Soundness and completeness of the Cirquent calculus system CL6 for computability

WENYAN XU and SANYANG LIU, Department of Mathematics, Xidian University, Xi’an, 710071, PR China. E-mail: [email protected]; [email protected] Downloaded from

Abstract Computability logic is a formal theory of computability. The earlier article ‘Introduction to cirquent calculus and abstract resource ’ by Japaridze proved soundness and completeness for the basic fragment CL5 of computability logic. The present article extends that result to the more expressive cirquent calculus system CL6, which is a conservative extension of http://jigpal.oxfordjournals.org/ both CL5 and classical propositional logic.

Keywords: Cirquent calculus, computability logic.

1 Introduction Computability logic (CoL), introduced by G. Japaridze [1, 2, 6], is a semantical and mathematical platform for redeveloping logic as a formal theory of computability. Formulas in CoL represent

interactive computational problems, understood as games between a machine and its environment at CSU Fresno on July 29, 2015 (symbolically named as and ⊥, respectively); logical operators stand for operations on such problems; ‘’ of a problem/game means existence of an algorithmic solution, i.e. ’s effective winning strategy; and of a logical formula is understood as such truth under every particular interpretation of atoms. The approach induces a rich collection of (old or new) logical operators. Among those, relevant to this article are ¬ (negation), ∨ (parallel disjunction) and ∧ (parallel conjunction). Intuitively, ¬ is a role switch operator: ¬A is the game A with the roles of and ⊥ interchanged (’s legal moves and wins become those of ⊥, and vice versa). Both A∧B and A∨B are games playing which means playing the two components A and B simultaneously (in parallel). In A∧B, is the winner if it wins in both components, while in A∨B winning in just one component is sufficient. The symbols and ⊥, together with denoting the two players, are also used to denote two special (the simplest) sorts of games. Namely, is a moveless (‘elementary’) game automatically won by the player , and ⊥ is a moveless game automatically won by ⊥. Cirquent calculus is a refinement of . Unlike the more traditional proof theories that manipulate tree-like objects (formulas, sequents, hypersequents, etc.), cirquent calculus deals with graph-style structures termed cirquents, with its main characteristic feature thus being allowing to explicitly account for sharing subcomponents between different components. The approach was introduced by Japaridze [4] as a new deductive tool for CoL, and was further developed in [5, 7, 8]. The article [4] constructed a cirquent calculus system CL5 for the basic (¬,∧,∨)-fragment of CoL, and proved its soundness and completeness with respect to the semantics of CoL.

Vol. 20 No. 1, © The Author 2011. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/jigpal/jzr043 Advance Access published 29 October 2011

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The atoms of CL5 represent computational problems in general, and are said to be general atoms. The so-called elementary atoms, representing computational problems of zero degree of interactivity (such as the earlier-mentioned games and ⊥) and studied in other pieces of literature on CoL, are not among them. Thus, CL5 only describes valid computability principles for general problems. This is a significant limitation of expressive power. For example, the formula A→A∧A is not valid in CoL when A is a general atom, but becomes valid (as any classical tautology for that matter) when A is elementary.1 So the language of CL5 naturally calls for an extension. Japaridze [4] claimed without a proof that the soundness and completeness result for CL5 could be extended to the more expressive cirquent calculus system CL6 (reproduced later), which is a conservative extension of both CL5 and classical propositional logic. This article is devoted to a soundness and completeness proof for system CL6, thus contributing to the task of extending the

cirquent-calculus approach so as to accommodate incrementally expressive fragments of CoL. Downloaded from

2 Preliminaries This article primarily targets readers already familiar with Japaridze [4], and can essentially be treated as a technical appendix to the latter. However, in order to make it reasonably self-contained, http://jigpal.oxfordjournals.org/ in this section we reproduce the basic concepts from [4, 6]. An interested reader may consult (the extremely well written) [4] and [6] for additional explanations, illustrations, examples and insights.

2.1 Games and strategies The letter ℘ is used as a variable ranging over {,⊥}, with ¬℘ meaning ℘’s adversary. A move is a finite string over the standard keyboard alphabet. A labmove is a move prefixed (‘labeled’) with or ⊥. A run is a finite or infinite sequence of labmoves, and a position is a finite run. Runs are at CSU Fresno on July 29, 2015 usually delimited by ‘’ and ‘’, with  thus denoting the empty run. For any run , ¬ is the same as , with the only difference that every label ℘ is changed to ¬℘. A game2 is a pair A=(LrA,WnA), where: (i) LrA is a set of runs satisfying the condition that a finite or infinite run  is in LrA iff so are all of ’s non-empty finite initial segments.3 If ∈LrA, then  is said to be a legal run of A; otherwise  is an illegal run of A. A move α is a legal move for a player ℘ in a position  of A iff ,℘α∈LrA; otherwise α is an illegal move. When the last move of the shortest illegal initial segment of  is ℘-labeled,  is said to be a ℘-illegal run of A. (ii) WnA is a function that sends every run  to one of the players or ⊥, satisfying the condition that if  is a ℘-illegal run of A, then WnA=¬℘. When WnA=℘,  is said to be a ℘-won run of A. Let A be a game. The depth of A is the smallest integer d such that the length of every legal run of A is ≤d. A is said to be elementary iff the depth of it is 0. There are exactly two elementary games, for which the same symbols and ⊥ are used as for the two players. We identify these with the two propositions of : (true) and ⊥ (false). ‘Snow is white’ is thus a moveless game automatically won by the machine, while ‘Snow is black’ is automatically lost. This is exactly what eventually makes classical propositional logic a natural fragment of CoL.

