game semantics intuitions

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)

Linear Logic

Recap yesterday:

Joinet: why analyticity of proofs is important and how theCurry-Howard isomorphism takes us from ND proofs to simply typedcalculus

Miquel: System F, because if we want a logical view of integers andrecursion, then Church numerals are logical but not don’t deliver onfunctions, Godel’s T delivers on functions but integers and recursorsare not very logical and System F is just right (good logical propertiesplus good computational intuitions ’compressing proofs’), especially ifwe consider Coherent Spaces as their denotational semanticsVaux: how to see Coherence spaces as a model of a formal system(Linear Logic) that isn’t System F, but a refinement of IntuitionisticLogic (also second-order version), with good involutive propertiesBeffara: how the sequent calculus version of Linear Logic can beshown to have good syntactic and semantic properties, independentlyof coherent spaces.


Linear Logic

Recap yesterday:

Joinet: why analyticity of proofs is important and how theCurry-Howard isomorphism takes us from ND proofs to simply typedcalculusMiquel: System F, because if we want a logical view of integers andrecursion, then Church numerals are logical but not don’t deliver onfunctions, Godel’s T delivers on functions but integers and recursorsare not very logical and System F is just right (good logical propertiesplus good computational intuitions ’compressing proofs’), especially ifwe 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 IntuitionisticLogic (also second-order version), with good involutive propertiesBeffara: how the sequent calculus version of Linear Logic can beshown to have good syntactic and semantic properties, independentlyof coherent spaces.


Linear Logic

Recap yesterday:

Joinet: why analyticity of proofs is important and how theCurry-Howard isomorphism takes us from ND proofs to simply typedcalculusMiquel: System F, because if we want a logical view of integers andrecursion, then Church numerals are logical but not don’t deliver onfunctions, Godel’s T delivers on functions but integers and recursorsare not very logical and System F is just right (good logical propertiesplus good computational intuitions ’compressing proofs’), especially ifwe consider Coherent Spaces as their denotational semanticsVaux: how to see Coherence spaces as a model of a formal system(Linear Logic) that isn’t System F, but a refinement of IntuitionisticLogic (also second-order version), with good involutive properties

Beffara: how the sequent calculus version of Linear Logic can beshown to have good syntactic and semantic properties, independentlyof coherent spaces.


Linear Logic

Recap yesterday:

Joinet: why analyticity of proofs is important and how theCurry-Howard isomorphism takes us from ND proofs to simply typedcalculusMiquel: System F, because if we want a logical view of integers andrecursion, then Church numerals are logical but not don’t deliver onfunctions, Godel’s T delivers on functions but integers and recursorsare not very logical and System F is just right (good logical propertiesplus good computational intuitions ’compressing proofs’), especially ifwe consider Coherent Spaces as their denotational semanticsVaux: how to see Coherence spaces as a model of a formal system(Linear Logic) that isn’t System F, but a refinement of IntuitionisticLogic (also second-order version), with good involutive propertiesBeffara: how the sequent calculus version of Linear Logic can beshown to have good syntactic and semantic properties, independentlyof coherent spaces.


Linear Logic

Recap yesterday:

Today things are not so organized.

I tend to think in terms of very pedestrian ’categorical models’ and theyled me to a version of Linear Logic that is off the beaten track anddidn’t show up in the fragments discussed by Beffara yesterday,Full Intuitionistic Linear Logic (FILL) introduced by Martin Hyland andmyself in 1993.Looking for a notion of game semantics appropriate for FILL led me togames that are more similar to the 50’s logicians versions of games.Lorenzen/Lorenz/Hintikka notions of games gave rise to a hugeamount of work on Games for Programming Languages (several PhDtheses, several conferences, huge numbers of systems andvariations...) 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.


Linear Logic

Categorical 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 todiscuss it, and in modeling Linear Logic using games (and intuitionsabout categories).


Linear Logic

Proof 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 themselvesGentzen: inference rules

the way mathematicians thinkNatural Deduction and Sequent Calculus


Linear Logic

Proofs 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


Linear Logic

The quest for proofs...

Traditional proof theory, to the extent that it relieson models, uses algebraic structures such as Boolean algebras,Heyting algebras, Kripke models

(or even Phase Spaces as we sawyesterday....) These models lose one important dimension. In thesemodels different proofs are not represented at all.Provability, the fact that Γ a collection of premisses A1, . . . ,Ak entailsA, is represented by the less or equal ≤ relation in the model. Thisdoes not give us a way of representing the proofs themselves. We onlyknow if a proof exists Γ ≤ A or not. All proofs are collapsed into theexistence of this relation.


