introduced in 2006 by japaridze , cirquent calculus is a refinement of classical sequent calculus .the advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources . unlike the more traditional proof - theoretic approaches that manipulate tree - like objects ( formulas , sequents , etc . ) , cirquent calculus is based on circuit - style structures called _ cirquents _ , in which different `` peer '' ( sibling , cousin , etc . ) substructures may share components .it is this resource sharing mechanism to which cirquent calculus owes its novelty ( and its virtues ) .cirquents come in a variety of forms . one way to characterizethe sort of cirquents studied in this paper in familiar terms is to say that a cirquent is a multiset of sequents ( called `` groups '' ) where each formula more precisely , each occurrence of a formula may simultaneously belong to more than one sequent .this explains the origin of the word `` cirquent '' , which is a hybrid of `` circuit '' and `` sequent '' . from its inception ,cirquent calculus has been paired with an abstract resource semantics .this semantics allows for reasoning about the interaction between a resource provider and a resource user , where resources are understood in the their most general and intuitive sense .interpreting resources in a more restricted computational sense has made cirquent calculus instrumental in axiomatizing various fundamental fragments of _ computability logic _ , a formal theory of ( interactive ) computability .the so - called classical " rules of cirquent calculus , in the absence of the particularly troublesome _ contraction _ rule , produce a sound and complete system cl5 for computability logic . born in , computability logic ( col )is an ambitious research program aimed at developing a formal theory of interactive computability . to this end ,formulas of col represent computational problems modeled at games .the notion of truth " for such formulas becomes synonymous with the notion of computability " for the computational problems they represent .while cl5 has shown to be sound and complete with respect japridze s abstract resource semantics , it has also shown to validate a strictly larger class of formulas than affine logic ( sequent calculus without the contraction rule ) , the latter being well studied and sound as a logic of resources . however , due to no shortcomings in effort , a complete resource aware semantics has not been found for affine logic . this had lead japridze to conclude that cl5 rather than affine logic adequately materializes the resource philosophy traditionally associated with the latter " .indeed , the semantics before syntax " philosophy upon which cl5 was conceived , together with its completeness result provide compelling evidence in this direction .  in parallel with the development of cirquent calculus , recent work has introduced sound and complete axiomizations for several fragments of col , the most fundamental of these being cl4 , which contains the propositional connectives ( negation ) , ( parallel disjunction ) , ( parallel conjunction ) , ( choice disjunction ) and ( choice conjunction ) as well as the choice " quantifiers and and the blind " quantifiers and . for a full discussion of these operators and the semantics of col ,see .while the semantics of col differs drastically from that of classical logic , the set of valid formulas of cl4 is identical to that of classical logic when restricted to its operators and the so - called elementary " sort of atoms .thus , the presence of the quantifiers and within cl4 s language guarantees its undecidability . in , however , the ,-free fragment of cl4 was shown to be decidable in polynomial space . a pspace - completeness proof was later given in .the deductive apparatus of cl4 is far removed from traditional gentzen or hilbert style axiomizations .further , the system does not extend very naturally to other fragments of computability logic .a rectification to this shortcoming came with the advent of cirquent calculus .as mentioned previously , the fragment of cirquent calculus not containing the _ contraction _ rule , produces a sound and complete system for the fragment of computability logic known as cl5 .this system contains the operators , and and in this paper we show it to be -complete .we further investigate the complexity of the logical system described by the rules of cl5 , but with the absence of the _ duplication _ rule .we call this logic cl5 and show that every theorem in cl5 has a polynomial size proof .this effectively places the problem of deciding cl5 provability in . a reduction from the vertex cover problem to cl5 provabilityis then given , solidifying the logic as np - complete . in some sense, cl5 is the col counterpart " to multiplicative linear logic , the former being a proper extension of the latter . here , cl5 lies strictly between multiplicative affine logic ( linear logic with weakening ) and cl5 . among the virtues of multiplicative affine logicis that , unlike the conp - complete classical logic , it is in np .cl5 thus presents a nice and natural extension of multiplicative affine logic that gets closer " to classical logic while still remaining in np .  with the tight relationship between cirquent calculus and computability logic, one should not overlook the potential impact of cirquent calculus on other areas of logic and proof theory . in ,xu constructs a cirquent calculus based system for the propositional fragment of independence friendly ( if ) logic , allowing one to account for independence from propositional connectives in the same vein that traditional if logic accounts for independence from quantifiers . in cirquent calculus based proof system was developed which , among other benifits , yields polynomial size proofs for all instances of the pigeonhole principle . to date , this is the first proof system to achieve such a result without leveraging cut , extension or substitution .this paper is organized as follows . in section[ 2 ] , we provide an introduction to the rules of cirquent calculus and describe the systems cl5 and cl5 . section [ 3 ] studies the system cl5 and proves its np - completeness . in the final section, we show the full cl5 to be -complete by a reduction from the tqbf- problem .the present section provides the technical definitions and information necessary for understanding the results of this paper . for a full discussion of the concepts of cirquent calculus ,see .as noted previously , a * formula * of the language of cirquent calculus is syntactically identical to that of classical propositional logic , where formulas are built from the connectives , and as well as non - logical * atoms * ( also known as propositional letters ) . the only difference between the two being that the language of cirquent calculus does not contain logical atoms such as or ; the latter can simply be understood as abbreviations of and , respectively , for some ( whatever ) atom .our language further mandates that can only be applied to atoms . as always ,a * literal * is an atom with or without the prefix .a -ary * pool * , for , is a sequence pl = of formulas .the formulas in a pool are not required to be unique .we refer to a particular occurrence of a formula as an * oformula*. a -ary * structure * , for , is a finite sequence st = for , where each , called a * group * of st , is a subset of .it is permitted that for and hence we use the term * ogroup * to refer to a particular occurrence of a group .we are now ready to give our definition for a cirquent . a -ary ( ) * cirquent * is a pair = ( st , pl ) , where st , called the structure of , is a -ary * structure * , and pl , called the * pool * of , is a -ary pool .  for example, let pl = and st = .here , has 4 oformulas and 3 ogroups and is typically represented by the diagram below .( 98,54 ) ( 0,44)(1,0)98 ( 0,32) ( 30,32) ( 60,32) ( 90,32)  ( 20,10)(-2,3)13 ( 20,10)(4,5)15 ( 49,10)(-4,5)15 ( 49,10)(4,5)15 ( 78,10)(4,5)15 ( 20,10 ) ( 49,10 ) ( 78,10 )  following , we will adopt the notion that a diagram simply is " ( rather than just represents " ) a cirquent .when an ogroup is connected with an arc to an oformula , we say that * contains * .  before defining the rules for our inference system ,let us first give some additional terminology necessary for the definition of our rules .for a given cirquent , two oformulas and are called * adjacent * if is positioned immediately to the right of . in such a case, it is said that * immediately precedes * and * immediately follows * . *merging two adjacent ogroups * and in a given cirquent means replacing in the two ogroups and by a single ogroup .the rightmost cirquent below represents the cirquent that results from merging the first and second ogroups in the leftmost cirquent .( 230,54 )  ( 0,44)(1,0)98 ( 0,32) ( 30,32) ( 60,32) ( 90,32)  ( 20,10)(-2,3)13 ( 20,10)(4,5)15 ( 49,10)(-4,5)15 ( 49,10)(4,5)15 ( 78,10)(4,5)15 ( 20,10 ) ( 49,10 ) ( 78,10 )  ( 140,44)(1,0)92 ( 140,32) ( 170,32) ( 197,32) ( 224,32)  ( 173,10)(-5,4)25 ( 173,10)(5,4)25 ( 173,10)(0,1)19 ( 212,10)(4,5)15 ( 173,10 ) ( 212,10 )  * merging two adjacent oformulas * and into a new oformula is the result of replacing an by and redirecting to it all of the arcs that were pointing to or .the rightmost cirquent below represents the cirquent obtained from merging , in the leftmost cirquent , ( the first ) and into a single oformula .( 230,54 )  ( 0,44)(1,0)98 ( 0,32) ( 30,32) ( 60,32) ( 90,32)  ( 20,10)(-2,3)13 ( 20,10)(4,5)15 ( 49,10)(-4,5)15 ( 49,10)(4,5)15 ( 78,10)(4,5)15 ( 20,10 ) ( 49,10 ) ( 78,10 )  ( 158,44)(1,0)80 ( 170,32) ( 200,32) ( 230,32) ( 162,10)(2,3)12 ( 189,10)(-4,5)15 ( 189,10)(4,5)15 ( 218,10)(4,5)15 ( 162,10 ) ( 189,10 ) ( 218,10 )  with these definitions in mind , we are now equipped to present the so - called core " cirquent calculus rules .the rules we give here are only those relevant to our language and are by no means exhaustive . in its full generality ,col encompasses numerous other logical operators all accompanied by a deep and meaningful semantics .as predicted earlier in , recent works have produced cirquent calculus axiomazations for logics containing some of the more powerful operators of computability logic .see , for example , .axioms are rules with an empty set of premises . in cirquent calcus , they come in two forms : the * _ empty cirquent axiom _ * and the * _ identity axiom_*. both varieties of the axiom rule are illustrated below .it is important to note that the _ identity axiom _ is actually a scheme of infinitely many axioms , as stands for an arbitrary formula .( 209,76 ) ( 0,63)_empty cirquent axiom _( 28,46)(1,0)36 ( 69,44)a  ( 150,63)_identity axiom _( 160,46)(1,0)36 ( 200,44)a ( 160,33) ( 188,33) ( 180,10)(-3,5)11 ( 180,10)(3,5)11 ( 180,10 )  as will be our convention throughout , the letter placed to the right of the horizontal line represents the rule by which the conclusion was obtained .   the _ mix _ rule takes two premises .its conclusion is obtained by placing each of the two premise cirquents side by side in a single cirquent , as illustrated below .