1This will be explained later. 2The concept of a game considered in CoL is more general than the one defined here, with games in the present sense called constant games. For simplicity, in this article we only consider constant games, omitting the word ‘constant’ and just saying ‘game’. 3This condition can be seen to imply that the empty run  is always in LrA.

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Let  be a run and α be a move. The notation

α

will be used to indicate the result of deleting from  all moves (together with their labels) except those that look like αβ for some move β, and then further deleting the prefix ‘α’ from such moves. For instance, 1.α,⊥2.β,1.γ,⊥2.δ1. =α,γ. The game operations outlined intuitively in the preceding section are defined below, where A, A1, A2 are arbitrary games, α ranges over moves, i ∈{1,2},  is an arbitrary run, and is any legal run of the game that is being defined. (1) ¬A (negation) is defined by: Downloaded from (i) ∈Lr¬A iff ¬∈LrA. (ii) Wn¬A = iff WnA¬ =⊥. (2) A1 ∧A2 (parallel conjunction) is defined by: ∧ . (i) ∈LrA1 A2 iff every move of  is i.α for some i,α and, for both i, i ∈LrAi . ∧ . (ii) WnA1 A2  = iff, for both i, WnAi  i =. http://jigpal.oxfordjournals.org/ (3) A1 ∨A2 (parallel disjunction) is defined by: ∨ . (i) ∈LrA1 A2 iff every move of  is i.α for some i,α and, for both i, i ∈LrAi . ∨ . (ii) WnA1 A2  = iff, for some i, WnAi  i =.

Intuitively, making a move i.α in A1 ∧A2 or A1 ∨A2 means making the move α in the Ai component of the game; and i. (respectively i.) is the subrun of  (respectively ) that took place in that component. In what follows, we explain—formally or informally—several additional basic concepts of the

CoL semantics. at CSU Fresno on July 29, 2015 (1) Static games: CoL restricts its attention to a special yet very wide subclass of games termed ‘static’. Intuitively, static games are interactive tasks where the relative speeds of the players are irrelevant, as it never hurts a player to postpone making moves. A formal definition of this concept can be found in [6], which we will not reproduce here as nothing in this article relies on it. The only relevant for us fact is that all elementary games are static, and that, as proven in [1, 6], the class of static games is closed under the operations ¬,∧,∨ (as well as any other game operations studied in CoL). (2) HPMs: CoL understands ’s effective strategies as interactive machines. Several sorts of such machines have been proposed and studied in CoL, all of them turning out to be equivalent in computing power once we exclusively consider static games. The most basic sort of such machines is called the hard-play machine (HPM). This is a kind of a Turing machine with the additional capability of making moves, and has two tapes4: the ordinary read/write work tape and the read-only run tape. The run tape serves as a dynamic input, at any time (‘clock cycle’) spelling the current position: every time one of the players makes a move, that move—with the corresponding label—is automatically appended to the content of this tape. The environment’s moves are the only non-deterministic events from the machine’s perspective, and there are no restrictions/assumptions on how frequently such moves may occur.

4Often there is also a third tape called the valuation tape. Its function is to provide values for the variables on which a game may depend. However, as we remember, in this article we only consider constant games—games that do not depend on any variables. This makes it possible to safely remove the valuation tape (or leave it there but fully ignore), as this tape is no longer relevant.

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(3) Algorithmic solutions: let M be an HPM. A configuration of M is a full of the current state of the machine, the contents of its two tapes, and the locations of the corresponding two scanning heads. The initial configuration is the configuration where M is in its start state and both tapes are empty. A configuration C is said to be an successor of a configuration C if C can legally follow C in the standard sense, based on the (deterministic) transition function of the machine and accounting for the possibility of non-deterministic updates of the content of the run tape through environment’s moves. A computation branch of M is a sequence of configurations of M where the first configuration is the initial configuration, and each other configuration is a successor of the previous one. Each computation branch B of M incrementally spells a run  on the run tape, which is called the run spelled by B. An algorithmic solution (’s winning strategy) for a given game A is understood as an HPM M such that, whenever

B is a computation branch of M and  the run spelled by B,  is a -won run of A. When the Downloaded from above is the case, we say that M wins A.

