Quantitative game semantics for linear logic

Annals of Pure and Applied Logic, 1992

Blass, A., A game semantics for linear logic, Annals of Pure and Applied Logic 56 (1992) 183-220. We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard's linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier 'almost' will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective @ is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godel's Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989) fits with game semantics.

Annals of Pure and Applied Logic, 1997

I present a semantics for the language of first order additive-multiplicative linear logic, i.e. the language of classical first order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sen-tences=games, sentences=resources or sentences=problems, where "truth" means existence of an effective winning (resource-using, problem-solving) strategy. The paper introduces a decidable first order logic ET in the above language and gives a proof of its soundness and completeness (in the full language) with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic. The semantics presented here is very similar to Blass's game semantics (A.Blass, "A game semantics for linear logic", APAL, 56). Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be adapted to the logic induced by Blass's semantics to show its decidability (via equality to ET), which was a major problem left open in Blass's paper. The reader needs to be familiar with classical (but not necessarily linear) logic and arithmetic.

Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies -that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of "atomic" strategies and that the equality between strategies generated in such a way admits a finite axiomatization. These generators satisfy laws which are a variation of bialgebras laws, thus bridging algebra and denotational semantics in a clean and unexpected way. † This work has been supported by the ANR Invariants algébriques des systèmes informatiques (INVAL). Physical address:Équipe PPS, CNRS and Université Paris 7, 2 place Jussieu, case 7017,

Theoretical Computer Science, 2011

We give a semantic account of the execution time (i.e. the number of cut elimination steps leading to the normal form) of an untyped M ELL net. We first prove that: 1) a net is head-normalizable (i.e. normalizable at depth 0) if and only if its interpretation in the multiset based relational semantics is not empty and 2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then give a semantic measure of execution time: we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the untyped lambda-calculus.

2006

We show that context semantics can be fruitfully applied to the quantitative analysis of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proof-net as a measure of its inherent complexity: it is both an upper bound to normalization time (modulo a polynomial overhead, independently on the reduction strategy) and a lower bound to the number of steps to normal form (for certain reduction strategies). Weights are then exploited in proving strong soundness theorems for various subsystems of linear logic, namely elementary linear logic, soft linear logic and light linear logic. * The author is partially supported by PRIN project FOLLIA (2004) and ANR project NOCOST (2005).

Nordic Journal of Philosophical Logic, 1999

Semantic games are an important evaluation method for a wide range of logical languages, and are frequently resorted to when traditional methods do not easily apply. A case in point is a family of independence-friendly (IF) logics, allowing regulation over information flow in formulas, ...

Studia Logica

We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al. (in: M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of $$\text{ SeqGZL } $$ SeqGZL (A. Ciabattoni, and T. Vetterlein, Fuzzy Sets and Systems 161(14):1941–1958, 2010) and in a sequent-o...

We briefly present a new representation theory for game semantics which is very concrete: instead of playing in an arena game in which P plays the innocent strategy given by a term, the same game is played out over (a souped up version of) the abstract syntax tree of the term itself. The plays that are thus traced out are called traversals. More abstractly, traversals are the justified sequences that are obtained by performing parallel-composition less the hiding. After stating and explaining a number of Path-Traversal Correspondence Theorems, we present a tool for game semantics based on the new representation.

Lecture Notes in Computer Science, 1998

We introduce a semantics of Logic Programming based on classical Game Theory, which is proven to be sound and complete w.r.t. traditional semantics like the minimum Herbrand model and the s-semantics. This AND compositional game semantics allows a very simple characterization of the solution set of a logic program in term of approximations of the value of the game associated to it, which can also be used to capture in a very simple way the traditional "negation as failure" extensions. This novel approach to semantics opens the way to a better understanding of the mechanisms at work in parallel implementations of logic programs, and is of great pedagogical value.

Proc. 11th Intl. Workshop on …, 2006

Defeasible Logic Programming (DeLP) is a general argumentation based system for knowledge representation and reasoning. Its proof theory is based on a dialectical analysis where arguments for and against a literal interact in order to determine whether this literal is believed by a reasoning agent. The semantics GS is a declarative trivalued game-based semantics for DeLP that is sound and complete for DeLP proof theory. Complexity theory is an important tool for comparing different formalism and for helping to improve implementations whenever it is possible. In this work we address the problem of studying the complexity of some important decision problems in DeLP. Thus, we characterize the relevant decision problems in the context of DeLP and GS, and we define data and combined complexity for DeLP. Since DeLP computes every argument from a set of defeasible rules, it is of central importance to analyze the complexity of two decision problems. The first one can be defined as "Is a set of defeasible rules an argument for a literal under a defeasible logic program?". We prove that this problem is P-complete. The second decision problem is "Does there exist an argument for a literal under a defeasible logic program?". We prove that this problem is in NP. Furthermore, we study data complexity of query answering in the context of DeLP. As far as we know, data complexity has not been introduced in the context of argumentation systems.