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Linear Logic Game Semantics Intuitions(from Categorical Models to Linear Logic) Valeria de Paiva Proof Theory in Paraty 2012 August 2012 Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicRecap yesterday: Joinet: why analyticity of proofs is important and how the Curry-Howard isomorphism takes us from ND proofs to simply typed calculus Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicRecap yesterday: Joinet: why analyticity of proofs is important and how the Curry-Howard isomorphism takes us from ND proofs to simply typed calculus Miquel: System F, because if we want a logical view of integers and recursion, then Church numerals are logical but not don’t deliver on ¨ functions, Godel’s T delivers on functions but integers and recursors are not very logical and System F is just right (good logical properties plus good computational intuitions ’compressing proofs’), especially if we consider Coherent Spaces as their denotational semantics Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicRecap yesterday: Joinet: why analyticity of proofs is important and how the Curry-Howard isomorphism takes us from ND proofs to simply typed calculus Miquel: System F, because if we want a logical view of integers and recursion, then Church numerals are logical but not don’t deliver on ¨ functions, Godel’s T delivers on functions but integers and recursors are not very logical and System F is just right (good logical properties plus good computational intuitions ’compressing proofs’), especially if we consider Coherent Spaces as their denotational semantics Vaux: how to see Coherence spaces as a model of a formal system (Linear Logic) that isn’t System F, but a reﬁnement of Intuitionistic Logic (also second-order version), with good involutive properties Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicRecap yesterday: Joinet: why analyticity of proofs is important and how the Curry-Howard isomorphism takes us from ND proofs to simply typed calculus Miquel: System F, because if we want a logical view of integers and recursion, then Church numerals are logical but not don’t deliver on ¨ functions, Godel’s T delivers on functions but integers and recursors are not very logical and System F is just right (good logical properties plus good computational intuitions ’compressing proofs’), especially if we consider Coherent Spaces as their denotational semantics Vaux: how to see Coherence spaces as a model of a formal system (Linear Logic) that isn’t System F, but a reﬁnement of Intuitionistic Logic (also second-order version), with good involutive properties Beffara: how the sequent calculus version of Linear Logic can be shown to have good syntactic and semantic properties, independently of coherent spaces. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicRecap yesterday: Today things are not so organized. I tend to think in terms of very pedestrian ’categorical models’ and they led me to a version of Linear Logic that is off the beaten track and didn’t show up in the fragments discussed by Beffara yesterday, Full Intuitionistic Linear Logic (FILL) introduced by Martin Hyland and myself in 1993. Looking for a notion of game semantics appropriate for FILL led me to games that are more similar to the 50’s logicians versions of games. Lorenzen/Lorenz/Hintikka notions of games gave rise to a huge amount of work on Games for Programming Languages (several PhD theses, several conferences, huge numbers of systems and variations...) but I know very little about those games. I want to talk to you about Lorenzen’s initial intuitions on logical games. and how to modify them for FILL, building up on Blass’ work. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicCategorical Models? When studying logic one can concentrate on: its models (Model Theory) its proofs (Proof Theory) on foundations and one favorite version (Set Theory) on computability and its effective versions (Recursion Theory). In this talk: we’re interested in Proof Theory, in using games models to discuss it, and in modeling Linear Logic using games (and intuitions about categories). Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicProof Theory: Proofs as Mathematical Objects of Study Frege: quantiﬁers! but also ﬁrst to use abstract symbols to write proofs Hilbert: proofs are mathematical objects of study themselves Gentzen: inference rules the way mathematicians think Natural Deduction and Sequent Calculus Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicProofs as ﬁrst class objects? Programme: elevate proofs to “ﬁrst class” logical objects. Instead of asking ‘when is a formula A true’, ask ‘what is a proof of A?’ Sometimes I call this programme Categorical Proof Theory. Sometimes I call it simply Proof Semantics, thinking in general terms Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicThe quest for proofs... Traditional proof theory, to the extent that it relies on models, uses algebraic structures such as Boolean algebras, Heyting algebras, Kripke models Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicThe quest for proofs... Traditional proof theory, to the extent that it relies on models, uses algebraic structures such as Boolean algebras, Heyting algebras, Kripke models (or even Phase Spaces as we saw yesterday....) These models lose one important dimension. In these models different proofs are not represented at all. Provability, the fact that Γ a collection of premisses A1 , . . . , Ak entails A, is represented by the less or equal ≤ relation in the model. This does not give us a way of representing the proofs themselves. We only know if a proof exists Γ ≤ A or not. All proofs are collapsed into the existence of this relation. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicThe quest for proofs... By contrast in categorical proof theory we think and write a proof as Γ →f A where f is the reason why we can deduce A from Γ, a name for the proof we are thinking of. Thus we can observe and name and compare different derivations. Which means that we can see subtle differences in the logics. (Some people prefer to think of the proofs as lambda-terms instead of morphisms, but we know that–with a bit of care– they are the same) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicDialogue/Games Classical logic is based on truth values. (To understand a sentence is to know under which circumstances it is true. The meaning of a propositional connective is explained by saying how the truth value of a compound formula is obtained from the truth values of its constituents.) Intuitionistic logic is based on proofs. (To understand a sentence is to know what constitutes a proof of it. The meaning of a propositional connective is given by describing the proofs of a compound formula, assuming that we know what constitutes a proof of a constituent.) Lorenzen(1959): semantics for both logics based on games. (To understand a sentence is to know the rules for attacking and defending it in a debate. The meaning of a propositional connective is given by saying how to debate a compound formula, assuming that one knows how to debate its constituents.) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLinear Logic Dialogue/Games? Blass was the ﬁrst to suggest a semantics for Linear Logic based on Lorenzen games in 1992. This started a whole new area of research: Games for semantics of programming languages: Main schools: Abramsky, Jaghadeesan, Malacaria (AJM) Hyland, Ong (HO) How to compare all those programming language games to Hintikka ı ´ Games, Lorenzen Dialogues/Dialogical games, Ehrenfeucht-Fra¨sse games, Conway games, Economic games, etc..? Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLinear Logic Jean-Yves Girard: “[...]linear logic comes from a proof-theoretic analysis of usual logic.” Also ”from a semantical analysis of the models of System F – or polymorphic lambda calculus–” Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLinear Logic Jean-Yves Girard: “[...]linear logic comes from a proof-theoretic analysis of usual logic.” Also ”from a semantical analysis of the models of System F – or polymorphic lambda calculus–” (the coherent semantics that we saw yesterday...) Linear logic is a resource-conscious logic, or a logic of resources. The resources in Linear Logic are premises, assumptions and conclusions, as they are used in logical proofs. Resource accounting: each meaning used exactly once, unless specially marked by ! Win: account for resources when you want. only when you want... Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLinear Implication and (Multiplicative) Conjunction Traditional implication: A, A → B B A, A → B A∧B Re-use A Linear implication: A, A −◦ B B A, A −◦ B A⊗B Cannot re-use A Traditional conjunction: A ∧ B A Discard B Linear conjunction: A⊗B A Cannot discard B Of course: !A A⊗ !A Re-use ! (A) ⊗ B B Discard Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLinear Implication and (Multiplicative) Conjunction Tensor is NOT a product: no duplication of objects A A ⊗ A no discarding of objects A I Game Semantics Intuitions (from Categorical A → Linear But objects !A are comonoids (both maps above) plus natural !Models toA Logic) Valeria de Paiva
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Linear LogicMultiplicative FILL(Hyland/dePaiva) Many people assume the multiplicative disjunction “par” does NOT make sense outside Classical Linear Logic. But the sequent calculus below works. (plus axiom A A and complicated rule for implication right...) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicMultiplicative FILL A A implies A A, ⊥ but cannot do A, A → ⊥ (multiplicative excluded middle) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicMore FILL... FILL (1993) comes from categorical semantics, Dialectica categories (1990). Have symmetric monoidal closed category (smcc) for modeling tensor and linear implication. Plus a tensor-like bifunctor for ‘par’. Tensor and par related by weak distributivities (Cockett Seely) Unit of par is the dualizing object ⊥, but only have A → ¬¬A, no isomorphism. FILL cut-elimination proved 1996 (Brauner/deP) ¨ FILL Mu-calculus-style 2006 (deP/Ritter) Also: Intuitionistic version not linear (FIL) 2005 with Luiz Carlos Pereira Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicThe Modality ! The sequent calculus rules are intuitive: Duplicating and erasing formulae preﬁxed by “!” correspond to the usual structural rules of weakening and contraction: ∆ B ∆, !A, !A B ∆, !A B ∆, !A B The rules for introducing the modality are more complicated, but familiar from Prawitz’s work on S4. !∆ B ∆, A B !∆ !B ∆, !A B (Note that !∆ means that every formula in ∆ starts with a ! operator.) objects !A are comonoids (!A →!A⊗!A and !A → I) plus natural !A → A and !A →!!A Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLinear Logic Games Blass: (1992) A game semantics for linear logic, (1994) Some Semantical Aspects of Linear Logic A game (or dialogue) semantics in the style of Lorenzen (1959) for Girard’s Linear Logic. Lorenzen: the (constructive) meaning of a proposition φ is speciﬁed by telling how to conduct a debate between a proponent P who asserts φ and an opponent 0 who denies φ. Technically, a dialogue is a sequence of signed expressions, alternatively supported by P and 0 and satisfying at each step some argumentation form/game rule for the connectives. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicBlass Linear Logic Games Propositions are interpreted as games, connectives as operations on games, and validity as existence of a winning strategy for proponent P. Afﬁne logic (linear logic plus weakening) is sound for this interpretation. Blass: a completeness theorem for the additive fragment of afﬁne logic Completeness fails for the multiplicative fragment... This work is seminal, spawned a whole area of research, because the semantics is intuitive and applications seem endless. BUT a big problem to begin with: strategies do not compose associatively, no category?... Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues Basic idea: To understand a (logical) sentence is to know the rules for attacking and defending it in a debate. Dialogue, debate and game are used synonymously. The meaning of a propositional connective is given by explaining how to debate a compound formula, assuming that one knows how to debate its constituents. A game (or an argument, dialogue or protocol) consists of two players, one (the Proponent or Player P) seeking to establish the truth of a formula under consideration (trying to prove it) while the other (the Opponent O) disputing it, trying to prove it false. The two players alternate, attacking and defending their positions. The essence of the semantics consists of rules of the debate between the players. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogue Intuitions To attack a conjunction A ∧ B, O may select either conjunct, and P must then defend that conjunct. To attack a disjunction A ∨ B, O may demand that P select and defend one of the disjuncts. To attack a negation ¬A, O may assert and defend A, with P now playing the role of opponent of A. To attack an implication A → B O may assert A; then P may either attack A, or assert and defend B. to challenge an implication essentially amounts to providing a proof of the antecedent and claiming that the other player will fail to build a proof of the consequent from it. The defense against such an attack then consists of a proof of the consequent. (Negation can be viewed as the special case of implication where the consequent is an indefensible statement.) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogue Intuitions The simplicity of this description of the connectives is deceptive. Lorenzen needs supplementary rules to obtain a game semantics for constructive logic. Lorenzen games were developed by Lorenz, Felscher and Rahman and co-authors, who established a collection of games for speciﬁc (non-classical) logics, including Linear Logic. This general framework was named Dialogic. By contrast programming language games will develop semantics for PCF, Idealized Algol, Concurrency features, etc... based on afﬁne logic, but worrying more about programming than about the logic. ` (Mellies ) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Supplementary Intuitions 1 Atomic formulas are never attacked or defended, but P may assert them only if they have been previously asserted by O. Why? semantics is to describe logical validity, not truth in a particular situation. so a winning strategy for P should succeed independently of any information about atomic facts. Thus P can safely assert such a fact only if O is already committed to it. (A consequence of this rule governing atomic statements is that negating a formula does not fully interchange the roles of the two players, because it is still P who is constrained by the rule.) Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Supplementary Intuitions 2 Nothing so far distinguishes ¬(¬A ∧ A), which is constructively valid, from A ∨ ¬A, which is not. Supplementary rules also govern repeated attacks on or defenses of the same assertion. Considering the excluded middle A ∨ ¬A, P must choose and assert one of A and ¬A. He cannot assert A, an atomic formula that O has not yet asserted. So he asserts ¬A, and O attacks by asserting A. Now that O has asserted A, P would like to go back and revise his defense of A ∨ ¬A by choosing A instead of ¬A, but the Lorenzen rules do not accept it. for constructive logic. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Supplementary Intuitions 3 Consider the debate about ¬(¬A ∧ A)). O attacks by asserting ¬A ∧ A. P can attack this assertion by demanding that O assert the conjunct A. Then, P re-attacks the same assertion by demanding that O assert the other conjunct ¬A. After O does so, P wins by attacking ¬A with A, which O has already asserted. The difference between the two debates is that Lorenzens supplementary rules allow P to re-attack ¬A ∧ A but not to re-defend A ∨ ¬A. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicBlass Games vs. Lorenzen Games: two differences? 1.Blass plays take inﬁnitely many moves and in particular, assertions of atomic formulas are not terminal positions in the games but are debatable like any other formula. when games are allowed to be inﬁnitely long, it is possible that neither player has a winning strategy. We can use such undetermined games as the interpretations of atomic formulas, to model the idea that the players (particularly P) do not know whether an atomic formula is true or not. 2. two kinds of conjunction, two kinds of disjunction solve the issue of re-attacking and re-defending formulas. Earlier discussion shows that, if A is an undetermined game, then P has no winning strategy in ¬A ∨ A. On the other hand, P always has a winning strategy (copy-cat) in the par game (¬A ` A), namely to start with the subdebate where O moves ﬁrst, to switch subdebates at every move, and to copy in the new subdebate the move that O just made in the other subdebate. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues Language L: standard ﬁrst-order logic connectives ∧, ∨, →, ¬ Two labels, O (Opponent) and P (Proponent). Special force symbols: ?... and !... , where ? stands for attack (or question) and ! stands for defense (or answer). The set of rules in dialogic is divided into particle rules and structural rules. Particle rules describe the way a formula can be attacked and defended, according to its main connective. Structural rules specify the general organization of the game. The difference between classical and intuitionistic logic is in the structural rules. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues: Structural Rules for Classical Logic [S1] Proponent P may assert an atomic formula only after it has been asserted by opponent O. [S4] A Proponent P-assertion may be attacked at most once. [S5] Opponent O can react only upon the immediately preceding proponent P-statement. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues: Structural Rules for Intuitionistic Logic [S1] P may assert an atomic formula only after it has been asserted by O. [S2] If p is an P-position, and if at round n − 1 there are several open attacks made by O, then only the latest of them may be answered at n (and the same with P and O reversed). [S3] An attack may be answered at most once. [S4] A P-assertion may be attacked at most once. [S5] O can react only upon the immediately preceding P-statement. The problem with the structural rules is that it is not clear which modiﬁcations can be made to them without changing the set of provable formulas. The second problem is that notions of context and its splitting will need to be accounted for. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues for Linear Logic Rahmans deﬁnitions of Lorenzen games for Linear logic start from the sequent formulation of the logic. From the dialogical point of view, assumptions are the Opponents concessions, while conclusions are the Proponents claims. In Linear Logic each occurrence of one formula in a proof must be taken as a distinct resource for the inference process: we must use all and each formula that has been asserted throughout the dialogue. And we cannot use one play more than once. Linear dialogues are contextual, the ﬂow of information within the proof is constrained by an explicit structure of contexts, ordered by a relation of subordination. How contexts are split is essential information for the games. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues for multiplicative Linear Logic Rules for connectives (particle rules): Linear implication −◦ and linear negation ()⊥ are the same as for Intuitionistic and Classical Logic: to attack linear implication, one must assert the antecedent, hoping that the opponent cannot use it to prove the consequent. The defence against such an attack then consists of a proof of the consequent. The only way to attack the assertion A⊥ is to assert A, and be prepared to defend this assertion. Thus there is no proper defence against such an attack, but it may be possible to counterattack the assertion of A. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues for multiplicative Linear Logic Rules for connectives (Particle rules): Multiplicative conjunction: Tensor The rule for tensor introduction shows that context splitting for tensor occurs when it is asserted by the Proponent. so the dialogical particle rule will let the challenger (Opponent) choose the context where the dialogue will proceed. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicLorenzen Dialogues for multiplicative Linear Logic Rules for connectives (Particle rules): Multiplicative Disjunction: Par The multiplicative disjunction par will generate a splitting of contexts when asserted by the Opponent, thus the particle rule will let the defender choose the context. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicMultiplicative FILL Assertion Attack Response A⊗B ⊗L A in context µ in subcontext ν of µ in ν A⊗B ⊗R B in context µ in subcontext ν of µ in ν A`B ? A in context µ in µ in subcontext ν of µ A`B ? B in context µ in µ in subcontext ν of µ A B A B in context µ in subcontext ν of µ in subcontext ν of µ ¬A A − in context µ in subcontext ν of µ in subcontext ν of µ Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicFull Intuitionistic Linear Logic Important to notice that while Rahman deals exclusively with classical linear logic, he does mention that an intuitionistic structural rule could be used instead. This is what we want to do for FILL. ¨ Why? Martin-Lof’s question 1991, Sweden. Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicGames for FILL Games particle rules for tensor, par and linear implication identical to Rahmans for Linear Logic. But we use Rahmans intuitionistic structural rules. Soundness and completeness should follow as for Rahman and Keiff. Calculations still to be checked, I’m afraid... Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicConclusions/Further Work discussed Logical Games for Linear Logic in two traditions: Blass-style and Lorenzen-style games (as introduced by Rahman, Keiff). Emphasis given to the interpretation of linear implication, (instead of involutive negation) plus structural intuitionistic rule. These Lorenzen games give us the ability to model full intuitionistic linear logic more easily. Preliminary work: still need to prove soundness and completeness More importantly: need to make sure that strategies are compositional, to make sure we can produce categories of Lorenzen games. Blass problem here too? Want only a fragment of Linear Logic, FILL (full intuitionistic linear ` logic). Will a bit of luck this will coincide with Mellies and Tabareau ”tensor logic” ? Need to check also Hyland and Schalk’s Abstract Games, which also arise from cat semantics... Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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Linear LogicThanks! Valeria de Paiva Game Semantics Intuitions (from Categorical Models to Linear Logic)
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