Linear Logic


Traditional proof theory, to the extent that it relieson models, uses algebraic structures such as Boolean algebras,Heyting algebras, Kripke models (or even Phase Spaces as we sawyesterday....) These models lose one important dimension. In thesemodels different proofs are not represented at all.Provability, the fact that Γ a collection of premisses A1, . . . ,Ak entailsA, is represented by the less or equal ≤ relation in the model. Thisdoes not give us a way of representing the proofs themselves. We onlyknow if a proof exists Γ ≤ A or not. All proofs are collapsed into theexistence of this relation.


Linear Logic


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 theproof we are thinking of. Thus we can observe and name andcompare different derivations. Which means that we can see subtledifferences in the logics.(Some people prefer to think of the proofs as lambda-terms instead ofmorphisms, but we know that–with a bit of care– they are the same)


Linear Logic

Dialogue/Games

Classical logic is based on truth values.(To understand a sentence is to know under which circumstances it istrue. The meaning of a propositional connective is explained by sayinghow the truth value of a compound formula is obtained from the truthvalues 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 theproofs of a compound formula, assuming that we know whatconstitutes 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 anddefending it in a debate. The meaning of a propositional connective isgiven by saying how to debate a compound formula, assuming thatone knows how to debate its constituents.)


Linear Logic

Linear Logic Dialogue/Games?

Blass was the first to suggest a semantics for Linear Logic based onLorenzen 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 HintikkaGames, Lorenzen Dialogues/Dialogical games, Ehrenfeucht-Fraıssegames, Conway games, Economic games, etc..?


Linear Logic

Linear Logic

Jean-Yves Girard: “[...]linear logic comes from aproof-theoretic analysis of usual logic.” Also ”from a semanticalanalysis of the models of System F – or polymorphic lambdacalculus–”

(the coherent semantics that we saw yesterday...)

Linear logic is a resource-conscious logic, or a logic of resources. Theresources in Linear Logic are premises, assumptions and conclusions,as they are used in logical proofs.Resource accounting: each meaning used exactly once, unlessspecially marked by !Win: account for resources when you want. only when you want...


Linear Logic

Linear Logic

Jean-Yves Girard: “[...]linear logic comes from aproof-theoretic analysis of usual logic.” Also ”from a semanticalanalysis of the models of System F – or polymorphic lambdacalculus–” (the coherent semantics that we saw yesterday...)

Linear logic is a resource-conscious logic, or a logic of resources. Theresources in Linear Logic are premises, assumptions and conclusions,as they are used in logical proofs.Resource accounting: each meaning used exactly once, unlessspecially marked by !Win: account for resources when you want. only when you want...


Linear Logic

Linear Implication and (Multiplicative) Conjunction

Traditional implication: A,A→ B ` BA,A→ B ` A∧ B Re-use A

Linear implication: A,A−◦ B ` BA,A−◦ B 6` A⊗ B Cannot re-use A

Traditional conjunction: A∧ B ` A Discard B

Linear conjunction: A⊗ B 6` A Cannot discard B

Of course: !A ` A⊗!A Re-use!(A)⊗ B ` B Discard


Linear Logic

Linear Implication and (Multiplicative) Conjunction

Tensor is NOT a product:no duplication of objects A 6` A⊗ Ano discarding of objects A 6` IBut objects !A are comonoids (both maps above) plus natural !A→ Aand !A→!!A


Linear Logic

Multiplicative FILL(Hyland/dePaiva)

Many people assume the multiplicative disjunction “par” does NOTmake sense outside Classical Linear Logic. But the sequent calculusbelow works. (plus axiom A ` A and complicated rule for implicationright...)


Linear Logic

Multiplicative FILL

A ` A implies A ` A,⊥ but cannot do ` A,A→⊥ (multiplicativeexcluded middle)


Linear Logic

More FILL...

FILL (1993) comes from categorical semantics, Dialectica categories(1990).Have symmetric monoidal closed category (smcc) for modeling tensorand linear implication.Plus a tensor-like bifunctor for ‘par’. Tensor and par related by weakdistributivities (Cockett Seely)Unit of par is the dualizing object ⊥, but only have A→¬¬A, noisomorphism.