( 166,102 ) ( 0,92)(1,0)26 ( 100,92)(1,0)40 ( 100,79) ( 132,79) ( 0,79) ( 17,79) ( 52,10 ) ( 52,10)(0,1)16 ( 52,10)(-1,1)16 ( 68,10 ) ( 84,10 ) ( 100,10 ) ( 68,10)(0,1)16 ( 84,10)(-1,1)16 ( 84,10)(1,1)16 ( 100,10)(0,1)16  ( 0,46)(1,0)142 ( 145,44)m ( 65,33) ( 97,33) ( 32,33) ( 49,33) ( 20,56 ) ( 20,56)(0,1)16 ( 20,56)(-1,1)16 ( 103,56 ) ( 119,56 ) ( 135,56 ) ( 103,56)(0,1)16 ( 119,56)(-1,1)16 ( 119,56)(1,1)16 ( 135,56)(0,1)16  the remaining rules described in subsections [ exch ] to [ cintro ] all take a single premise cirquent .   the _ exchange _ rule comes in two varieties , * _ oformula exchange _ * and * _ ogroup exchange_*. the conclusion of the _ oformula exchange _rule is obtained by swapping the positions of two adjacent oformulas in the premise cirquent .the _ ogroup exchange _rule is similar in that it allows two adjacent ogroups in a premise cirquent to exchange positions in the conclusion . in both varieties of the rule ,the arcs from each ogroup to its oformulas should be preserved .below is an example of _ oformula _ ( resp ._ ogroup _ ) _ exchange _ in which the oformulas f and g ( resp .ogroups and ) are swapped .( 182,124 ) ( -4,109)_oformula exchange _( 10,92)(1,0)53 ( 10,79) ( 33,79) ( 55,79) ( 14,56)(0,1)19 ( 14,56)(5,4)23 ( 14,56 ) ( 37,56 ) ( 59,56 ) ( 37,56)(0,1)18 ( 59,56)(0,1)18  ( 10,46)(1,0)53 ( 67,44)e ( 10,33) ( 33,33) ( 55,33) ( 14,10)(0,1)18 ( 14,10)(5,2)45 ( 14,10 ) ( 37,10 ) ( 59,10 ) ( 37,10)(0,1)18 ( 59,10)(0,1)18  ( 122,109)_ogroup exchange _ ( 130,92)(1,0)53 ( 130,79) ( 153,79) ( 175,79) ( 134,56)(0,1)19 ( 134,56)(5,4)23 ( 134,56 ) ( 157,56 ) ( 179,56 ) ( 157,56)(0,1)19 ( 179,56)(0,1)18  ( 130,46)(1,0)53 ( 187,44)e ( 130,33) ( 153,33) ( 175,33) ( 134,10)(5,4)23 ( 134,10 ) ( 157,10 ) ( 157,10)(-4,3)23 ( 157,10)(0,1)18 ( 179,10 ) ( 179,10)(0,1)18   the _ weakening _ rule has two forms ; * _ ogroup weakening _ * and * _ pool weakening_*. in the case of _ ogroup weakening _ , the conclusion is obtained from the premise by adding a new arc between a pre - existing ogroup and oformula pair . in a application of _pool weakening _, the conclusion is obtained by inserting a new oformula at any position in the pool of the premise .( 195,124 ) ( 0,109)_ogroup weakening _ ( 20,92)(1,0)31 ( 20,79) ( 43,79) ( 24,56)(0,1)18 ( 24,56 ) ( 47,56 ) ( 47,56)(0,1)18 ( 47,56)(-5,4)23  ( 20,46)(1,0)31 ( 55,44)w ( 20,33) ( 43,33) ( 24,10)(0,1)18 ( 24,10)(5,4)23 ( 24,10 ) ( 47,10 ) ( 47,10)(0,1)18 ( 47,10)(-5,4)23  ( 140,109)_pool weakening _ ( 142,92)(1,0)54 ( 142,79) ( 188,79) ( 169,56 ) ( 169,56)(-5,4)23 ( 169,56)(5,4)23  ( 142,46)(1,0)54 ( 198,44)w ( 142,33) ( 165,33) ( 188,33) ( 169,10)(-5,4)23 ( 169,10)(5,4)23 ( 169,10 )   the _ duplication _ rule can be applied in one of two ways , called * _ downward duplication _ * and * _ upward duplication_*. the conclusion of the _ downward duplication _rule is obtained from its premise by replacing an ogroup with two adjacent ogroups that each have arcs to exactly the same oformulas as the original ogroup .an application of _ upward duplication _works in the opposite direction in that the premise cirquent is obtained by replacing an ogroup in the conclusion by two adjacent ogroups that are both identical to the original ogroup .( 238,122 ) ( 0,107)_downward duplication _ ( 19,92)(1,0)53 ( 19,79) ( 42,79) ( 64,79) ( 23,56)(0,1)19 ( 23,56)(5,4)23 ( 23,56 ) ( 68,56 ) ( 68,56)(0,1)18  ( 19,46)(1,0)53 ( 76,44)d ( 19,33) ( 42,33) ( 64,33) ( 23,10)(0,1)18 ( 23,10)(5,4)23 ( 23,10 ) ( 46,10 ) ( 69,10 ) ( 46,10)(-5,4)22 ( 46,10)(0,1)18 ( 69,10)(0,1)18  ( 156,107)_upward duplication _ ( 169,46)(1,0)53 ( 169,33) ( 192,33) ( 214,33) ( 173,10)(0,1)19 ( 173,10)(5,4)23 ( 173,10 ) ( 218,10 ) ( 218,10)(0,1)18  ( 169,92)(1,0)53 ( 226,44)d ( 169,79) ( 192,79) ( 214,79) ( 173,56)(0,1)18 ( 173,56)(5,4)23 ( 173,56 ) ( 196,56 ) ( 219,56 ) ( 196,56)(-5,4)22 ( 196,56)(0,1)18 ( 219,56)(0,1)18   the _ contraction _ rule takes a premise cirquent which contains two adjacent and identical oformulas .the conclusion of an application of this rule is obtained from the premise by merging two adjacent and identical oformulas and into a single oformula .( 266,102 )  ( 95,92)(1,0)73 ( 95,79) ( 117,79) ( 139,79) ( 161,79) ( 120,56)(0,1)19 ( 120,56 ) ( 119,56)(-1,1)19 ( 146,56)(1,1)19 ( 145,56 ) ( 165,56 ) ( 165,56)(0,1)19 ( 100,56 ) ( 100,56)(0,1)19 ( 143,56)(0,1)19  ( 95,46)(1,0)73 ( 172,44)c ( 97,33) ( 128,33) ( 159,33) ( 121,10 ) ( 143,10 ) ( 121,10)(-1,1)19 ( 143,10)(1,1)19 ( 121,10)(3,5)11 ( 143,10)(-3,5)11 ( 102,10)(0,1)19 ( 102,10 ) ( 162,10)(0,1)19 ( 162,10 )   the conclusion of this rule is obtained from the premise by merging two adjacent oformulas and into a single oformula such that all arcs pointing to either or now point to . two examples of an application of this ruleare given below .( 200,110 )  ( 20,92)(1,0)49 ( 20,79) ( 40,79) ( 60,79) ( 23,56 ) ( 43,56 ) ( 63,56 ) ( 23,56)(0,1)19 ( 43,56)(0,1)19 ( 43,56)(-1,1)19 ( 63,56)(0,1)19  ( 20,46)(1,0)49 ( 71,44)  ( 20,33) ( 40,33) ( 23,10 ) ( 43,10 ) ( 63,10 ) ( 23,10)(0,1)19 ( 43,10)(1,2)10 ( 43,10)(-1,1)19 ( 63,10)(-1,2)10  ( 120,92)(1,0)69 ( 120,79) ( 140,79) ( 160,79) ( 180,79) ( 123,56 ) ( 143,56 ) ( 163,56 ) ( 123,56)(0,1)19 ( 143,56)(0,1)19 ( 143,56)(-1,1)19 ( 143,56)(1,1)19 ( 163,56)(0,1)19 ( 163,56)(-1,1)19 ( 163,56)(1,1)19 ( 184,56 ) ( 183,56)(0,1)19  ( 120,46)(1,0)69 ( 191,44) ( 120,33) ( 140,33) ( 180,33) ( 123,10 ) ( 143,10 ) ( 164,10 ) ( 123,10)(0,1)19 ( 143,10)(1,2)10 ( 143,10)(-1,1)19 ( 164,10)(-1,2)10 ( 164,10)(1,1)19 ( 184,10 ) ( 183,10)(0,1)19   this rule takes a premise cirquent that contains two adjacent oformulas and such that no ogroup contains both and and every ogroup that contains ( resp . ) is immediately followed ( resp . preceded ) by an ogroup containing ( resp . ) .the conclusion in an application of this rule is obtained from its premise by merging each ogroup that contains with the ogroup containing that immediately follows it .the oformulas and should then merge into .we again give two examples below .( 200,100 ) ( 20,92)(1,0)49 ( 20,79) ( 40,79) ( 60,79) ( 23,56 ) ( 23,56)(0,1)19 ( 43,56 ) ( 43,56)(0,1)18 ( 43,56)(-1,1)19 ( 64,56 ) ( 64,56)(0,1)18  ( 20,46)(1,0)49 ( 71,44)  ( 20,33) ( 40,33) ( 23,10 ) ( 23,10)(0,1)19 ( 53,10 ) ( 53,10)(0,1)18 ( 53,10)(-3,2)30  ( 120,92)(1,0)69 ( 120,79) ( 120,33) ( 120,46)(1,0)69 ( 140,79) ( 160,79) ( 180,79) ( 180,33) ( 140,33) ( 123,56 )  ( 123,56)(0,1)19 ( 143,56 ) ( 143,56)(0,1)18 ( 142,56)(-1,1)19 ( 143,56)(2,1)40 ( 166,57)(-5,2)43 ( 164,56)(1,1)19 ( 164,56 ) ( 164,56)(0,1)18 ( 183,56)(0,1)20 ( 184,56 )  ( 191,44)  ( 123,10 ) ( 123,10)(0,1)19 ( 153,10 ) ( 153,10)(0,1)18 ( 153,10)(-3,2)30 ( 153,10)(3,2)30 ( 184,10 ) ( 183,10)(0,1)20   the cirquent calculus system built from all eight of the above rules has been aptly named classical cirquent calculus " , or * ccc*. by removing the _ contraction _ rule from ccc , we get the system * cl5*. we further let * * cl5** denote the system that results from additionally removing the _ duplication _rule from cl5 .let be one of the cirquent calculus systems * ccc * , * cl5 * or * * cl5** . a * proof * of a cirquent in is a tree of cirquents whose root is where each node follows from its children by one of the rules of .when we say a formula is provable in , we mean the cirquent containing a single oformula with one ogroup and arc is provable in . in this paper , for simplicity , we are only interested in proving formulas , even though all of our results almost straightforwardly extend from formulas ( as special cases of cirquents ) to all cirquents . correspondingly , we agree that , unless suggested otherwise by the context , provability " means provability of formulas " .it was shown in that the provable formulas of * ccc * coincide exactly with those of classical propositional logic .interestingly enough , the provable formulas of * cl5 * can also be described in a very natural way . because we will rely on this result and its associated concepts in several of our later proofs, we shall give the relevant details here .  a * substitution * for a formula is a function that maps every atom in to some formula .if is an atom for every in , then we say that is an * atomic - level substitution*. this notion can be extended to all formulas by requiring that , and .let and be formulas . is said to be an * instance * of iff there exists a substitution such that .if is an atomic level substitution , then is an * atomic - level instance * of .a formula is called * binary * iff no atom has more than two occurrences in it .a binary formula is said to be * normal* iff , whenever an atom occurs twice in the formula , one occurrence is positive and the other is negative .the following theorem is a combination of theorem 12 and lemma 9 of .[ thm : cl5bin ] ( japaridze ) a formula is provable in cl5 iff it is an instance of a binary tautology iff it is an atomic - level instance of a normal binary tautology .  at this point, a semantical characterization of the provable formulas of * * cl5** has yet to be given . in light of the results of this paper , such a finding would perhaps be very interesting .in this section , we show that deciding provability for a formula of the logic * * cl5** is np - complete .this result is achieved by a combination of two theorems , the first of which states that every formula provable in * * cl5** has a proof whose size is polynomial in the size of . towards this goal , we begin by presenting a set of technical lemmas .  [lem : bintautology ] every formula provable in cl5 is an atomic - level instance of a normal binary tautology .this follows immediately from the  " direction of theorem [ thm : cl5bin ] , given that the theorems of cl5 form a subset of the theorems of cl5 .[ lem : width ] let be a formula provable in cl5 with positive occurrences of atoms . no cl5 proof of can contain a cirquent with more than ogroups .let and be as in the condition of the lemma and let be a proof tree for . assume for a contradiction that has a cirquent that contains more than ogroups .now let be the subtree of rooted at .a careful examination of the rules of cl5 will show that , no matter which series of rules are applied in , the total number of ogroups in all the leaves of must be greater than or equal to the number of ogroups in .this is because the premise cirquent(s ) in the application of a cl5 rule must , together , contain at least as many ogroups as the conclusion .every leaf in must be derived by either the _ identity axiom _ or the _ empty cirquent axiom _ and thus can not contain more than 1 ogroup .this means has at least leaves with a single ogroup , each of which must follow by the _ identity axiom_. to be a consequence of the _ identity axiom _ , a cirquent must contain exactly two oformulas and .but the total number of positive occurrences of atoms in oformulas in the leaves of can not exceed .this is because , by our assumption , the conclusion of contains a single oformula with positive literals .further , the oformulas of the premise cirquent(s ) in the application of a cl5 rule can not contain more positive occurrences of atoms than the oformulas of the conclusion .this means at most applications of the _ identity axiom _ are possible , contradiction .the * length of a formula * is defined as the total number of occurrences of literals and connectives in .the * size of a group * is the total number of oformulas it contains .the * size of a cirquent * is then the sum of the lengths of the oformulas in its pool plus the sum of the sizes of each ogroup in its structure . naturally , the * size of a proof * is the sum of the sizes of the cirquents it contains .