2.2 Formulas and their

The language of CL6 is more expressive than that of CL5 in that, along with the old atoms of CL5 http://jigpal.oxfordjournals.org/ called general, it has an additional sort of atoms called elementary, including non-logical elementary atoms and logical atoms and ⊥. On the other hand, all general atoms are non-logical. So in this article, a ‘non-logical atom’ refers to a non-logical elementary atom or a general atom. We use the uppercase letters P,Q,R,S as metavariables for general atoms, and the lowercase p,q,r,s as metavariables for non-logical elementary atoms. A CL6-formula is built from atoms in the standard way using the connectives ¬,∨,∧, with F →G understood as an abbreviation for ¬F ∨G and ¬ limited only to non-logical atoms, where ¬¬F is understood as F, ¬(F ∧G)as¬F ∨¬G, ¬(F ∨G) ¬ ∧¬ ¬ ⊥ ¬⊥ ¬

as F G, as , and as . An atom P (respectively p) and its negation P (respectively at CSU Fresno on July 29, 2015 ¬p) is called a literal, and the two literals are said to be opposite. A CL6-formula is said to be elementary iff it does not contain general atoms. Throughout the rest of this article, unless otherwise specified, by an ‘atom’ or a ‘formula’ we mean one of the language of CL6. An interpretation is a function ∗ that sends every non-logical elementary atom p to an elementary game p∗ and every general atom P to a static game P∗, and extends to all formulas by seeing the logical connectives as the same- game operations. A formula F is uniformly valid iff there is an HPM M, called a uniform solution of F, such that, for every interpretation ∗, M wins F∗. The soundness and completeness of system CL6, promised in the title of the present article, is in the sense that uniformly valid formulas are precisely those provable in CL6.5 Remembering that elementary games are nothing but classical propositions, it is rather obvious that, semantically, the operations ¬, ∧, ∨ are conservative generalizations of the corresponding Boolean connectives. That is, when applied to elementary games, these operations behave exactly like the classical negation, conjunction and disjunction, respectively. So, for instance, the formula p→p∧p is valid for exactly the same as in classical logic. At the same time, the formula P →P∧P (whose atom P, unlike p, may be interpreted as a non-elementary game), is not uniformly valid. Indeed, let A be the game where ⊥ and only ⊥ is (legally) allowed to make one single

5Another sort of validity studied in CoL is multiform validity. A formula F is multiformly valid if, for every interpretation ∗, there is a machine that wins F∗. However, for this concept to be logically well-behaved, one typically needs to consider non-constant games along with constant ones (this is the why we are not attempting to officially state a definition). As proven in [3], multiform validity extensionally coincides with uniform validity. This means that our present soundness and completeness proof equally goes through for both concepts of validity. Yet, for simplicity, we will explicitly talk only about uniform validity.

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move, which should be either a or b; the game is won by ⊥ iff it made the move a. Let B be the same game, with the only difference that it is won by ⊥ iff it made the move b. Next, let A be an interpretation such that PA =A, and B be an interpretation such that PB =B. Now consider an arbitrary HPM M. With a little thought, one can see that M inevitably loses at least one of the two games (P →P∧P)A or (P →P∧P)B. Namely, consider the scenario where ⊥ makes the move a in one of the two conjuncts of the (game represented by the) consequent, and makes the move b in the other conjunct.6 If makes the move a in the antecedent,7 then it loses the game (P →P∧P)B; if makes the move b in the antecedent,8 then it loses the game (P →P∧P)A; and if does not move at all, then it loses both games. Thus, M is not a uniform solution of the formula P →P∧P: whatever M does, there is an interpretation that makes it the loser. On the other hand, the ‘similar’ yet resource-balanced formula P →P, i.e. ¬P∨P, is uniformly

valid. To get some intuitive insights into the reasons that create a dramatic difference between the Downloaded from latter and ¬P∨(P∧P) (i.e. P →P∧P), compare the games ¬Chess∨Chess and ¬Chess∨(Chess∧ Chess), where Chess is the ordinary game of chess readjusted so that draw outcomes are ruled out. The game ¬Chess∨Chess is, in fact, a simultaneous play on two boards, where on the right board the machine plays white, and on the left board plays black. There is a simple strategy for the machine

that guarantees success in this game. All that the machine needs to do is to mimic in Chess the moves http://jigpal.oxfordjournals.org/ made by the adversary in ¬Chess, and vice versa. However, such a copy-cat strategy obviously fails to achieve success in the three-board game ¬Chess∨(Chess∧Chess), for here the best the machine can do is to ‘synchronize’ ¬Chess with one of the two conjuncts of Chess∧Chess. It is possible that then ¬Chess and the unmatched Chess are both lost, in which case the whole game is also lost.

2.3 Cirquents and system CL6 at CSU Fresno on July 29, 2015 Where k ≥0, a k−ary pool is a sequence F1,F2,...,Fk  of k formulas. Since we may have Fi =Fj for some i =j in such a sequence, we use the term oformula to refer to a formula together with a particular occurrence of it in the pool. For example, the pool E,F,G,E has three formulas but four oformulas. Similarly, the terms ‘oliteral’,‘oatom’, etc. will be used in this article to refer to the corresponding entities together with particular occurrences. A k−ary structure is a finite sequence St=1,...,m, where m≥0 and each i, said to be a group of St, is a subset of {1,...,k}. Again, to differentiate between a group as such and a particular occurrence of a group in the structure, we use the term ogroup for the latter. For example, the structure {2,3},{2,3},{1,4},∅ has three groups but four ogroups. A k-ary (k ≥0) cirquent is a pair C =(StC ,PlC ), where StC , called the structure of C,isak-ary structure, and PlC , called the pool of C,isak-ary pool. An ogroup of such a C will mean an ogroup of StC , and an oformula of C will mean an oformula of PlC . Usually, we understand the groups of a cirquent as sets of its oformulas rather than sets of the corresponding ordinal numbers. Thus, if PlC =E,F,G,E and ={2,4}, we would think of  simply as the set {F,E}, and say that  contains F and E. When both the pool and the structure of a cirquent C are empty, i.e. C =(,), we call it the empty cirquent. Throughout this article, we identify each formula F with the cirquent ({1},F).

