow must be shoveled away .a total effort of unity must be invested to this task .two players may cooperate to share the cost so that each pays , or one player may cover the full cost .otherwise , both players miss .cooperation benefits not only the opponent but also the focal player itself .different from the prisoner s dilemma in which defection is beneficial , in the snowdrift game , cooperation can deserve even when the opponent defects .if the participation is free , the payoff matrix of the snowdrift game is given by in this case , heterogeneous networks reinforce evolution of cooperation as in the prisoner s dilemma . without dismissing the structure of the snowdrift game , the participation cost translates the payoff matrix to the following form :   as shown in supplementary figure [ fig : supp2 ] , the participation cost does not influence on the regular random graph .the fraction of cooperators converges to a value close to the theoretical estimate , where is the cost - to - benefit ratio .if cooperation is relatively costly with a small ( large ) , cooperators decrease in number .  on heterogeneous networks ,how the fraction of cooperators varies as a function of depends on the participation cost .we again find three regimes as shown in fig .[ fig : sg ] . the scale - free networks host more cooperators than the regular random graph only when is near zero or negative ( regime ( i ) ) .the advantage of the scale - free networks is neutralized by intermediate ( roughly speaking , ) , which defines regime ( ii ) .note that the reduction of cooperation is not as much as that for the prisoner s dilemma . with large ( roughly speaking , ), is rather insensitive to the local payoff structure due to long transients ( regime ( iii ) ) .   with the participation cost incorporated , general symmetrical two - person games with two strategiesare represented by in accordance with the previous sections , we denote by cooperation ( defection ) the strategy corresponding to the first ( second ) row and column . as increases, players are tempted to defect , and decreases .as decreases , players would refuse cooperation to avoid exploitation by defectors .therefore , decreases .the prisoner s dilemma , the snowdrift game , and the stag hunt game , are defined by , , and , respectively .the prisoner s dilemma usually accompanies another condition so that mutual cooperation is more beneficial than alternating unilateral cooperation .  multiplying each element of eq .( [ eq : full ] ) by a common constant modifies just the time scale of evolution . accordingly, there are three free parameters in the payoff matrix , which are chosen to be , , and , while we set and .  in fig .[ fig : t_and_s](a ) , for is plotted in the - parameter space for the regular random graph . as expected , the number of cooperators decreases with and increases with .the results are independent of . for the scale - free networks , we plot in fig .[ fig : t_and_s](b - e ) for four values of ( also see supplementary figure [ fig : supp3 ] for the direct comparison of for the scale - free networks and for the regular random graph ) .we confirm the existence of the three regimes , extending the results shown in figs .[ fig : pd ] and [ fig : sg ] .first , as shown in fig .[ fig : t_and_s](b ) , scale - free networks promote evolution of cooperation when the participation is costless .cooperation is strengthened in the prisoner s dilemma ( , ) , the snowdrift game ( , ) , and also the stag hunt game ( , ) .second , the advantage of the heterogeneity is lost for a wide range of and when ( fig .[ fig : t_and_s](c ) ) and ( fig .[ fig : t_and_s](d ) ). third , the transient is very long when participation is costly ( ) .this allows defectors to survive for a long time even without dilemma ( ) and considerable cooperators to survive under the prisoner s dilemma ( fig .[ fig : t_and_s](e ) ) .we have discovered that the participation cost , which is irrelevant in well - mixed populations and on homogeneous networks including the regular lattices , casts a dramatic effect on evolutionary dynamics on heterogeneous networks . when participation is nearly free ( regime ( i ) ) ,heterogeneous networks promote cooperation .this is because the cooperation on hot players is robust against invasion of whatever strategies of cold players . even if a cold player is good at exploiting cooperators in the neighborhood, the aggregated payoff would be much smaller than that of a hot player that would earn a lot just by playing the game many times .hot players are leaders , and cold players are myopic followers .however , this phenomenon is not robust against variation in participation costs , which is consistent with the loss of cooperation under positive affine transformation of the payoff matrix . when the participation cost is intermediate ( regime ( ii ) ) , cooperators do not really increase and even decrease on heterogeneous networks .when participation is costly ( regime ( iii ) ) , hot players myopically follow cold players . in regimes ( i ) and( iii ) , not local payoff structure but network structure governs evolution .the local payoffs are relevant only in regime ( ii ) , for which cooperation is not enhanced by heterogeneous networks .regimes ( ii ) and ( iii ) , which have been largely unexplored , may be relevant in , for example , environmental problems , political conflicts , and human relationships . in these situations ,players are often forced to play and incur participation costs . in other words ,the best one could get may be the least disastrous , but not really wonderful , solution .  