[ lem : depth ] let be a formula provable in cl5 and let be the length of .there exists a cl5 proof of using applications of rules .let be a formula and let be a cl5 proof tree for .given , we show that a new proof can be obtained from such that proves using applications of rules .we begin by noting that any proof containing applications of the _ empty cirquent axiom _ can be transformed into one in which no applications of the rule are made .indeed , observe that the empty cirquent can only be a ( dummy " ) premise of _ mix_. in such a case , the conclusion of _ mix _ is simply the same as its other premise .therefore , the empty cirquent can be deleted and the application of _ mix _ can be skipped .this can be done for all empty cirquents contained in the proof until none remain .let be the proof that results from removing all applications of the _ empty cirquent axiom _ in the preceding manner .observe now that must contain exactly one application of _ conjunction introduction _ or _ disjunction introduction _ for each occurrence of or in , respectively .this is because every application of either rule introduces , in its conclusion , a single or connective that can not be later removed by the application of any cl5 rule .the number of positive occurrences of atoms in also bounds the number of applications of the _ identity axiom_. this is because , in a bottom up view of a proof , no cl5 rule allows formulas to be removed or merged . with the number of applications of _ conjunction introduction _ , _ disjunction introduction _ and the _ identity axiom _ all bounded by in , we can also bound the number of applications of the _ mix _ rule . to see this , first note that because contains no applications of the _ empty cirquent axiom _ , every leaf node must be derived by the _ identity axiom_. that is , there are paths from root to leaf in . viewing the proof tree in a top down fashion , _ mix _ is the only cl5 rule which allows the proof tree to branch " , given it is the only rule with more than one premise .thus , each application of _ mix _ , which must take two premises , increases the number of paths from root to leaf by 1 . because the number of such paths is bounded by ,so too must be the number of applications of the _ mix _ rule .each application of _ pool weakening _ introduces , in its conclusion , an oformula that must be present ( perhaps only as a proper subformula of some oformula ) in the conclusion of .this is because no cl5 rule can remove , in its conclusion , an oformula contained in its premise cirquent .thus we have an immediate bound of for the number of applications of _ pool weakening _ in .observe that this also implies a bound of on the number of oformulas contained in any cirquent . to bound the number of possible applications of _ ogroup weakening _ in , notice that no cirquent in can contain more than arcs .this is because , by lemma [ lem : width ] , bounds the maximum number of ogroups in any cirquent and , as noted previously , bounds the number of oformulas in any cirquent .further , _conjunction introduction _ and _ disjunction introduction _ are the only cl5 rules whose conclusion can contain fewer arcs than its premise .the number of applications of these rules in does not exceed and each application can remove no more than the maximum arcs contained in any premise cirquent .thus , no more than total arcs can be removed from premise to conclusion for all applications of _ conjunction _ and _ disjunction introduction _ in .each application of _ ogroup weakening _ creates a single arc in its conclusion that was not present in its premise . as the conclusion of contains a single arc , no more than arcs can be created from premise to conclusion in by applications of _ ogroup weakening_.  every leaf of must be derived by the _ identity axiom _ , which is applied at most times in .that is , there are at most paths from root to leaf in .each such path for contains at most applications of cl5 rules other than _ oformula _ or _ogroup exchange_. thus , any can contain at most sequences for where each ( for ) is a sequence of cirquents such that every cirquent follows from by _ exchange _ ( of either sort ) for all . in ,each can be of arbitrary length .we show , however , that a new proof can be obtained from such that each uses no more than applications of _ oformula _ and _ ogroup exchange_. this is because , the last cirquent in any can always be obtained from using no more than applications of _ oformula _ and _ ogroup exchange_. by lemma [ lem : width ] , the first cirquent can contain at most ogroups . moving all of these ogroups to an arbitrary position in can be done with applications of _ ogroup exchange_. additionally , moving each of possible oformulas in to an arbitrary position in also requires no more than applications of _ oformula exchange_. this means that every sequence in can be replaced with a new sequence such that uses no more than applications of _ oformula _ or _ ogroup exchange_. let be the cl5 proof tree that results from replacing each in by . must again contain paths from root to leaf .each such path now has a bound of applications of _ oformula _ or _ ogroup exchange _ , resulting in a total bound of applications of _ exchange _ in all branches of .it should be clear that if proves , then will also prove .the number of applications of non-_exchange _ rules does not change from to and remains .further , uses no more than applications of _exchange_. totaling the number of applications of every rule in , we obtain a bound of .[ np ] let be a formula provable in cl5 and let be the size of .there exists a cl5 proof of whose size is polynomial in .  by lemma [ lem : depth ], there exists a cl5 proof of that uses a polynomial number of rule applications .further , by lemma [ lem : width ] , the size of any cirquent in must be polynomial in .thus , the maximum size of each cirquent multiplied by the maximum number of cirquents in yields a polynomial bound on the size of .the result of theorem [ np ] effectively places the provability problem for cl5 formulas in np .we further this result by additionally showing that the problem is np - complete . before giving the proof ,we solidify some standard concepts that will be used within it .a * graph * is an ordered pair made up of a set of vertices ( ) and edges ( ) , with each edge being an unordered pair of vertices . the * degree * of a vertex in , denoted deg(v ) , is defined as the number of edges incident to . given a graph , a * vertex cover * is such that every edge of is incident to at least one vertex in .in complexity theory , the * vertex cover problem * can be stated as a decision problems as follows .given a graph and number , does have a vertex cover , i.e. a vertex cover using at most vertices ?it is well known that this problem is np - complete .deciding provability for the logic cl5 is np - complete .it follows from theorem [ np ] that cl5 provability is in np . to see it is np - hard , we give a polynomial time reduction to it from the vertex cover problem .fix some arbitrary graph and some .the reduction , borrowed from , follows .        above , is a new atom that differs from all other atoms in the formula . in , and the vertices on the endpoints of edge for .it is also important to note that each for represents both an atom in as well as the label of a vertex in .obviously the mapping is computable in polynomial time .it now remains to show that a graph has a vertex cover if and only if the formula is provable in cl5 .   " our reduction is identical to that of ( section 5.2 ) for multiplicative affine ( direct ) logic . as a direct consequence of theorem 3 from ,cl5 proves every formula provable in multiplicative affine logic , where cl5 is the system cl5 with the limitation that cirquents contained in proofs do not have groups that share oformulas . because _ upward __ downward _ ) _ duplication _ is only applicable when the premise ( resp .conclusion ) of the rule is a cirquent in which groups share oformulas , all cirquents provable in cl5 are also provable in cl5 .] , so the result follows immediately .   " assume is provable in cl5 .we need to show that there exists a vertex cover of . by theorem [ thm : cl5bin ], there exists a normal binary tautology and an atomic - level substitution such that .let be the subformula of such that .let be the set of the atoms of , and let be the set of those members of that have ( not only positive but also ) negative occurrences of . for each ,let be the conjunct in the component of that contains . by our construction, each takes the form for .define a vertex cover for as the set of all vertices in labeled where is contained in for some .  to see that is indeed a vertex cover for , note first that is an atomic - level substitution , and hence each positive literal must be mapped to a unique positive occurrence of in . by the definition of ,however , there are exactly positive occurrences of in .this means contains no more than atoms and subsequently the vertex cover derived from contains no more than vertices .  because is an atomic - level instance of , both formulas take exactly the same form , the only difference being in the atoms they are built from .let us now define a model for .for each let  .for any atom with the property that , let  .further , for each atom of where for some , let  .all remaining atoms of should be interpreted as  .note that every atom of the earlier defined is interpreted as , meaning that .next let be the subformula of such that .each disjunct of takes the form for some positive integer .if then  and  , making the entire subformula evaluate to under . if then  and  .note , however , that when then where and .this means  and  , again making the subformula evaluate to under .because each disjunct of evaluates to under , we have  .finally , let be the subformula of such that . because is a tautology where  and  , we have  . each conjunct of takes the form such that and are the endpoints of an edge in .further , by our construction , every edge in is represented by such a conjunct in .observer that , by our definition of , we have  for an atom of only when .thus , is true only when every edge in has an endpoint in .that is , is a vertex cover of .this section contains our main result , namely , the -completeness of the decision problem for provability of formulas in * cl5*.   