6Precisely, this means that ⊥ makes the two moves ‘2.1.a’ and ‘2.2.b’ (or, instead, ‘2.2.a’ and ‘2.1.b’). 7That is makes the move ‘1.a’. 8That is makes the move ‘1.b’.

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Rather than writing cirquents as ordered tuples in the above-described style, we prefer to represent them through (and identify them with) diagrams. Below is such a representation for the cirquent whose pool is E,F,G,H and whose structure is {1,2},{2},{3,4}.

EFGH @ @ @ @• @• @•

The top level of a diagram thus indicates the oformulas of the cirquent, and the bottom level gives its ogroups. An ogroup  is represented by a •, and the lines connecting  with oformulas, called arcs, are pointing to the oformulas that  contains. Finally, we put a horizontal line at the top of the Downloaded from diagram to indicate that this is one cirquent rather than two or more cirquents put together. A model is a function M that assigns a truth value — true (1) or false (0)—to each (elementary or general) atom, with being always assigned true and ⊥ false, and extends to compound formulas in the standard classical way. Let M be a model, and C a cirquent. We say that a group  of C is true in M iff at least one of its oformulas is so. And C is true in M if every group of C is so. Otherwise, http://jigpal.oxfordjournals.org/ C is false. Finally, C or a group  of it is a tautology iff it is true in every model. A is a function σ that sends every general atom P to some formula σ(P), and sends every elementary atom to itself. If, (for every general atom P), such a σ(P) is an atom, then σ is said to be an atomic-level substitution. Let A and B be cirquents. We say that B is an instance of A iff B=σ(A) for some substitution σ, where σ(A) is the result of replacing in all oformulas of A every (general or elementary) atom α by σ(α); and B is an atomic-level instance of A iff B=σ(A) for some atomic-level substitution σ.

A positive occurrence (respectively negative occurrence) of an atom is one that is not (respec- at CSU Fresno on July 29, 2015 tively is) in the scope of ¬. A cirquent is said to be binary iff no general atom has more than two occurrences in it. A binary cirquent is said to be normal iff, whenever it has two occurrences of a general atom, one occurrence is negative and the other is positive. A binary tautology (respectively normal binary tautology) is a binary (respectively normal binary) cirquent that is a tautology. The set of rules of CL6 is obtained from that of CL5 by adding to it as an additional axiom, plus the rule of contraction limited only to elementary formulas. Below we reproduce those rules from [4], followed by illustrations. Axioms (A): axioms are ‘rules’ with no . There are three sorts of axioms in CL6. The first one is the empty cirquent. The second one is any cirquent that has exactly two oformulas F and ¬F, for some arbitrary formula F, and an ogroup that contains F and ¬F. In other words, this is the cirquent ({1,2},F,¬F). The third one is a cirquent that has exactly one oformula and one ogroup that contains , i.e. the cirquent ({1},). Mix (M): according to this rule, the conclusion can be obtained by simply putting any two cirquents (premises) together, thus creating one cirquent out of two. Exchange (E): this rule comes in two versions: oformula exchange and ogroup exchange. The conclusion of oformula exchange is obtained by interchanging in the two adjacent oformulas E and F, and redirecting to E (respectively F) all arcs that were originally pointing to E (respectively F). Ogroup exchange is the same, with the only difference that the objects interchanged are ogroups.9

9Exchange is thus a cirquent version of the commutativity rule of regular sequents. Note that the presence of this rule makes it possible to see both the pool and the structure of a cirquent as a multiset rather than a sequence.

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Weakening (W): this rule also comes in two versions: ogroup weakening and pool weakening. A conclusion of ogroup weakening is obtained by adding in the premise a new arc between an existing ogroup and an existing oformula. As for pool weakening, a conclusion is obtained through inserting a new oformula anywhere in the pool of the premise. Duplication (D): a conclusion of this rule is obtained by replacing in the premise some ogroup  by two adjacent ogroups that, as groups, are identical with . Contraction (C): according to this rule, if a cirquent (a premise) has two adjacent elementary oformulas F (the first), F (the second) that are identical, then a conclusion can be obtained by merging F,F into F and redirecting to the latter all arcs that were originally pointing to the first or the second F. ∨−introduction (∨): for the convenience of description, we explain this rule in the bottom-up

view. According to this rule, if a cirquent (the conclusion) has an oformula E ∨F that is contained Downloaded from by at least one ogroup, then the premise can be obtained by splitting the original E ∨F into two adjacent oformulas E and F, and redirecting to both E and F all arcs that were originally pointing to E ∨F. ∧−introduction (∧) this rule, again, is more conveniently described in the bottom-up view.