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


Linear Logic

The Modality !

The sequent calculus rules are intuitive: Duplicating and erasingformulae prefixed by “!” correspond to the usual structural rules ofweakening and contraction:

∆ ` B

∆, !A ` B

∆, !A, !A ` B

∆, !A ` B

The rules for introducing the modality are more complicated, butfamiliar from Prawitz’s work on S4.

!∆ ` B

!∆ `!B

∆,A ` 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


Linear Logic

Linear 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) forGirard’s Linear Logic.

Lorenzen: the (constructive) meaning of a proposition φ is specified bytelling how to conduct a debate between a proponent P who asserts φand 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 someargumentation form/game rule for the connectives.


Linear Logic

Blass Linear Logic Games

Propositions are interpreted as games, connectives as operations ongames, and validity as existence of a winning strategy for proponent P.

Affine logic (linear logic plus weakening) is sound for thisinterpretation.

Blass: a completeness theorem for the additive fragment of affine logicCompleteness fails for the multiplicative fragment...

This work is seminal, spawned a whole area of research, because thesemantics is intuitive and applications seem endless.BUT a big problem to begin with: strategies do not composeassociatively, no category?...


Linear Logic

Lorenzen Dialogues

Basic idea: To understand a (logical) sentence is to know the rules forattacking and defending it in a debate.

Dialogue, debate and game are used synonymously.

The meaning of a propositional connective is given by explaining howto debate a compound formula, assuming that one knows how todebate 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 aformula under consideration (trying to prove it) while the other (theOpponent O) disputing it, trying to prove it false.The two players alternate, attacking and defending their positions. Theessence of the semantics consists of rules of the debate between theplayers.


Linear Logic

Lorenzen Dialogue Intuitions

To attack a conjunction A∧ B, O may select either conjunct, and Pmust then defend that conjunct.

To attack a disjunction A∨ B, O may demand that P select and defendone of the disjuncts.

To attack a negation ¬A, O may assert and defend A, with P nowplaying the role of opponent of A.

To attack an implication A→ B O may assert A; then P may eitherattack 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 theconsequent is an indefensible statement.)


Linear Logic

Lorenzen Dialogue Intuitions

The simplicity of this description of the connectives is deceptive.Lorenzen needs supplementary rules to obtain a game semantics forconstructive logic.Lorenzen games were developed by Lorenz, Felscher and Rahmanand 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 forPCF, Idealized Algol, Concurrency features, etc... based on affinelogic, but worrying more about programming than about the logic.(Mellies )


Linear Logic

Lorenzen Supplementary Intuitions 1

Atomic formulas are never attacked or defended, but P may assertthem only if they have been previously asserted by O.Why? semantics is to describe logical validity, not truth in a particularsituation.so a winning strategy for P should succeed independently of anyinformation about atomic facts.Thus P can safely assert such a fact only if O is already committed toit.(A consequence of this rule governing atomic statements is thatnegating a formula does not fully interchange the roles of the twoplayers, because it is still P who is constrained by the rule.)


Linear Logic


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 ofthe same assertion.

Considering the excluded middle A∨ ¬A, P must choose and assertone of A and ¬A.He cannot assert A, an atomic formula that O has not yet asserted. Sohe asserts ¬A, and O attacks by asserting A. Now that O hasasserted A, P would like to go back and revise his defense of A∨ ¬Aby choosing A instead of ¬A, but the Lorenzen rules do not accept it.for constructive logic.


Linear Logic


Consider the debate about ¬(¬A∧ A)).

O attacks by asserting ¬A∧ A. P can attack this assertion bydemanding that O assert the conjunct A. Then, P re-attacks the sameassertion by demanding that O assert the other conjunct ¬A. After Odoes so, P wins by attacking ¬A with A, which O has alreadyasserted.

The difference between the two debates is that Lorenzenssupplementary rules allow P to re-attack ¬A∧ A but not to re-defendA∨ ¬A.


Linear Logic

Blass Games vs. Lorenzen Games: two differences?