the following is a algorithm that , if view of theorem [ thm : cl5bin ] , decides provability of a formula in cl5 . on input , existentially guess a binary formula such that is an instance of .then , universally guess a truth assignment for .if is true under , accept .otherwise , reject .let tqbf- be the problem of deciding truth for a quantified boolean formula of the form , where and are sequences of variables and is a quantifier - free boolean formula all of whose variables are among or .as shown in theorem 4.1 of , this problem is -complete .we say an atom is * isolated * in a formula if there is only a single ( positive or negative ) occurrence of in .otherwise it is * non - isolated*.   by lemma [ lem : ptime_algorithm ] , the problem of deciding provability for cl5 is in . to show it is -hard, we construct a polynomial time mapping reduction from tqbf- to cl5-provability .we can safely restrict our attention to instances of tqbf- where the boolean portion of the formula is in disjunctive normal form , as the complexity of the problem is not reduced with these limitations .our reduction follows .let be a formula that takes the form with and being sets of variables and being a boolean formula in disjunctive normal form all of whose variables are among .the following steps are used to construct the corresponding cl5 formula .  1 .  for each , let and be the number of positive occurrences of the literals and in , respectively. define where , and are all fresh . ]pairwise distinct variables , uniquely chosen for . here , if ( resp . ) is 0 , the second ( resp .third ) conjunct should be omitted .now let and let be the formula . here is a formula derived from such that , for each , every positive occurrence of in is replaced by a unique literal from and every occurrence of in is replaced by a unique literal from .2 .  consider any .let be the number of positive occurrences of in , and be the number of negative occurrences .for each pair with and , we choose a fresh and unique variable .now , define to be the result of replacing in , for each , every positive occurrence of the literal by and every ( positive ) occurrence of the literal by where ( resp . ) is unique for each replacement of an occurrence of ( resp . ) .   " assume is true. then there exists some truth assignment such that is true under any truth assignment that extends to . by our assumption, is in disjunctive normal form and hence takes the form where each is either a positive or negative literal . now let and represent the antecedent and consequent of the outermost implication in such that .our construction guarantees that takes the form where each is a disjunction of literals ( such a disjunction " may have only a single disjunct " ) .that is , is obtained from by replacing each oliteral by , where is a disjunction of literals .we will henceforth use to represent a unique position in and to represent the corresponding position in .we want to show that is provable in cl5 . by theorem [ thm : cl5bin ], it suffices to show that is an instance of a binary tautology .we construct a binary tautology of which is an instance .namely , we let be the formula obtained from as follows . for each , if ( resp . ) replace the third ( resp .second ) occurrence of by an atom such that does not occur elsewhere in and is unique for each .it should be easy to see that is a quantifier free binary formula and is an instance of .we need only show that is a tautology . againnotice that takes form where is the same as in .    for any in where , if  then  , and hence .if  , then all of the literals in the consequent of the implication in with antecedent literal must be true under , otherwise we will have  and again  .thus , we need only guarantee under truth assignmentssuch that , for each conjunct in , and every literal in the consequent of the implication containing antecedent literal are true under .let be the formula that results from replacing in every positive occurrence of or , where , by its truth value under .for example , if  , replace by and by .it should be easy to see that is a tautology . for any position that contains ( resp . ) in and in , the position in must contain a single literal ( resp . ) such that ( resp . ) if .this is because , by our construction , if  ( resp .  ) , every for ( resp . for ) must occur in the consequent of the implication in with non - isolated antecedent . for reasons already discussed ,such a literal must be true under any truth assignment where .define as the formula such that , for each occurrence of or in position of , the literal in position of is replaced by the same value ( or ) as .note that only substitutes the logical atom in position of when contains a positive or negative occurrence of an atom in . since our goal is to show that is true under truth assignments that make true , we need only show that is a tautology .  for a contradiction ,assume is not a tautology. then there exists some truth assignment defined on the variables of such that  .we define a truth assignment for as follows . if a variable is such that , for some with , , we let ; otherwise we let .it is not hard to see that , if a subformula of in a position is false under , then the subformula of in the corresponding position is false under ( but not necessarily vice versa ) .this , in view of the monotonicity of and , obviously implies that , because and have the same -structures .now we are dealing with a contradiction , because the tautological can not be false .   " assume is provable in cl5 . by theorem [ thm :cl5bin ] , for some normal binary tautology there exists an atomic level substitution such that  .let , where and represent the antecedent and consequent of the outermost implication in .we want to define a truth assignment such that is true for any truth assignment that extends to . in what follows ,we define such a partial truth assignment for while concurrently defining a partial truth assignment for ._ procedure 1 _ - consider a and let be the atom of such that the first occurrence of in * originates * from a ( i.e. , a gets replaced by ) when transitioning from to ._ case 1 _ : the second and third occurrences of in originate from and , respectively ( for some in ) .define , and .the consequent of the second occurrence of in should take the form for some positive literals in andthe consequent of should take the form for some negative literals .let all of and be true under ._ case 2 _ : the second and third occurrences of in originate from and , respectively ( for some in ) .define , and .the consequent of the second occurrence of in should take the form for some negative literals and the consequent of should take the form for some positive literals .let all of and be false under ._ case 3 _ : there are only two occurrences of in , both of which originate from .if the consequent of the second occurrence of in takes the form for some positive literals then define and .further , let all of be true under .if the consequent of the second occurrence of in takes the form for some negative literals then define and .further , let all of be false under .  _ case 4 _ : if none of the cases are satisfied , then the second and third occurrences of in must originate from and , respectively ( where and ) .define  , and  .further , let every atom in the consequent of evaluate to under and every atom in the consequent of evaluate to under .as we remember , in disjunctive normal form , taking the form where each is a literal .it is also the case that takes the form where each is a disjunction of literals ( possibly with just a single disjunct " ) .thus , we will use our previous convention in which represents a unique position in and represents the corresponding position in .our construction guarantees that for each position in containing some positive ( resp .negative ) literal ( resp . ) such that , contains a positive ( resp .negative ) literal ( resp . ) . by _ procedure 1_ we have . as in the previous direction ,let be the formula that results from replacing , in , for every , all positive occurrences of the literals and by their truth values under .let be the formula such that , for each position in containing a logical atom , the literal in the corresponding position of is replaced by .the interpretation defined as part of _ procedure 1 _ is such that .because is a tautology , any extension of defined on all atoms of must make true .we also know that for each position in that contains a logical atom ( resp . ) , the literal in position of evaluates to ( resp . ) under .this means that is true regardless of how its non - logical atoms are interpreted and is a tautology . our goal is to show that is a tautology as well .pick some arbitrary truth assignment for .we show . for each position in containing a positive literal , the corresponding position in contains a disjunction of positive literals ( ) such that ,  , for some .similarly , for each position in containing a negative literal , the corresponding position in contains a disjunction of negative literals ( ) such that , , , for some .we now define a truth assignment for . if , let be such that all of the corresponding literals , for every evaluate to false under .if , let be such that all corresponding literals , for every evaluate to false under . with some thought , one can see that whenever has a false ( under ) literal in a position , has a false ( under ) disjunction of literals in the corresponding position .so , if , then .that is , ( which is the case due to the tautologicity of ) implies .

Abstract: introduced in 2006 by japaridze , cirquent calculus is a refinement of sequent calculus . the advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources . unlike the more traditional proof - theoretic approaches that manipulate tree - like objects ( formulas , sequents , etc . ) , cirquent calculus is based on circuit - style structures called _ cirquents _ , in which different `` peer '' ( sibling , cousin , etc . ) substructures may share components . it is this resource sharing mechanism to which cirquent calculus owes its novelty ( and its virtues ) . from its inception , cirquent calculus has been paired with an abstract resource semantics . this semantics allows for reasoning about the interaction between a resource provider and a resource user , where resources are understood in the their most general and intuitive sense . interpreting resources in a more restricted computational sense has made cirquent calculus instrumental in axiomatizing various fundamental fragments of _ computability logic _ , a formal theory of ( interactive ) computability . the so - called classical " rules of cirquent calculus , in the absence of the particularly troublesome _ contraction _ rule , produce a sound and complete system cl5 for computability logic . in this paper , we investigate the computational complexity of cl5 , showing it is -complete . we also show that cl5 without the _ duplication _ rule has polynomial size proofs and is np - complete .