According to this rule, if a cirquent (the conclusion) has an oformula E ∧F that is contained by at http://jigpal.oxfordjournals.org/ least one ogroup, then the premise can be obtained by splitting the original E ∧F into two adjacent oformulas E and F, and splitting every ogroup  that originally contained E ∧F into two adjacent ogroups E and F , where E contains E (but not F), and F contains F (but not E), with all other (=E ∧F) oformulas of  contained by both E and F . Below we provide illustrations for all rules, in each case an abbreviated name of the rule standing next to the horizontal line separating the premises from the conclusions. Our illustrations for the axioms (the ‘A’ labeled rules) are specific cirquents or schemate of such; our illustrations for all other rules are merely examples chosen arbitrarily. Unfortunately, no systematic ways for schematically representing cirquent calculus rules have been elaborated so far. This explains why we appeal to at CSU Fresno on July 29, 2015 examples instead.

A A A F ¬F @ @• •

oformula exchange ogroup exchange E FG EFG EFG @ Q Q Q Q • @• • Q• Q• • Q• Q• M E E E FG FEGP EFG QPP QQ • @• •••Q P •••Q Q

ogroup weakening pool weakening FG H FHP EFG  PP Q •• • • P• Q• • W W D FG H FGHP EFG Q PP Q •• Q• • P• •••Q

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F required to be elementary EFF HE FH FEFG Q ¨H¨H@  ¨ ¨ • Q• • ¨ H@• ••JJ¨¨AA ••¨ • C ∨ ∧ ∨ ∧ EF HE¨HFH FE¨ FG  ¨ H ¨  • • • • • ••

, ,..., =

The above are all eight rules of CL6. Let C A1 An (possibly n 0) be any cirquents. A deriva- Downloaded from tion of C from A1,...,An in CL6 is a tree of cirquents with C at its root, where each node is a cirquent that either follows from its children by one of the rules of CL6, or is among A1,...,An (and has no children). A proof of C in CL6 is a derivation of C in CL6 from the empty set of cirquents. Since we identify each formula F with the cirquent ({1},F), a proof of a formula F in CL6 is stipulated to { }, 

be a proof of the cirquent ( 1 F ). A formula or cirquent X is provable in CL6 iff it has a proof http://jigpal.oxfordjournals.org/ in CL6. The following is an example of a proof of p→p∧p, i.e. ¬p∨(p∧p), in CL6.

A A p ¬p ¬pp @ • @ • M p ¬p ¬pp

@ • @ • at CSU Fresno on July 29, 2015 C p ¬pp Q Q  Q• Q• E ¬ppp QXX  Q••XX ∧ ¬pp∧p H  HH • ∨ ¬p∨(p∧p) •

On the other hand, CL6 does not prove P →P∧P, i.e. ¬P∨(P∧P). The above-displayed proof does not go through with P instead of p, because contraction can no longer be applied. A routine yet long syntactic analysis could convince us that any other attempted proof would fail, too. However, there is no need for that: the unprovability of P →P∧P immediately follows from our soundness theorem for CL6 and the earlier observed fact that P →P∧P is not uniformly valid. As a warm-up exercise, the reader may try to verify that CL6 proves

(p→Q)∧(p∧Q →R)→(p→R),

and then observe that the same proof does not go through with P instead of p.

[17:18 13/1/2012 jzr043.tex] Paper Size: a4 paper Job: JIGPAL Page: 324 317–330 Cirquent calculus system 325 2.4 System CL2 The earlier mentioned system CL5 differs from CL6 in that the −axiom and the contraction rules are absent there. Also, as noted, the language of CL5 does not allow elementary atoms. In next section, we will see that our proofs are carried out purely syntactically, based on the soundness and completeness of system CL2 (introduced in Japaridze [3]) with respect to uniform validity. That is to say we in fact do not directly use the semantics of CoL. Below we explain what the language of CL2 and its rules are. The language of CL2 is more expressive than the one in which formulas of CL6 are written because, on top of ¬,∨,∧, it has the binary connectives  and , called choice operators. The CL2- formulas are built from atoms (including general atoms and elementary atoms) in the standard way using the connectives ¬,∨,∧,,. As in the case of CL6-formulas, the operator ¬ is only allowed to be applied to non-logical atoms. A CL2-formula is said to be elementary iff it contains neither general Downloaded from atoms nor ,. A surface occurrence of a subformula of a CL2-formula is an occurrence that is not in the scope of ,. A general literal is P or ¬P, where P is a general atom. The elementarization of a CL2-formula A is the result of replacing in A every positive surface occurrence of each general literal by ⊥, every surface occurrence of each −subformula by ⊥, and every surface occurrence of each −subformula by . A CL2-formula is said to be stable iff its elementarization is a tautology http://jigpal.oxfordjournals.org/ of classical logic. CL2 has the following three rules. −→ −→ Rule (a): H →F, where F is stable and H is the smallest set of formulas such that, whenever −→ F has a surface occurrence of a subformula G1 G2, for both i∈{1,2}, H contains the result of replacing that occurrence in F by Gi. Rule (b): H →F, where H is the result of replacing in F a surface occurrence of a subformula  G1 G2 by G1 or G2. at CSU Fresno on July 29, 2015 Rule (c): H →F, where H is the result of replacing in F two—one positive and one negative— surface occurrences of some general atom by a non-logical elementary atom that does not occur in F. −→ The set H of the premises of Rule (a) may be empty, in which case the rule (its conclusion, i.e.) acts like an axiom. Otherwise, the system has no (other) axioms.

