An infinite-game semantics for well-founded negation in logic programming (original) (raw)

General Logic Programs as Infinite Games

M.H. van Emden introduced a simple game semantics for definite logic programs 3 . Recently [RW05,GRW05], the authors extended this game to apply to logic programs with negation. Moreover, under the assumption that the programs have a finite number of rules, it was demonstrated in [RW05,GRW05] that the game is equivalent to the well-founded semantics of negation. In this paper we present workin-progress towards demonstrating that the game of [RW05,GRW05] is equivalent to the well-founded semantics even in the case of programs that have a countably infinite number of rules. We argue however that in this case the proof of correctness has to be more involved. More specifically, in order to demonstrate that the game is correct one has to define a refined game in which each of the two players in his first move makes a bet in the form of a countable ordinal. Each ordinal can be considered as a kind of clock that imposes a "time limit" to the moves of the corresponding player. We argue that this refined game can be used to give the proof of correctness for the countably infinite case.

An Infinite-Valued Semantics for Logic Programs with Negation

Lecture Notes in Computer Science, 2002

We give a purely model-theoretic (denotational) characterization of the semantics of logic programs with negation allowed in clause bodies. In our semantics (the first of its kind) the meaning of a program is, as in the classical case, the unique minimum model in a programindependent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between False (the minimum element) and True (the maximum), with a Zero element in the middle. The truth values below Zero are ordered like the countable ordinals. The values above Zero have exactly the reverse order. Negation is interpreted as reflection about Zero followed by a step towards Zero; the only truth value that remains unaffected by negation is Zero. We show that every program has a unique minimum model MP , and that this model can be constructed with a TP iteration which proceeds through the countable ordinals. Furthermore, collapsing the true and false values of the infinite-valued model MP to (the classical) True and False, gives a three-valued model identical to the well-founded one.

A constructive game semantics for the language of linear logic

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.

Classical Negation and Game-Theoretical Semantics

Notre Dame Journal of Formal Logic, 2014

Typical applications of Hintikka's game-theoretical semantics (GTS) give rise to semantic attributes-truth, falsity-expressible in the † 1 1-fragment of second-order logic. Actually a much more general notion of semantic attribute is motivated by strategic considerations. When identifying such a generalization, the notion of classical negation plays a crucial role. We study two languages, L 1 and L 2 , in both of which two negation signs are available: + and. The latter is the usual GTS negation which transposes the players' roles, while the former will be interpreted via the notion of mode. Logic L 1 extends independence-friendly (IF) logic; + behaves as classical negation in L 1. Logic L 2 extends L 1 , and it is shown to capture the † 2 1-fragment of third-order logic. Consequently the classical negation remains inexpressible in L 2 .

Well founded semantics for logic programs with explicit negation

European Conf. on Artificial Intelligence (ECAI'92), 1992

The aim of this paper is to provide a semantics for general logic programs (with negation by default) extended with explicit negation, subsuming well founded semantics 22]. The Well Founded semantics for extended logic programs (WFSX) is expressible by a default theory semantics we have devised 11]. This relationship improves the cross{fertilization between logic programs and default theories, since we generalize previous results concerning their relationship 3, 4, 7, 1, 2], and there is an increasing use of logic programming with explicit negation for nonmonotonic reasoning 7, 15, 16, 13, 23]. It also clari es the meaning of logic programs combining both explicit negation and negation by default. In particular, it shows that explicit negation corresponds exactly to classical negation in the default theory, and elucidates the use of rules in logic programs. Like defaults, rules are unidirectional, so their contrapositives are not implicit; the rule connective, , is not material implication, but has rather the avour of an inference rule, like defaults. It is worth noting that existing top{down procedures for well-founded semantics without explicit negation 24, 14] can be easily adapted to the semantics we are proposing for extended programs. This issue is only brie y considered here and will be the subject of a forthcoming report.

Extended Game-Theoretical Semantics

Between Logic and Reality, 2011

A new version of Game-Theoretical Semantics (GTS) is put forward where game rules are extended to the non-logical constants of sentences. The resulting theory, together with a refinement of our criteria of identity for functions, provide the technical basis for a game-based conception of linguistic meaning and interpretation.

A 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.

Presentation of a Game Semantics for First-Order Propositional Logic

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,

Games and Trees in Infinitary Logic: A Survey

Quantifiers: Logics, Models and Computation, 1995

We describe the work and underlying ideas of the Helsinki Logic Group in infinitary logic. The central idea is to use trees and Ehrenfeucht-Fraïssé games to measure differences between uncountable models. These differences can be expressed by sentences of so-called infinitely deep languages. This study has ramified to purely set-theoretical problems related to properties of trees, descriptive set theory in ω 1 ω 1 , detailed study of transfinite Ehrenfeucht-Fraïssé games, new constructions of uncountable models, non-well-founded induction, infinitely deep languages, non-structure theorems, and stability theory. The aim of this paper is to give an overview of the underlying ideas of this reasearch together with a survey of the main results.

A purely model-theoretic semantics for disjunctive logic programs with negation

2007

We present a purely model-theoretic semantics for disjunctive logic programs with negation, building on the infinite-valued approach recently introduced for normal logic programs [9]. In particular, we show that every disjunctive logic program with negation has a non-empty set of minimal infinite-valued models. Moreover, we show that the infinite-valued semantics can be equivalently defined using Kripke models, allowing us to prove some properties of the new semantics more concisely.