oil from spills , red tides , flotsam and other suspended and surface tracers approach the nearshore , carried by winds and currents .it is not uncommon , however , that these debris and tracers slow down and park themselves , somewhere beyond the break zone ( see figure [ red ] ) ; eventually , a portion of these reach the beach zone by the action of turbulence and tidal effects , in combination with inertial effects on the debris .  the tendency of tracers to park themselves in certain areas of the great barrier reef has been noted . , who reported the phenomenon , and denoted it as sticky waters . "though we will not be discussing estuarine environments and the mechanism at play in the great barrier reef situation may be different from the nearshore case , we will borrow this terminology and refer to the phenomenon we investigate in this paper as nearshore sticky waters . "  of obvious environmental , economic , and social importance ,understanding why nearshore sticky waters occur is also fundamental to improved environmental assessments of coastal settings .moreover , as part of a larger research agenda aimed at improving models for pollutant transport in ocean general circulation models , nearshore sticky waters offers a field - verifiable problem with which to test contaminant advection reaction and dispersion models .the focus is on tracer transport phenomena , with length scales several times larger than the depth and temporal scales of hours , weeks .that is , we are mostly concerned with large - scale pollution disasters , " such as large - scale red tides , significant oil spills , etc .although we consider long time and space scales , we can not ignore depth dependent features of the flow and the transport of tracers .the tracer may be buoyant but not necessarily entirely residing on the surface of the ocean ; we therefore consider a layered ( instead of simply depth averaged ) model for the tracer and account for the vertical structure of the advective velocity .we defer consideration of tracers with non - trivial inertial effects to a separate study . obviously , tracers advect and diffuse in the alongshore direction as well as in the cross - shore direction .in fact , advection / diffusion in the longshore direction is usually more intense in many non - estuarine environments .however , if we consider a situation where the longshore variations of the tracer concentrations are small , the divergence of the flux in the longshore direction is negligible , and it is appropriate to consider a one - dimensional problem in the cross - shore direction .nearshore sticky waters will refer to the slowing down or the parking of the tracer approaching the shore . in a sticky water situationthe center of mass of the incoming tracer that is approaching the shore at advective speeds will experience a partial or total slowing down . whatever tracer amounts reach the shore will do so by the action of dispersive effects , usually higher inside the breakzone than in deeper waters . in the cross - shore direction ,large scale currents are typically weak , close to the shore . in a wave - dominated nearshoresetting , the typical advection velocity would be the residual flow due to the waves , the stokes drift velocity .the length scales are those of the long waves , _i.e. _ , waves which have wavelengths that are large when compared to the depth .the diffusive length scale is typified by large - scale eddies ; if the break zone is a significant source of mixing , the length scale would be the distance between the start of the breaking of the waves and the shore .when advective and diffusive effects are those in balance , the diffusive time scale is large , in the order of hours .there is consensus that in wave - dominated beaches the dissipation of waves is different inside and outside of the breakzone , the latter being considerably smaller than the former . , chapter 10 , describes the theoretical development of a model for wave action dissipation , based upon dimensional analysis and homogeneous turbulence concepts ( see also , for further developments ) .the model used in is that of , which is one of several based upon hydraulic jump parametrizations .analysis of field data of the dispersion of tracers in the nearshore suggest that the diffusivity is much higher in the break zone than outside .dispersion estimates based upon the dimensional analysis model of are off by orders of magnitude , when compared to field data ( see ) .a possible explanation for the discrepancy might lie in the fact that the dimensional parametrization is based upon homogeneous turbulence conditions and is more typical of the smaller scale vertical diffusion , rather than the larger eddy - scale transverse diffusion ( feddersen , private communication ) .the basic depth - averaged hydrodynamics , appropriate to these scales , are captured by the similarly scaled vortex force model in ( see for background and for a comparison of this `` vortex force '' model and the `` radiation stress '' alternative . see also ) . in the following form, the model has been used to study nearshore problems , such as longshore currents ( in ) , and rip currents ( in ) .the depth - averaged momentum balance reads : ^\bot  + { \bf n } , \label{mom}\ ] ] where is the vorticity of the depth - averaged velocity , and encompasses bottom drag , wind forcing , and dissipation ; it also encompasses momentum transfers from wave breaking to the current momentum ( see ) .the vortex force is the second term on the right hand side , which couples the residual flow due to the waves to the rotation in the current .the depth - averaged stokes drift velocity is denoted by ; the operator is used to obtain ^\bot = ( -v^{st},u^{st}) ] .this crude model is similar to the one proposed in , which is inspired by field measurements . in the examples that follow m and is the width of the transition of the sigmoid ./s is the background eddy diffusivity , and m/s the enhanced diffusivity in the nearshore due to turbulence and wave breaking ( figure [ velob ] ) .fick s law gives a diffusive flux on the surface . in the subsurface region ,the diffusive flux is driven by gradients of the bulk concentration but its effect on is found via the equation of state , ( [ eqstate ] ) .fick s law along with integration in the depth of the mixed layer gives a diffusive flux in the bulk .the last process we model is the exchange of material between the surface and the bulk due to wave - mixing and buoyancy . on long time scales corresponding to averaging over many waves , a simple model for the net flux from the slick into the suspension is the linear expression , where is a parameter which sets the relative proportions of the oil in the slick and in suspension for vertical equilibrium , _ viz ._ ,  is the time scale for the vertical mixing .it is a measure of the total kinetic energy ( tke ) in the fluctuating velocity field .consequently dimensional analysis suggests that , the diffusion time scale over the typical depth of the mixed layer .( see , and references contained therein , for an alternative formulation of this exchange flux and for more details on the mixing / buoyancy physics ).  neglecting sources and sinks of oil , the conservation laws for and , obtained from the above fluxes are : } { \partial x } & = -\frac{(1-\gamma ) s - \gamma p b}{\tau(x ) } + \frac{\partial}{\partial x}\left[d(x ) \frac{\partial s}{\partial x } \right ] , \label{oilsl } \\\frac{\partial b}{\partial t } +  \frac{\partial[u_b(x )  b]}{\partial x } & =  \frac{(1-\gamma ) s - \gamma p b}{\tau(x ) }  \\ & + \frac{\partial}{\partial x } \left [ \xi(x )  d(x ) \frac{\partial}{\partial x }  b\right ] , \nonumber \\\implies \frac{\partial b}{\partial t } +  \frac{\partial[v(x )  b]}{\partial x } & = \frac{(1-\gamma ) s - \gamma p b}{\tau(x ) }  + \frac{\partial}{\partial x } \left [ d(x ) \frac{\partial b}{\partial x } \right ] , \quad \label{altoilb}\end{aligned}\ ] ] where we have used and ( see figure [ veloc ] ) . note that the `` effective '' velocity of the oil in suspension is not the depth - averaged velocity of the flow ; rather it has a contribution that depends on the eddy diffusivity and also the bottom topography if , the maximum depth of the mixed layer .  to conserve total mass we specify zero flux conditions at and : eqs ., and , together with specified initial conditions and give a complete mathematical model which can be solved numerically .we denote this model as the _ pde model_.by specifying appropriate functional forms for the diffusivity and the vertical mixing time in the pde model eqs . , and , we can describe a variety of scenarios .conversely , a natural question is the extent to which simple choices for the functions , and , as motivated in the previous section , yield results that are valid for real physical flows . in this sectionwe demonstrate that the results we obtain from the pde model are robust , and depend only on gross features of and , but not on fine details of these functions .  to this endwe first derive a reduced finite dimensional model that describes the evolution of gaussian pulse initial conditions .the ratio of the vertical and horizontal time scales is given by for to 6 m , even in the deep ocean where is small .thus , the vertical tracer distribution equilibrates very rapidly .we can determine and in terms of the total oil by setting the exchange flux to zero yielding substituting in the equations for and , we obtain , \nonumber \\u_{e}(x ) & = \frac{\gamma p u_{s } + ( 1-\gamma ) \xi v(x)}{\gamma p + ( 1-\gamma ) \xi}. \label{totaloil}\end{aligned}\ ] ] observe that these equations no longer depend on , showing that as long as the mixing in the vertical direction is rapid on the scale of the horizontal transport , the precise choice of the mixing time is irrelevant.   setting } ] m. since , and are never zero ( _ cf ._ , figure [ veloc ] ) . the steady state solutions for and have their maxima at , andas one would expect the maxima for both and jump instantaneously from the interior to the boundary .the center of mass of the oil slick , should approach the center of mass of the steady distribution , and thus stays away from the shore .we also plot the ode model trajectories of the maximum and the center of mass .the agreement between the pde and ode trajectories is excellent .* case ii : * .as is decreased gets closer to and , which are smaller in magnitude than .this causes a slowdown of the center of mass of compared to the ballistic trajectory .( see figure [ evogamma - a ] . ) the oil , which started in the slick , transitions to the interior and the bulk of the oil in short order .we note the and for this case and the previous case are strictly negative , because ( see figure [ veloc ] ) , so the maximum for and in the steady state is at ( see figure [ pl2-a ] . )thus we expect the maxima to jump to zero , and this is borne out by simulations from the full model and also the reduced model , as depicted in figure [ evogamma - a ] .* cases iii and iv : * and respectively . in these cases , at m ( inside the break zone ) and m ( outside the break zone ) , respectively .the effective velocity is positive on the shallow side of , as shown in figure [ veloc ] and the effective velocity changes sign .the maximum for the steady states of are now in the interior of the domain , while the maximum for the steady states of remain at ( see figures [ pl2-b]- ) .we thus predict that the maxima for will jump instantaneously , but the maxima for will remain away from the shore .this is indeed the case , for both the full and the reduced models as shown in figures [ evogamma - b]- .  since , the initial pulse of oil in the slick transitions quickly to the interior , and remains there .also , the maximum of stays away from the shore , so this is an example of _ nearshore sticky waters_. the space - time evolution of is shown in figure [ pl1 ] , showing the slowing and parking of the tracers in the bulk away from the shore .we wish to draw attention to the agreement between the pde and the ode models in figures [ evogamma - b]- , even in situations where the ballistic and actual trajectories are very different .the model we propose describes , in a simple way , how buoyant contaminants that are impervious to inertial effects , may slow down as they approach nearshore environments .evidence of unusually long residence times for tracers , measured in breakzone field experiments , is reported from time to time ( see for example , ) .the mechanism depends on vertical variations of the tracer density and the cross - shore component of the velocity , and thus on buoyant stratification and topographic features , rather than on blocking or stationary structures in the lagrangian trajectories induced by the velocity field ( see , for details on lagrangian coherent structures ) .it is also different from the surface - wave deceleration mechanism outside of the surf zone , that is explored in .  