3 Soundness and completeness of CL6 In what follows, we may use such as (AME) to refer to the subsystem of CL6 consisting only of the rules whose names are listed between the parentheses. So, (AME) refers to the system that only has axioms, mix and exchange. The same notation can be used next to the horizontal line separating two cirquents to indicate that the lower cirquent (‘conclusion’) can be obtained from the upper cirquent (‘premise’) by whatever number of applications of the corresponding rules. The following Lemmas 1, 2, 3, 4 are precisely Lemmas 4, 5, 10 and 11 of [4], so we state them without proofs (such proofs are given in [4]).

LEMMA 1 All of the rules of CL6 preserve truth in the top-down direction. Taking no premises, (the conclu- sion of) axioms are thus tautologies.

LEMMA 2 The rules of mix, exchange, duplication, contraction, ∨-introduction and ∧-introduction preserve truth in the bottom-up direction as well.

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LEMMA 3 The rules of mix, exchange, duplication, ∨-introduction and ∧-introduction preserve binarity and normal binarity in both top-down and bottom-up directions.

LEMMA 4 Weakening preserves binarity and normal binarity in the bottom-up direction.

LEMMA 5 If CL6 proves a cirquent C, then it also proves every instance of C.

PROOF. Let T be a proof tree of an arbitrary cirquent C, C be an arbitrary instance of C, and σ be a substitution with σ(C)=C . Replace every oformula F of every cirquent of T by σ(F). It is not hard to see that the resulting tree T , which uses exactly the same rules as T does, is a proof of C . Downloaded from LEMMA 6 Contraction preserves binarity and normal binarity in both top-down and bottom-up directions.

PROOF. This is so because contraction limited to elementary formulas can never affect what general atoms occur in a cirquent and how many times they occur. http://jigpal.oxfordjournals.org/ LEMMA 7 A cirquent is provable in CL6 iff it is an instance of a binary tautology.

PROOF.(⇒) Consider an arbitrary cirquent A provable in CL6. By induction on the height of its proof tree, we want to show that A is an instance of a binary tautology. The above is obvious when A is an axiom. Suppose now A is derived by exchange from B. Let us just consider oformula exchange, with ogroup exchange being similar. By the induction hypothesis, B is an instance of a binary tautology

B . Let A be the result of applying exchange to B ‘at the same place’ as it was applied to B when at CSU Fresno on July 29, 2015 deriving A from it, as illustrated in the following example:

Ps¬PP∨r ¬P∧¬r Es¬ER∨r ¬R∧¬r  "  " :  c " :  c " B  c " B  c " ss c" ss c" E E P ¬PsP∨r ¬P∧¬r E ¬Es R∨r ¬R∧¬r H " H "     A: HH " A : HH " ss HH"" ss HH""

Obviously A will be an instance of A . It remains to note that, by Lemmas 1 and 3, A is a binary tautology. The rules of duplication, ∨-introduction and ∧-introduction can be handled in a similar way. Next, suppose A is derived from B and C by mix. By the induction hypothesis, B and C are instances of some binary tautologies B and C , respectively. We may assume that no general atom P occurs in both B and C , for otherwise, in one of the cirquents, rename P into another general atom Q different from everything else. Let A be the result of applying mix to B and C . By Lemmas 1 and 3, A is a binary tautology. And, as in the cases of the other rules, it is evident that A is an instance of A . Suppose A is derived from B by weakening. If this is ogroup weakening, A is an instance of a binary tautology for the same reason as in the case of exchange, duplication, ∨-introduction or ∧-introduction. Assume now we are dealing with pool weakening, so that A is the result of inserting

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a new oformula F into B. By the induction hypothesis, B is an instance of a binary tautology B . Let P be a general atom not occurring in B . And let A be the result of applying weakening to B that inserts P ‘at the same place’ into B as the above application of weakening inserted F into B when deriving A. Obviously A inherits binarity from B ; by Lemma 1, it inherits from B tautologicity as well. And, for the same reason as in all previous cases, A is an instance of A . Finally, suppose A is derived from B by contraction. Then the contracted formula F should be elementary. By the induction hypothesis, B is an instance of a binary tautology B . Let σ be a =σ substitution such that B (B ). And let F1, F2 be two oformulas in B ‘at the same place’ as F, F σ = σ = δ are in B, with (F1) F and (F2) F. Let be the substitution such that, for any general atom δ =σ δ = δ =δ = P, (P) (P)ifP occurs in F1 or F2, and (P) P otherwise. Thus, (F1) (F2) F. And let B =δ(B ). Obviously—for the same reasons as in classical logic—substitution does not destroy

tautologicity, so B is a tautology because B is so. Further, the substitution δ does not introduce any Downloaded from new occurrences of general atoms, so it does not destroy the binarity of B , either. To summarize, B is a binary tautology. Also, of course, B is an instance of B . Notice that B has F and F where B has the contracted oformulas F and F. So, let A be the result of applying contraction to B ‘at the same place’ as it was applied to B when deriving A from it, as illustrated in the following example: http://jigpal.oxfordjournals.org/