for replicator - type and many other update rules in well - mixed populations ,evolution is invariant under uniform addition of a constant to the payoff matrix ( e.g. participation cost ) .then a game in regimes ( ii ) and ( iii ) can be translated into a game in regime ( i ) . however , this operation is disallowed for heterogeneous networks . because multiplying the payoff matrix by a positive constant alters just the evolutionary time scale in either case , there are two free parameters for two - person games in homogeneous populations , whereas there are three of them in heterogeneous populations ( e.g. , , and as used in fig .[ fig : t_and_s ] ) .the present results can be generalized in some aspects .first , two - person games can be asymmetrical .second , the update rule does not have to be of a replicator type as far as the reproduction rate is not extremely nonlinear in the generation payoff .third , large noise would blur but would not break down the three regimes .for example , irrational actions may cause hot cooperators , which are stable with small participation costs and small noise , to defect and elicit upsurges of defectors nearby .however , noise does not overturn the fact that hot ( cold ) players are better off for small ( large ) participation costs .fourth , network models can be arbitrary .for example , the scale - free networks based on the configuration model ( without growth and preferential attachment ) and the erds - rnyi random graph promote altruism in regime ( i ) , albeit to a lesser extent than the ba model .these heterogeneous networks as well as the ba model do not enhance altruism in regimes ( ii ) and ( iii ) ( data not shown ) .the present results also have limitations .first , we have only assumed memoryless strategies , namely , unconditional cooperators and unconditional defectors .second , the networks have been static . in coevolutionary dynamicsin which players form and sever links as well as play games , only regime ( i ) has been considered , a main conclusion being enhanced cooperation . in regime ( iii ) , for example , everybody must sever links to be loners ( for the role of loners , also refer to ) .third , we have used the additive payoff .an alternative is to use the average payoff , or division of the generation payoff of each player by the number of neighbors .the average payoff is not affected by the number of neighbors and hampers the enhanced altruism on heterogeneous networks .we have adopted the additive payoff scheme because its intuitive meaning is clearer than the average payoff .these topics warrant for future work .we thank hisashi ohtsuki and eizo akiyama for their valuable discussions and critical reading of the manuscript .kim , b. j. , trusina , a. , holme , p. , minnhagen , p. , chung , j. s. & choi , m. y. 2002 dynamic instabilities induced by asymmetric influence : prisoner s dilemma game in small - world networks .e _ * 66 * , 021907 .figure 1 : the prisoner s dilemma on the scale - free networks with participation costs .the payoff matrix is represented by eq .( [ eq : pd1-net ] ) .( a ) the final fraction of cooperators .( b ) for the scale - free networks shown in ( a ) minus for the regular random graph shown in supplementary figure [ fig : supp1]a .figure 2 : the effect of the number of neighbors in the prisoner s dilemma on the scale - free networks .( a ) the average generation payoff and ( b ) the average number of flips are plotted in terms of the number of neighbors that a player has .note that the log scale is used for the abscissa of ( b ) for clarity .the lines correspond to ( thinnest line ) , 0.2 , 0.23 , 0.24 , 0.25 , 0.3 , and 0.5 ( thickest line ) .the payoff matrix is given by eq .( [ eq : pd1-net ] ) with .figure 3 : the snowdrift game on the scale - free networks with participation costs .( a ) , and ( b ) for the scale - free networks shown in ( a ) minus for the regular random graph shown in supplementary figure [ fig : supp2 ] .supplementary figure 1 : ( a ) the final fraction of cooperators in the prisoner s dilemma on the regular random graph .the payoff matrix is represented by eq .( [ eq : pd1-net ] ) .the results for the payoff matrix eq .( [ eq : pd2-net ] ) are shown in ( b - d ) .( b ) for the regular random graph , ( c ) for the scale - free networks , and ( d ) for the scale - free networks minus for the regular random graph .supplementary figure 3 : for the scale - free networks ( fig .[ fig : t_and_s](b - e ) ) minus for the regular random graph ( fig .[ fig : t_and_s](a ) ) in the - space .( a ) , ( b ) , ( c ) , and ( d ) .

Abstract: real social interactions occur on networks in which each individual is connected to some , but not all , of others . in social dilemma games with a fixed population size , heterogeneity in the number of contacts per player is known to promote evolution of cooperation . under a common assumption of positively biased payoff structure , well - connected players earn much by playing frequently , and cooperation once adopted by well - connected players is unbeatable and spreads to others . however , maintaining a social contact can be costly , which would prevent local payoffs from being positively biased . in replicator - type evolutionary dynamics , it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation . in this situation , more connected players earn less so that they are no longer spreaders of cooperation . instead , those with fewer contacts win and guide the evolution . the participation cost , or the baseline payoff , is irrelevant in homogeneous populations but is essential for evolutionary games on heterogeneous networks .   keywords : altruism , evolutionary game , complex networks , scale - free networks