the phenomenon and the dynamics detailed here should have relevance to the long - term transport of other contaminants and biological material in which inertial effects are negligible . in environmental mitigation problems the timing of landfall of pollutants and the manner in which these make it to land are crucial .we hope this simple model motivates efforts to improve forecasting circulation models for nearshore flows and transport , with better the modeling of breakzone boundary conditions , and nearshore subscale transport of pollutants .the proposed model ( section [ sec : pde ] ) is a set of coupled pdes and , despite its relative simplicity , it has multiple , potentially relevant parameters ( see table 1 ) . in order to gain further insight into the dynamics of the pde model, we derived a finite dimensional reduction in section [ sec : ode ] .we gain significant insight from studying the reduced model , namely we can explain the evolution of the vertical stratification of the oil density , the evolution of the oil density from symmetric to shoreward skewed distributions , the eventual fate of the maximum and the center of mass of the oil distribution , both in the surface and in the interior including an explanation of stickiness phenomenon , and a prediction for the time and space scales for the complex dynamics of the oil distribution .most importantly , the reduced model demonstrates that nearshore stickiness is a robust phenomenon , and helps identify the key parameters ( or combinations thereof ) which govern the gross behavior of the system .the determining factors are the two dimensionless quantities these quantities indicate the outcomes would change if the topographical parameters or the magnitude of the , , , and/or are changed .a general conclusion from our analysis is that increasing and/or increasing ( equivalently decreasing ) leads to more stickiness , _i.e _ , the oil approaches the shore more slowly and stalls near the break zone for much longer .the parameter depends on the topography of the nearshore , the sea conditions and the nature of the pollutant , as all of these influence .locations where the topography drops steeply even near the shore , such as in the southern california bight , will have small and are thus less sticky . "conversely a typical situation in the gulf of mexico or on the mid and southern portions of the atlantic us coast would have large when compared to , even at considerable distances from the shore , i.e. large , so the nearshore is considerably more sticky . "the parameter depends on the enhanced turbulent diffusivity in the breakzone as well as the parameter which governs the fraction of the oil which is in the slick .increasing the turbulent eddy diffusivity or decreasing the fraction of the total oil in the slick both lead to increased stickiness .the models we have developed have the virtue of simplicity and being analytically tractable , but they lack many of the ingredients of more realistic physical models . in particular we do not include inertial effects , and the transient setup and setdown process ( see ) .one expects the mixed layer to have a diffuse boundary unlike our simple model with a sharply discontinuous density at a constant layer depth .the mixed layer can depend on the sea state and the oil or pollutant constituents .for instance , the mixed layer depth and the concentration of the oil in suspension will tend to increase with langmuir turbulence and wave breaking activity .we use a kinematically specified flow with a parabolic profile , but a more realistic model should include a dynamical model for the ocean with a realistic vertical mean lagrangian velocity structure , at least close to the shore , as well as a careful treatment of the form - stress for the coupling of wind effects , which are recognized as very important in the case of surface oil slick dynamics ( see ) . finally , we have a simplified description of buoyancy and vertical mixing effects .these are important issues which we consider in a sequel to this paper .while these effects are relevant , we believe they will not change the basic conclusions we have drawn from the simplified model , and it gives a robust description of the mechanisms which can lead to the the observed apparent stalling of incoming pollutants approaching a shore .we received funding from gomri / bp and from nsf dms grant 1109856 .jmr wishes to thank the statistics and applied mathematical sciences institute , an nsf funded institute , in which some of this research was done .jmr also thanks the j. tinsley oden fellowship program at the university of texas for its support .scv was also supported in part by the nsf dms grant 0807501 .j. c. mcwilliams is acknowledged for bringing to our attention the problem of sticky waters in the great barrier reef , which lead us to ask whether sticky waters occur in the nearshore setting .we are also very grateful to prof .f. feddersen for sharing with us his expertise on dispersive processes in the surfzone .his suggestions considerably improved this paper .grant , s. b. , kim , j. h. , jones , b. h. , jenkins , s. a. , wasyl , j. , cudaback , c. , 2005 .surf zone entrainment , along - shore transport , and human health implications of pollution from tidal outlets .journal of geophysical research 110 .ohlmann , j. c. , fewings , m. r. , melton , c. , 2012 .lagrangian observations of inner - shelf motions in southern california : can surface waves decelerate shoreward - moving drifters just outside the surf zone ?journal of physical oceanography 42 , 13131326 .reniers , a. j. h. m. , macmahan , j. h. , thornton , e. b. , stanton , t. p. , henriquez , m. , brown , j. w. , brown , j. a. , gallagher , e. , 2009 .surf zone surface retention on a rip - channeled beach .journal of geophysical research 114 .rippy , m. , franks , p. , feddersen , f. , guza , r. , moore , d. , 2013 .physical dynamics controlling variability in nearshore fecal pollution : fecal indicator bacteria as passive particles .marine pollution bulletin 66 , 151157 .weir , b. , uchiyama , y. , lane , e. , restrepo , j. m. , mcwilliams , j. c. , 2011 . a vortex force analysis of the interaction of rip currents and surface gravity waves . journal of geophysical research 116 , c050001 .

Abstract: wind- and current - driven flotsam , oil spills , pollutants , and nutrients , approaching the nearshore will frequently appear to slow down / park just beyond the break zone , where waves break . moreover , the portion of these tracers that beach will do so only after a long time . explaining why these tracers park and at what rate they reach the shore has important implications on a variety of different nearshore environmental issues , including the determination of what subscale processes are essential in computer models for the simulation of pollutant transport in the nearshore . using a simple model we provide an explanation for the underlying mechanism responsible for the parking of tracers , not subject to inertial effects , the role played by the bottom topography , and the non - uniform dispersion which leads , in some circumstances , to the eventual landing of all or a portion of the tracers . we refer to the parking phenomenon in this environment as nearshore sticky waters .  oil slick , pollutant transport , shallow water flows , mixing and dispersion , , waves and currents


multi - state attractor neural networks in which the units ( neurons ) can be in more than two states are , in general , more flexible and efficient biological or artificial devices than networks of binary units .much work has been done over some time on the retrieval problem in multi - state networks of various architectures , with either simple or hierarchical patterns in more than two states .the retrieval problem consists in the recognition of patterns that have been stored in a network by means of a learning ( or training ) rule , when the network is set in an appropriate initial state to start its operating stage .thus , the retrieval problem deals with the memorization ability of a network .the networks that have been considered are the dilute , the layered feed - forward and the fully connected networks .more recently , some work has been done on the categorization problem in multi - state attractor networks , following extensive studies of the problem in binary networks .the categorization problem consists in the spontaneous recognition of a level of hierarchical patterns other than those stored in the training process of a network .the problem deals with the ability to create a representation for concepts when the network is only exposed to examples in the training stage .some of the questions that one may ask are the following .first , one is interested in the minimal structure of the training patterns , and their number , in order to achieve a satisfactory recognition of a macroscopic number of hierarchically related ancestors .second , one would like to know the recognition rate ( number of patterns per neuron ) of these ancestors and how stable they are as attractors of the network dynamics .the recognition quality is of primary interest and one may also want to check on the robustness of the recognition process to various kinds of noise .  the simplest , and most studied case of the categorization problem , consists in the recognition of ancestors of a two - level hierarchy of ancestors and descendents trained only with the latter according to a specific learning rule .the hierarchical patterns that are generated trough a stochastic procedure lead to correlations between patterns in different levels as well as correlations between patterns in the same level . as a consequence, there is a complex structure of attractors in a network with hierarchical patterns in which the attractors may neither coincide with the training patterns nor with the ancestors , and it is of interest to study under what conditions the latter become stable attractors .patterns in more than two states , which represent a gradual coding , may have a low activity which is biologically appealing .moreover , `` small '' patterns , in which a number of bits have been turned off , are patterns of low activity that can infer patterns of full size and thereby enhance the performance of a multi - state network , as demonstrated explicitly in works on both the retrieval problem and the categorization problem .the dynamics of the latter has been studied in an extremely dilute asymmetric three - state network with a monotonic neuron firing function and a generalized hebbian learning rule .the extremely dilute network requires a vanishingly small connectivity between neurons in order to allow for an exact solution of the network dynamics , and one may ask what the behavior would be for a network with full connectivity .the purpose of the present paper is to answer some of the questions raised above investigating the equilibrium , statistical mechanics behavior for the categorization problem in a fully connected multi - state network with hierarchical patterns of low activity in a two - level hierarchy of ancestors and descendents .our aim is to obtain the phase diagrams that describe the various regimes of performance of the network in terms of the relevant parameters : the activity of the training patterns , the dynamical activity of the firing units , the correlation between ancestors and descendents , the number of descendents , the multi - state threshold and the synaptic noise level , assuming a fixed activity of the ancestors .the quality of the performance of the network is described by so - called categorization curves that express the dependence of the categorization error on some of the parameters of the model .since it is known from the results on the retrieval problem in a -state network that the relevant phase diagrams become increasingly complex as one goes from the three to the four - state model , we consider a -state model and a -state ( graded response ) model .we make use of a generalized hebbian learning rule that has been used before .the outline of the paper is the following . in sectionii we present the general ising - state model for the categorization problem .the mean - field theory for that model is summarized in section iii . in sectioniv we present and discuss the results for , in the absence or presence of synaptic noise and in section v we present the results for the -state model .we conclude in section vi with a summary of the results .consider a network of nodes , . at the time step , the state of the node is described by the variable , that can be in any one of the ising states in the interval [ -1,1 ] , for .the task to be performed by the network is the recognition of a macroscopic set of concepts , with , where is finite . during the learning stage ,only a set of `` small '' examples of each concept are presented to the network . by `` small '' examples we mean that a macroscopic number of bits in each example are turned off .the concepts are assumed to be independent identically distributed random variables with zero mean and variance .the examples of the concept are generated through a stochastic process based on an appropriate probability distribution , given below , such that the properties of the distribution will be chosen in accordance with the states of the neurons , eq .