Pr∧sr∧s ¬PP∨q ¬P∧¬q Rr∧sr∧s ¬RQ∨q ¬Q∧¬q H Q " H Q " B: HH@  Q " B : HH@  Q " H@ss QQ"" H@ss QQ"" C C Pr∧s ¬PP∨q ¬P∧¬q Rr∧s ¬RQ∨q ¬Q∧¬q : H Q " : H Q " A HH  Q " A HH  Q " Hss QQ"" Hss QQ"" at CSU Fresno on July 29, 2015

Obviously A will be an instance of A . And, by Lemma 1 and Lemma 6, A is a binary tautology. (⇐) Consider an arbitrary cirquent A that is an instance of a binary tautology A . In view of Lemma 5, it would suffice to show that CL6 proves A . We construct a proof of A , in the bottom-up fashion, as follows. Starting from A , we keep applying ∨-introduction and ∧-introduction until we hit an essentially literal cirquent10 B. As in the proof of Theorem 6 of [4], such a cirquent B is guaranteed to be a tautology, and A follows from it in (∨∧). Furthermore, in view of Lemma 3, B is in fact a binary tautology. The tautologicity of B means that every ogroup of it contains either a or at least one pair of opposite (general or elementary) non-logical oliterals. For each ogroup of B that contains a , pick one occurrence of and apply to B a series of weakenings to first delete all arcs but the arc pointing to the chosen occurrence, and next delete all homeless oformulas if any such oformulas are present. For each ogroup of the resulting cirquent that contains a pair of opposite non-logical oliterals, pick one such pair, and continue applying a series of weakenings, as in the proof of Theorem 6 of [4], until a tautological cirquent C is hit with no homeless oformulas, where every ogroup only has either a or a pair of opposite non-logical oliterals. By Lemma 4, C remains binary. Our target cirquent A is thus derivable from C in (W∨∧). Apply a series of contractions to C to separate all shared and all shared elementary non-logical oliterals p or ¬p, as illustrated below; as a result, we get a cirquent D which is still a binary tautology, but whose ogroups no longer share any elementary oformulas.

10An essentially literal cirquent, defined in [4], is one every oformula of whose pool either is an oliteral or is homeless.

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¬ ¬ ¬ ¬ ¬ ¬ ¬ P Pr r rrr Q r s sQs¨ s A ¡ A ¡ @ ¨¨¨¨¨ D: @  ¨ ¨ ¨ s @@s AA¡¡ssA¡ s @s¨s¨¨s¨¨ s s s (C) P ¬Pr ¬rr¬Q ¬r ¬sQs  Q  C : @ A ¡A ¡ @  Q ¡A ss @@s A¡s A¡ ss  @ s QQs s ss¡ A

It is easy to see that the binarity of D implies that there are no shared general oliterals P or ¬P in it except the cases when they are shared by identical-content ogroups. Applying to D a series of duplications, as illustrated below, yields a cirquent E that no longer has identical-content ogroups and hence no longer has any shared oformulas. Downloaded from

P ¬Pr¬r ¬rrr¬Q ¬r ¬s ¬sQss  ¨ ¨ ¨ E : A ¡ A ¡ @ ¨¨¨¨¨¨ s AA¡¡ssA¡ s @s¨s¨¨s¨¨ s s s http://jigpal.oxfordjournals.org/ (D) P ¬Pr¬r ¬rrr¬Q ¬r ¬s ¬sQs s  ¨ ¨ ¨ D: @ A ¡ A ¡ @ ¨¨¨¨¨¨ s @@s AA¡¡ssA¡ s @s¨s¨¨s¨¨ s s s

A is thus derivable from E in (DCW∨∧). In turn, E is obviously provable in (AME). So, CL6 proves A .

LEMMA 8 at CSU Fresno on July 29, 2015 A cirquent is an instance of a binary tautology iff it is an atomic-level instance of some normal binary tautology.

PROOF. Our proof here almost literally follows the proof of Lemma 9 of [4]. The ‘if’ part is trivial. For the ‘only if’ part, assume A is an instance of a binary tautology B. Let P1,...,Pn be all of the general atoms of B that have two positive or two negative occurrences in B. Let Q1,...,Qn be any pairwise distinct general atoms not occurring in B. Let C be the result of replacing in B one of the two occurrences of Pi by Qi, for each i =1,...,n. Then obviously C is a normal binary cirquent, and B an instance of it. By transitivity, A (as an instance of B) is also an instance of C. We want to see that C is a tautology. Deny this. Then there is a classical model M in which C is false. Let M be the model such that:

• M agrees with M on all atoms that are not among P1,...,Pn,Q1,...,Qn;

• for each i ∈{1,...,n}, M (Pi)=M (Qi)=false if Pi and Qi are positive in C; and M (Pi)=

M (Qi)=true if Pi and Qi are negative in C. By induction on complexity, it can be easily seen that, for every subformula F of a formula of C, whenever F is false in M,soisitinM . This extends from (sub)formulas to groups of C and hence

C itself. Thus, C is false in M because it is false in M. But M does not distinguish between Pi

and Qi (any 1≤i ≤n). This clearly implies that C and B have the same truth value in M . That is, B is false in M , which is however impossible because B is a tautology. From this , we conclude that C is a (normal binary) tautology.

