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122
Computational Lambda-Calculus and Monads
, 1988
"... The λ-calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses fij-conversion to prove equivalence of programs, then a gross simplification is introduced, that may jeopardise the ap ..."
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Cited by 501 (6 self)
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The λ-calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses fij-conversion to prove equivalence of programs, then a gross simplification is introduced, that may jeopardise the applicability of theoretical results to real situations. In this paper we introduce a new calculus based on a categorical semantics for computations. This calculus provides a correct basis for proving equivalence of programs, independent from any specific computational model.
Logic Programming in a Fragment of Intuitionistic Linear Logic
, 1994
"... When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ..."
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Cited by 340 (44 self)
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When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪{D}. Thus during the bottom-up search for a cut-free proof contexts, represented as the left-hand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goal-directed interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, fromthe context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model.
Computational Interpretations of Linear Logic
- Theoretical Computer Science
, 1993
"... We study Girard's Linear Logic from the point of view of giving a concrete computational interpretation of the logic, based on the Curry-Howard isomorphism. In the case of Intuitionistic Linear Logic, this leads to a refinement of the lambda calculus, giving finer control over order of evaluati ..."
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Cited by 318 (3 self)
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We study Girard's Linear Logic from the point of view of giving a concrete computational interpretation of the logic, based on the Curry-Howard isomorphism. In the case of Intuitionistic Linear Logic, this leads to a refinement of the lambda calculus, giving finer control over order of evaluation and storage allocation, while maintaining the logical content of programs as proofs, and computation as cut-elimination.
Full Abstraction for PCF
- INFORMATION AND COMPUTATION
, 1996
"... An intensional model for the programming language PCF is described, in which the types of PCF are interpreted by games, and the terms by certain "history-free" strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable i ..."
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Cited by 254 (16 self)
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An intensional model for the programming language PCF is described, in which the types of PCF are interpreted by games, and the terms by certain "history-free" strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable in a certain simple extension of PCF. We then introduce an intrinsic preorder on strategies, and show that it satisfies some remarkable properties, such that the intrinsic preorder on function types coincides with the pointwise preorder. We then obtain an order-extensional fully abstract model of PCF by quotienting the intensional model by the intrinsic preorder. This is the first syntax-independent description of the fully abstract model for PCF. (Hyland and Ong have obtained very similar results by a somewhat different route, independently and at the same time.) We then consider the effective version of our model, and prove a Universality Theorem: every element of the effective extensional model is definable in PCF. Equivalently, every recursive strategy is definable up to observational equivalence.
Games and Full Completeness for Multiplicative Linear Logic
- JOURNAL OF SYMBOLIC LOGIC
, 1994
"... We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the den ..."
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Cited by 243 (26 self)
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We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass et al.
BI as an Assertion Language for Mutable Data Structures
, 2000
"... Reynolds has developed a logic for reasoning about mutable data structures in which the pre- and postconditions are written in an intuitionistic logic enriched with a spatial form of conjunction. We investigate the approach from the point of view of the logic BI of bunched implications of O'Hea ..."
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Cited by 191 (14 self)
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Reynolds has developed a logic for reasoning about mutable data structures in which the pre- and postconditions are written in an intuitionistic logic enriched with a spatial form of conjunction. We investigate the approach from the point of view of the logic BI of bunched implications of O'Hearn and Pym. We begin by giving a model in which the law of the excluded middle holds, thus showing that the approach is compatible with classical logic. The relationship between the intuitionistic and classical versions of the system is established by a translation, analogous to a translation from intuitionistic logic into the modal logic S4. We also consider the question of completeness of the axioms. BI's spatial implication is used to express weakest preconditions for object-component assignments, and an axiom for allocating a cons cell is shown to be complete under an interpretation of triples that allows a command to be applied to states with dangling pointers. We make this latter a feature, by incorporating an operation, and axiom, for disposing of memory. Finally, we describe a local character enjoyed by specifications in the logic, and show how this enables a class of frame axioms, which say what parts of the heap don't change, to be inferred automatically.
A games semantics for linear logic
- Ann. Pure Appl. Logic
, 1992
"... 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 ..."
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Cited by 160 (3 self)
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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.
Proof-nets: The parallel syntax for proof-theory
- Logic and Algebra
, 1996
"... The paper is mainly concerned with the extension of proof-nets to additives, for which the best known solution is presented. It proposes two cut-elimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponential ..."
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Cited by 126 (1 self)
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The paper is mainly concerned with the extension of proof-nets to additives, for which the best known solution is presented. It proposes two cut-elimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional proof-theory deals with cut-elimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However old-fashioned proof-theory, which is concerned with the ritual question: “is-that-theory-consistent? ” never really cared. The situation changed when subtle algorithmic aspects of cut-elimination became prominent: typically
Decision Problems for Propositional Linear Logic
- FOCS
, 1990
"... Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, ..."
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Cited by 110 (16 self)
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Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspace-complete. We also establish membership in np for the multiplicative fragment, np-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic.
New Foundations for the Geometry of Interaction
- Information and Computation
, 1993
"... this paper, we present a new formal embodiment of Girard's programme, with the following salient features. 1. Our formalisation is based on elementary Domain Theory rather than C --algebras. It exposes precisely what structure is required of the ambient category in order to carry out the inte ..."
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Cited by 73 (22 self)
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this paper, we present a new formal embodiment of Girard's programme, with the following salient features. 1. Our formalisation is based on elementary Domain Theory rather than C --algebras. It exposes precisely what structure is required of the ambient category in order to carry out the interpretation. Furthermore, we show how the interpretation arises from the construction of a categorical model of Linear Logic; this provides the basis for a rational reconstruction which makes the structure of the interpretation much easier to understand. 2. The key definitions in our interpretation differ from Girard's. Most notably, we replace the "execution formula" by a least fixpoint, essentially a generalisation of Kahn's semantics for feedback in dataflow networks [Kah77, KM77]. This, coupled with the use of the other distinctive construct of Domain theory, the lifting monad, enables us to interpret the whole of Linear Logic, and to prove soundness in full generality. 3. Our general notion of interpretation has simple examples, providing a suitable basis for concrete implementations. In fact, we sketch a computational interpretation of the Geometry of Interaction in terms of dataflow networks. Recall that computation in dataflow networks is asynchronous, i.e. "no global time", and proceeds by purely local "firing rules" that manipulate tokens. The further structure of this paper is as follows. In Section 2, we review the syntax of Linear Logic, and present the basic, and quite simple intuitions underlying the interpretation. In Section 3, we use these ideas to construct models of Linear Logic. In Section 4 we define the Geometry of Interaction interpretations, and how that they arise from the model constructed previously in a natural fashion. In Section 5, we give a computati...