( [ 1 ] ) . for finite ,say assumes the values , or depending , respectively , on the example being either in agreement with the concept , being turned off , or opposite to the concept at the site . in the case of continuous neurons , i. e. , , we assume that is a continuous variable in the interval $ ] . in either case , we assume that belongs to a set of independent random microscopic activities with mean and variance \delta_{ij}\delta_{\mu\nu } \label{3b}\ ] ] with .the symbol represents the kronecker delta . in consequence, we have the following relations , and \delta_{ij}\delta_{\mu\nu}\;. \label{5}\ ] ] the mean activity of the examples becomes for every and . according to eqs .( [ 2 ] ) and ( [ 3a ] ) , is the correlation between an example and the concept to which it belongs .the pure multi - state model can be obtained by taking the number of examples , the activity and the correlation . since ,the activity of the examples is not greater than the activity of the concepts . in this sense, we refer to `` small '' examples , with the effective `` size '' of the patterns being . in this model , the view point is that the small examples are samples of the full - activity concepts to be inferred .  in this workwe are interested in the capacity of the network to infer only large concepts of full activity from the set of examples and restrict ourselves , therefore , to binary concepts , with equal probability , that is to say , to the case .this task is considered to be successful if the _ categorization _ overlap between the concept and the network state approaches unity after the network has reached the equilibrium state . to quantify the performance of the network, we define the categorization error for the concept as thus , should be small in the categorization phase and in the disordered phase .next we pass to the discussion the dynamics of the model , following the steps of ref . and references therein . for a given configuration of the network , the local field on site is where the synapses are constructed from the examples , according to the modified hebb rule the state of each site is updated asynchronously according to a glauber ( single spin - flip ) dynamics in which the transition probabilities are given by }   { \sum_{l=1}^q\exp[\beta\epsilon_j(\sigma_l|h_j(\{s_i(t)\}))]}\ ; , \label{9}\ ] ] where is the inverse temperature and the single site energy , , is given by here , is a non - negative constant that favors local states of small dynamical activity . in the absence of stochastic noise , the deterministic evolution of the systemis ruled by where is the non - decreasing step function , for finite , \label{11}\ ] ] with and , in which , if and otherwise .the spin on site assumes the state given by eq .( [ 1 ] ) if the local field is bound by .the width of the intermediate states with constant for ( that is , excluding the limiting values of ) , is given by .thus , the width of the zero state for the three - state network studied below is . in the limit , the input - output function , eq .( [ 11 ] ) , becomes the piecewise linear function where means the minimum between and .the slope of the linear part in here is , which is the gain parameter of the continuous network .the equilibrium thermodynamic properties of the fully connected infinite network that follows from the above dynamics is described by the hamiltonian where the first sum is over all distinct pairs .the relevant order parameters , when the network is in the ordered sub - space of the phase space , are the _ retrieval _ overlaps between the actual state of the network and each one of the examples of each concept .the underlying idea in studying the categorization performance of the network is that when the number of correlated examples is higher than a critical value , for a given correlation strength , single examples are no longer local minima of the free - energy , but a mixed state having macroscopic symmetric overlap , for , with all the examples of a given concept , becomes a minimum .this state characterizes the categorization phase , and it yields a finite , macroscopic , overlap with concept . since we are interested mainly in the categorization ability of the network , we restrict ourselves , in what follows , to the study of configurations that have a macroscopic overlap of order with a mixture of a finite number of examples of a given concept . noting that the concepts are uncorrelated , one may concentrate on the overlap with anyone of them , say for .the free - energy per site follows as with the averages over examples and concepts in that order , as indicated , where is the canonical partition function in order to average over the quenched disorder , we employ the replica method , in which using the generalized hebb learning rule , eq .( [ 8 ] ) , and introducing a field , in order to generate an equation for the overlap , the hamiltonian eq .( [ 12 ] ) , for the replica , becomes introducing this expression in eq .( [ 132 ] ) , separating the first concept , we linearize the quadratic terms and obtain the replicated partition function \nonumber\\   & & \quad\times   \sum_{\{s_i^a\}}\left\langle\left\langle\exp(\beta pgn )   \right\rangle_{\{\lambda_i^{\mu\rho}\}}\right\rangle_{\{\xi_i^{\mu}\ } }   \left\langle\left\langle\exp\left\{\beta\sum_{ia}\left[\sum_{\rho }   m_{1\rho}^a\lambda_i^{1\rho}\xi_i^1s_i^a   \right.\right.\right.\right.\nonumber\\   & & \quad-\left.\left.\left.\left .\frac{1}{2n}\sum_{\rho}(\lambda_i^{1\rho}\xi_i^1   s_i^a)^2-\theta(s_i^a)^2+h_1\xi_i^1s_i^a\right]\right\ }     \right\rangle_{\{\lambda_i^{1\rho}\}}\right\rangle_{\{\xi_i^1\}}\ , , \label{135}\end{aligned}\ ] ] where involves the uncondensed examples .  in the thermodynamic limit , we obtain following ref . , where here , is a matrix in the space of replicas with elements given by and thus , is the spin - glass order parameter and is the dynamical activity of the network .whereas for the binary network in the replica - symmetric theory , in the case of multi - state networks one has , in general , that .introducing , as usual , the overlap parameter associated to the correlation between the overlaps of the examples and concepts that do not condense , and restricting our study to the replica - symmetric solution , in which we obtain that the replica - symmetric free - energy per site can be re - written as      \nonumber\\   & + & \frac{\alpha}{2}\left[\frac{\gamma_1 ^ 2qc}{(1-\gamma_1c)^2 }   + ( s-1)\frac{\gamma_2 ^ 2qc}{(1-\gamma_2c)^2}\right ]   -\frac{1}{\beta}\left\langle\left\langle\int{\cal d}z\ ,   \ln\sum_{\{s\}}{\rm e}^{\beta{\cal h}_{eff}}\right\rangle   _ { \{\lambda^{1\rho}\}}\right\rangle_{\{\xi^1\}}\ ; , \label{14}\end{aligned}\ ] ] where is a gaussian measure . the effective hamiltonian , , is given by where is an effective width of the intermediate states , as wil be seen below ( see eq .( [ 32 ] ) ) .eventually , depending on the state of the network specified by the dynamical activity and the spin - glass order parameter , may become negative , favoring an order with large absolute values for .although eqs .( [ 14])-([16 ] ) follow from the assumption of replica symmetry , we believe that such an order will exist , in general , albeit in a small region of the phase space . here , represents the susceptibility of the network .the parameter is given by the algebraic saddle - point equation the remaining saddle - point equations determining the order parameters are  and in the above equations , the susceptibility remains finite , so that at zero temperature , that is , , while at finite temperature we have in general .the overlap with the first concept , which measures the categorization ability , is given by performing the configurational average in the saddle - point equations , we obtain for the symmetric overlap , and as well as the overlap with the concept , the effective transfer function is given by in the case of the three - state network , and    -\exp\left[-\phi^2_-(h_s,\theta')\right ] }   { { \rm erf}\left[-\phi_+(h_s,\theta')\right ]   -{\rm erf}\left[-\phi_-(h_s,\theta')\right]}\ , , \label{29a}\ ] ] where for .thus , is the effective gain parameter for the continuous network .the effective field for the symmetric solution , , is given by where the first term in eq .( [ 30 ] ) is a signal term , while the second term is the gaussian noise due to the macroscopic number of uncondensed examples and the presence of the symmetric mixture states .the latter , in which ( cf .( [ 15 ] ) ) , is reduced in the case of examples of low activity , .one should expect , thus , an enhancement of the categorization ability of the network in that case .the above equations are obtained under the assumption that the number of examples is large , so that the average over examples is given by a gaussian distribution . in the following sections we discuss the results based on the solutions of the saddle - point equations for both , the three - state and the continuous network .the limit of stability of the replica - symmetric solution comes from the study of quadratic fluctuations of the free - energy in the vicinity of the symmetric saddle - point . following the almeida and thouless ( at ) analysis , we obtain ^ 2   \right\rangle_{\{\lambda^{1\rho}\}}\right\rangle_{\{\xi^1\ } }   \leq 1\ , .\label{22}\ ] ] as the stability condition for the replica - symmetric solution .we begin by discussing the results for the categorization performance in three - state networks in the absence of retrieval noise .the probability distribution in this case is given by satisfying the conditions ( [ 3a ] ) and ( [ 3b ] ) .thus , the example has a probability to be aligned with the concept , while it has a probability to be turned off and a probability to be opposed to the concept .  the effective transfer function , eq .( [ 29 ] ) , at zero temperature becomes from eq .( [ 16 ] ) , we see that may become negative .since is algebraically the same as the network acts , in this case , as a binary network at .accordingly , eq . ([ 25 ] ) remains unchanged , while eqs .( [ 26 ] ) and ( [ 27 ] ) become and +\frac{1}{\sqrt{2\pi v } }   \exp\left[-\frac{(sm_sb-\theta'\theta(\theta'))^2}{2v}\right]\ , .\label{34}\ ] ] the overlap with the concept , eq .( [ 28 ] ) , is given by we show in fig . [ fig1 ] the categorization phase - diagram for the case where , . with the choice that , we are looking in a way for an optimal phase diagram in the sense that the training examples either coincide with the corresponding concepts , that is , for , or are zero , but they are never opposed to the concept . for other values of the parameters , similar diagrams are obtained , although with lower capacity .the categorization phase ( c ) , characterized by and is globally stable below the heavy solid line .it becomes only locally stable , while the spin - glass ( sg ) phase is globally stable , between the heavy solid and the light solid line , where the system always jumps discontinuously to the spin - glass phase .this is in distinction with known results for the categorization phase diagram in the dilute network  , where the transition to the spin glass phase is partly continuous and partly discontinuous . above the light solid line , and at the left of the dash - dotted line , where it disappears continuously , the spin - glass phase , with and , is stable . at the right of the dash - dotted line the paramagnetic ( p ) , or _zero_-phase , with and , is stable .note that , for large threshold , there is a direct transition from the categorization phase to the fully disordered p phase , at low .there exists also a retrieval phase of examples , without categorization , not shown in the figure . since we are dealing with a large number of examples ( thus favoring the categorization ) ,that phase is present in the phase diagram only at very small values of and . to the left of the dotted line ,the effective width is negative . here, every non - zero value of the local field is sufficient to access the neural states and , in consequence , the network behaves in this region as a binary network .the dashed line signals the optimal , i.e. , the value of the width parameter for which the categorization overlap reaches its maximum value .it is interesting to note that the present phase diagram is similar to that of ref . , for the retrieval problem , with the categorization phase taking the role of the retrieval phase in that problem .an important question addressed in this paper refers to the role played by the activity of the examples , , on the categorization ability of the network . in fig .