1.Blass plays take infinitely many moves and in particular, assertionsof atomic formulas are not terminal positions in the games but aredebatable like any other formula.when games are allowed to be infinitely long, it is possible that neitherplayer has a winning strategy. We can use such undetermined gamesas the interpretations of atomic formulas, to model the idea that theplayers (particularly P) do not know whether an atomic formula is trueor not.2. two kinds of conjunction, two kinds of disjunction solve the issue ofre-attacking and re-defending formulas.Earlier discussion shows that, if A is an undetermined game, then Phas no winning strategy in ¬A∨ A. On the other hand, P always has awinning strategy (copy-cat) in the par game (¬A ` A), namely to startwith the subdebate where O moves first, to switch subdebates at everymove, and to copy in the new subdebate the move that O just made inthe other subdebate.


Linear Logic

Lorenzen Dialogues

Language L: standard first-order logic connectives ∧,∨,→,¬Two labels, O (Opponent) and P (Proponent).Special force symbols: ?... and !... , where ? stands for attack (orquestion) and ! stands for defense (or answer).The set of rules in dialogic is divided into particle rules and structuralrules.Particle rules describe the way a formula can be attacked anddefended, according to its main connective.Structural rules specify the general organization of the game.

The difference between classical and intuitionistic logic is in thestructural rules.


Linear Logic

Lorenzen Dialogues: Structural Rules for Classical Logic

[S1] Proponent P may assert an atomic formula only after it has beenasserted by opponent O.[S4] A Proponent P-assertion may be attacked at most once.[S5] Opponent O can react only upon the immediately precedingproponent P-statement.


Linear Logic

Lorenzen Dialogues: Structural Rules for Intuitionistic Logic

[S1] P may assert an atomic formula only after it has been asserted byO.[S2] If p is an P-position, and if at round n− 1 there are several openattacks 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 whichmodifications can be made to them without changing the set ofprovable formulas. The second problem is that notions of context andits splitting will need to be accounted for.


Linear Logic

Lorenzen Dialogues for Linear Logic

Rahmans definitions of Lorenzen games for Linear logic start from thesequent formulation of the logic.From the dialogical point of view, assumptions are the Opponentsconcessions, while conclusions are the Proponents claims.In Linear Logic each occurrence of one formula in a proof must betaken as a distinct resource for the inference process: we must use alland 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 proofis constrained by an explicit structure of contexts, ordered by a relationof subordination.How contexts are split is essential information for the games.


Linear Logic

Lorenzen Dialogues for multiplicative Linear Logic

Rules for connectives (particle rules):Linear implication−◦ and linear negation ()⊥ are the same as forIntuitionistic and Classical Logic:to attack linear implication, one must assert the antecedent, hopingthat the opponent cannot use it to prove the consequent.The defence against such an attack then consists of a proof of theconsequent.The only way to attack the assertion A⊥ is to assert A, and beprepared to defend this assertion.Thus there is no proper defence against such an attack, but it may bepossible to counterattack the assertion of A.


Linear Logic


Rules for connectives (Particle rules):Multiplicative conjunction: TensorThe rule for tensor introduction shows that context splitting for tensoroccurs when it is asserted by the Proponent.so the dialogical particle rule will let the challenger (Opponent) choosethe context where the dialogue will proceed.


Linear Logic


Rules for connectives (Particle rules):Multiplicative Disjunction: ParThe multiplicative disjunction par will generate a splitting of contextswhen asserted by the Opponent, thus the particle rule will let thedefender choose the context.


Linear Logic

Multiplicative FILL

Assertion Attack ResponseA⊗ B ⊗L Ain context µ in subcontext ν of µ in ν

A⊗ B ⊗R Bin context µ in subcontext ν′ of µ in ν′

A ` B ? Ain context µ in µ in subcontext ν of µ

A ` B ? Bin context µ in µ in subcontext ν′ of µ

A ( B A Bin 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


Linear Logic

Full Intuitionistic Linear Logic

Important to notice that while Rahman deals exclusively with classicallinear logic, he does mention that an intuitionistic structural rule couldbe used instead.This is what we want to do for FILL.Why? Martin-Lof’s question 1991, Sweden.


Linear Logic

Games for FILL

Games particle rules for tensor, par and linear implication identical toRahmans 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...


Linear Logic

Conclusions/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 ofinvolutive negation) plus structural intuitionistic rule.These Lorenzen games give us the ability to model full intuitionisticlinear logic more easily.Preliminary work: still need to prove soundness and completenessMore 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 linearlogic). Will a bit of luck this will coincide with Mellies and Tabareau”tensor logic” ? Need to check also Hyland and Schalk’s AbstractGames, which also arise from cat semantics...


Linear Logic

Thanks!


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