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Let σ be a substitution such that A=σ(C). Let σ be a substitution such that, for each general atom P of C, σ (P) is the result of replacing in σ(P) each occurrence of each general atom by a new general atom in such a way that: no general atom occurs more than once in σ (P), and whenever P =Q, no general atom occurs in both σ (P) and σ (Q). Since C is a binary tautology and is its own instance, by Lemma 7, CL6 proves C. Then, by Lemma 5, CL6 proves σ (C) (an instance of C). In view of Lemma 1, we immediately get that σ (C) is a tautology. σ (C) can also be easily seen to be a normal binary cirquent, because C is so. Finally, with a little thought, A can be seen to be an atomic-level instance of σ (C).

LEMMA 9 A CL6-formula F is provable in CL2 iff it is an instance of a binary tautology. Downloaded from PROOF. Again, it should be acknowledged that the present proof very closely follows the proof of Lemma 27 of [4], even though there are certain differences. (⇒) Consider an arbitrary CL6-formula F provable in CL2. Fix a CL2-proof of F in the form of a sequence Fn,Fn−1,...,F1 of formulas, with F1 =F. We may assume that this sequence has no

repetitions or other redundancies. We claim that, for each i with 1≤i ≤n, the following conditions http://jigpal.oxfordjournals.org/ are satisfied: Condition 1: Fi does not contain ,. Condition 2: Whenever Fi contains an elementary atom not occurring in F, that atom is non- logical, and has exactly two—one positive and one negative—occurrences in Fi. Condition 3:Ifi

general atoms. Then it is not hard to see that H is binary and Fn is an instance of H . With Condition

3 in mind, by induction, one can further see that the formulas Fn−1, Fn−2, ... are also instances of H . Thus, F is an instance of H . It remains to show that H is a tautology. But this is indeed so because H results from the tautological G by replacing positive occurrences of ⊥ and replacing two—one positive and one negative—occurrences of elementary atoms by general atoms. It is known from classical logic that such replacements do not destroy truth and hence tautologicity of formulas.

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(⇐) Assume F is a CL6-formula which is an instance of a binary tautology T. In view of Lemma 8, we may assume that T is normal and F is an atomic-level instance of it. Let us call the general atoms that only have one occurrence in T single, and the general atoms that have two occurrences married. Let σ be the substitution with σ(T)=F. Let G be the formula resulting from T by the following steps: substituting each single atom P by σ(P); substituting each married atom Q by σ(Q)ifσ(Q) is elementary; substituting each married atom R by a non-logical elementary atom r not occurring in F if σ(R) is general. It is clear that then F can be derived from G by a series of applications of Rule (c), with each such application replacing two—a positive and a negative—occurrences of some non-logical elementary atom r by σ(R). So, in order to show that CL2 proves F, it would suffice to verify that G is stable and hence it can be derived from the empty set of premises by Rule (a). But G is indeed stable. To see this, consider the elementarization G of G. It results from T by replacing

the only occurrence of each single general atom by some elementary atom, and doing the same with Downloaded from both occurrences of each married general atom. In other words, G is an instance of T. Hence, as T is a tautology, so is G , meaning that G is stable.

THEOREM 10 A formula is provable in CL6 iff it is uniformly valid. http://jigpal.oxfordjournals.org/ PROOF. This theorem is an immediate corollary of Lemma 9, Lemma 7 and the known fact (proven in [3]) that CL2 is sound and complete with respect to uniform validity.

Funding This work was supported by the NNSF (60974082) of China.

References at CSU Fresno on July 29, 2015 [1] G. Japaridze. Introduction to computability logic. Annals of Pure and Applied Logic, 123, 1–99, 2003. [2] G.Japaridze. Computability logic: a formal theory of interaction. In Interactive Computation: The New Paradigm, D. Goldin, S. Smolka, and P. Wegner, eds, pp. 183–223. Springer, 2006. [3] G. Japaridze. Propositional computability logic II. ACM Transactions on Computational Logic, 7, 331–362, 2006. [4] G. Japaridze. Introduction to cirquent calculus and abstract resource semantics. Journal of Logic and Computation, 16, 489–532, 2006. [5] G. Japaridze. Cirquent calculus deepened. Journal of Logic and Computation, 18, 983–1028, 2008. [6] G. Japaridze. In the beginning was game semantics. In Games: Unifying Logic, Language, and Philosophy, O. Majer, A.-V. Pietarinen, T. Tulenheimo, eds, pp. 249–350. Springer, 2009. [7] G. Japaridze. From formulas to cirquents in computability logic. Logical Methods in Computer Science, 7, 1–55, 2011. [8] W.-y. Xu and S.-y. Liu. Deduction theorem for symmetric cirquent calculus, Quantitative Logic and Soft Computing 2010. Advances in Intelligent and Soft Computing, 82, 121–126, 2010.

Received 10 February 2011

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