[ fig2 ] the categorization error is shown as a function of the activity , for , , and for several values of .the results reveal that is a monotonically increasing function of .since this is the general behavior for other values of the parameters , the results confirm that for the connected , as well as for the dilute  networks , it is better to train the network with low - activity examples .this can be understood noting that the activity of the examples is decreased when a macroscopic number of bits of every example is turned off .but in keeping the overlap between examples and concepts fixed , the bits that are turned off in the examples must be those that are inverted with respect to the concepts .when the activity reaches its minimal , optimal value , , the only bits that are turned on in the examples are those that are aligned with the concepts , leading to the smallest categorization error . in this casethe categorization task of the network becomes similar to the reconstruction of a puzzle from loose pieces .finally , the figure also shows the discontinuous jump to the spin - glass ( sg ) phase , at the upper phase boundary of fig .[ fig1 ] .  the categorization error as a function of the number of examples , for , and andtwo different activities , namely ( all wrong bits in the examples are turned off ) and ( all wrong bits in the examples are included ) is shown in fig .[ fig3 ] . starting from the spin glass phase , with categorization error equal to , the network undergoes a discontinuous transition to the categorization phase as the number of examples increases above a critical value .the number of examples required for the jump to the categorization phase is considerable smaller for , than for .nevertheless , the final categorization error is similar for the two activities .this means that the network is able to overcome a higher amount of errors in the examples by a larger number of these examples . a higher value of is also required for a higher threshold , in order to reach a higher local field to attain the states with non - zero activity .we consider next the categorization performance obtained from , eq .( [ 29 ] ) , for finite .[ fig4 ] illustrates the influence of the temperature on the categorization error , for , , , and activity equal to and . to the left of the arrow in the curve corresponding to ,the categorization phase is the global minimum , while it is a local minimum to the right . in what concerns the present set of parameters ,the categorization phase for is a local minimum for all temperatures , whereas the spin - glass phase is the global minimum .thus , it is also advantageous for an enhancement of the performance of the network , in the presence of synaptic noise , to train the network with examples of low activity . fig .[ fig4 ] also shows the discontinuous transition to the spin - glass phase at an activity - dependent transition temperature .  the phase diagram for vs. presented in fig .[ fig5 ] for , and .the categorization phase is stable below the upper phase boundary , where it disappears discontinuously , becoming a global minimum below the lower phase boundary .at very small and there is a retrieval phase without categorization , not shown in the figure . the dashed line on the left is the locus of the at - line .the replica - symmetric solution for the categorization phase becomes unstable to replica - symmetry - breaking fluctuations at the left of this line .the re - entrant behavior of the upper phase boundary at low is associated to the instability of the replica - symmetric solution in this region .the spin - glass phase becomes a global minimum to the right of the heavy solid , and to the left of the dotted line , where it disappears continuously . at the right of the dotted line ,the paramagnetic phase is the global minimum .in this section we discuss the categorization properties of a network with continuous , monotonic neurons trained with continuous or discrete examples of binary concepts .the continuous limit is obtained by taking in eqs .( [ 1 ] ) and ( [ 11 ] ) .the following results are independent of the specific form of , provided that its mean and variance are given by eqs .( [ 3a ] ) and ( [ 3b ] ) , respectively .the general eqs .( [ 25])-([28 ] ) , for the saddle - points , apply also to this case . in the absence of noise , the effective transfer function , eq .( [ 29a ] ) , becomes the stepwise linear function in which is the effective gain parameter .consequently we obtain \ , , \label{42}\end{aligned}\ ] ]    \nonumber\\ & & + \frac{1}{2}\left[1+\frac{v}{2\theta'^2}\left(\frac{1}{2 }   + \frac{s^2m_s^2b^2}{2v}\right)\right]\left[{\rmerf}\left(m_-\right )   -{\rm erf}\left(m_+\right)\right ] \label{43}\end{aligned}\ ] ] and \ , , \label{44}\ ] ] where the zero - temperature phase diagram for vs. with and is shown in fig .the categorization phase exists below the light solid line , and it is the global minimum below the heavy solid line . at the left of the dotted line , is zero and the effective gain is infinite . in this region , the states are the only accessible states for non - zero local field and the network behaves as a binary network .in there , the critical for categorization assumes its value in the binary network for this set of parameters , i. e. , .when the network enters the multi - state , continuous regime , the categorization capacity starts to increase abruptly , and reaches its maximum value for .the dashed line signals the optimum for each .it is worth noting that for the optimal line coincides with the transition to the binary regime .this means that whenever there is a binary network capable to perform the categorization task , it will give the best categorization properties for low . only when the network with continuous neurons is expected to have a better performance .contrary to the case of finite , where at zero temperature the replica - symmetric solution is always unstable , there is here a region where it is stable , and this is the part of the phase diagram below the light dash - dotted line .the phase diagram illustrates that also the network of continuous neurons is robust to low gain in the states .the existence of a replica - symmetric stable phase at zero temperature was noticed in ref . , for the retrieval problem in a network of continuous neurons .finally , the heavy dash - dotted line represents the onset of the continuous spin glass transition .  in fig .[ fig7 ] we present the categorization error as a function of the activity of examples , for , , and to .since we deal with a non specified , the only restriction imposed is .we note from the figure that the categorization error is no longer a monotonic increasing function of the activity for all values of . for , is a decreasing function of , for small .the reason is that in the case of large threshold , the local field must be sufficiently high to overcome the threshold , and this is obtained through a moderate increase in the activity of the examples .  finally , we discuss the influence of the number of examples on the categorization ability of networks with continuous neurons .[ fig8 ] shows the categorization error as a function of for , threshold ranging from to and . as a result of the continuous nature of the units , for low thresholdthe categorization error decreases smoothly with the increasing number of examples .this is distinct to the previous case of discrete units , where an abrupt decrease in was observed even at ( see fig .[ fig1 ] ) .furthermore , the decreasing in is no longer monotonic for all values of the threshold .for example , for there is a local maximum in for .the categorization problem , that consists of the recognition of ancestors , when a network is trained only with their descendents , is studied in this work for multi - state fully connected neural network models , keeping in mind an application to either artificial or biological networks in which the training is with sparsely coded patterns .indeed , multi - state networks offer the possibility of recognizing full - sized patterns in networks trained with `` small '' patterns , in which a macroscopic number of bits have been reduced or , eventually , set to zero reducing thereby the activity of the encoded patterns .we found that a low activity can enhance the categorization ability of a fully connected network in a significant way , by changing the threshold for firing of the units .this confirms and extends earlier results on an extremely dilute network of -state neurons .the way the network works for the categorization task is the following .after training with correlated examples , the network searches for stable symmetric mixtures states , in place of pure examples .if these patterns have low activity , it will be less likely that they have bits with opposite sign to the corresponding concepts .the recognition of the latter from the common features of the examples will thereby be enhanced .we derived formal expressions , within replica - symmetric mean - field theory , for the free energy and the relevant order parameters for the categorization problem in a fully connected neural network model , with units in general ising - states and multi - state patterns belonging to a two - level hierarchy .training of the network was assumed to take place through a generalized hebbian learning rule involving only the descendents .these may be considered as corrupted examples of the ancestors ( concepts ) with a number of turned off or inverted bits .explicit results for the relevant phase diagrams and the categorization curves were then obtained for a -state model with a monotonic activation function and for a monotonic -state model . in the first case we also checked the robustness of the network performance to synaptic noise .our results are restricted to binary ancestors and multi - state descendents , although the case of multi - state ancestors has been considered in an extremely dilute network .the limit of validity of the replica - symmetric solution was established in this work looking for the almeida - thouless lines . for ,the replica - symmetric solution is unstable in the absence of synaptic noise ( ) and there is a re - entrant behavior for the ratio of recognized concepts , at small synaptic noise , in accordance with earlier results on the retrieval problem and on the categorization problem in connected networks of binary neurons .nevertheless , since the replica - symmetric solution stabilizes at very small , we argue that replica - symmetry breaking effects should be negligible , even at . on the other hand, there is a finite region of interest for the categorization performance domain where the replica - symmetric solution is stable , even at , in the case of the -state network , as demonstrated explicitly in this work .to summarize , we succeeded in studying a fully connected multi - state neural network model for the categorization problem of recognizing binary concepts when the network is trained with -state examples of low activity , in place of the full activity patterns of a binary network of states .the work presented here can be extended in various directions .first , to infer multi - state concepts in a network with full connectivity and to study the categorization performance for sparsely coded sequential examples . in order to come closer to biological networks , it would be interesting to consider the partial dilution of synapses .we are indebted to alba theumann for showing us how to find the almeida - thouless line for a network with hierarchically correlated patterns .we thank d. boll for comments and discussions , as well as for the kind hospitality of the institute for theoretical physics of the catholic university of leuven , where part of the work of drcd and wkt was done .this work was supported , in part , by cnpq ( conselho nacional de desenvolvimento cientfico e tecnolgico , brazil ) , and finep ( financiadora de estudos e projetos , brazil ) .

Abstract: the categorization ability of fully connected neural network models , with either discrete or continuous -state units , is studied in this work in replica symmetric mean - field theory . hierarchically correlated multi - state patterns in a two level structure of ancestors and descendents ( examples ) are embedded in the network and the categorization task consists in recognizing the ancestors when the network is trained exclusively with their descendents . explicit results for the dependence of the equilibrium properties of a -state model and a -state model are obtained in the form of phase diagrams and categorization curves . a strong improvement of the categorization ability is found when the network is trained with examples of low activity . the categorization ability is found to be robust to finite threshold and synaptic noise . the almeida - thouless lines that limit the validity of the replica - symmetric results , are also obtained .   epsf  [ sec : level1 ]