Showing papers on "Linear logic published in 1997"

Relational Methods in Computer Science

Abstract: I. Introduction.- 1 Background Material.- II. Algebras.- 2 Relation Algebras.- 3 Heterogeneous Relation.- 4 Fork Algebras.- III. Logics.- 5 Relation Algebra and Modal Logics.- 6 Relational Formalisation of Nonclassical Logics.- 7 Linear Logic.- IV. Programs.- 8 Relational Semantics of Functional Programs.- 9 Algorithms from Relational Specifications.- 10 Programs and Datatypes.- 11 Refinement and Demonic Semantics.- 12 Tabular Representations in Relational Documents.- V. Other Application Areas.- 13 Databases.- 14 Logic, Language, and Information.- 15 Natural Language.- Bibliography (compiled by Wolfram Kahl, Thomas Stroehlein).- Symbol Table.- Addresses of Contributors.

First-order logic with two variables and unary temporal logic

Abstract: We investigate the power of first-order logic with only two variables over /spl omega/-words and finite words, a logic denoted by FO/sup 2/. We prove that FO/sup 2/ can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "sometime in the past", a logic we denote by unary-TL. Moreover, our translation from FO/sup 2/ to unary-TL converts every FO/sup 2/ formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal. While satisfiability for full linear temporal logic, as well as for unary-TL, is known to be PSPACE-complete, we prove that satisfiability for FO/sup 2/ is NEXP-complete, in sharp contrast to the fact that satisfiability for FO/sup 3/ has non-elementary computational complexity. Our NEXP time upper bound for FO/sup 2/ satisfiability has the advantage of being in terms of the quantifier depth of the input formula. It is obtained using a small model property for FO/sup 2/ of independent interest, namely: a satisfiable FO/sup 2/ formula has a model whose "size" is at most exponential in the quantifier depth of the formula. Using our translation from FO/sup 2/ to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types.

A new deconstructive logic: linear logic

Abstract: The main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.

How to declare an imperative

Abstract: How can we integrate interaction into a purely declarative language? This tutorial describes a solution to this problem based on a monad. The solution has been implemented in the functional language Haskell and the declarative language Escher. Comparisons are given with other approaches to interaction based on synchronous streams, continuations, linear logic, and side effects.

Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories.

Abstract: This note applies techniques we h a ve developed to study coherence in monoidal categories with two tensors, corresponding to the tensor{par fragment o f linear logic, to several new situations, including Hyland and de Paiva's Full Intuitionistic Linear Logic (FILL), and Lambek's Bilinear Logic (BILL). Note that the latter is a non-commutative logicc we also consider the noncommutative v ersion of FILL. The essential diierence between FILL and BILL lies in requiring that a certain tensorial strength be an isomorphism. I n a n y FILL category, it is possible to isolate a full subcategory of objects (the ucleus") for which this transformation is an isomorphism. In addition, we deene and study the appropriate categorical structure underlying the MIX rule. For all these structures, we do not restrict consideration to the \pure" logic as we a l l o w non-logical axioms. We deene the appropriate notion of proof nets for these logics, and use them to describe coherence results for the corresponding categorical structures. 0. Introduction In CS91] we i n troduced the notion of \weakly distributive category", now renamed \lin-early distributive category", in order to study the pure proof theory of the cut rule for the sequent calculus with nite sequences of formulas on both sides of the turnstile. This is generally thought of as the \classical" sequent calculus, but in fact this proof theory is not truly \classical" in any real sense, and may be thought o f a s t h e tensor{par fragment of linear logic with no negation. We w i s h e d t o s h o w h o w features could be added in a modular fashion to this basic categorical setting, in order to model the more expressive fragments of linear logic: this program is now largely complete, see CS91, BCST, BCS92], and includes the subject matter of this paper. Crucial to this program was the provision of an intrinsic characterization of the par. In classical linear logic the negation was an obstruction, for it allowed the par to be viewed as merely the de Morgan dual of the usual tensor product, and so for its special

Linear Type Theories, Semantics and Action Calculi

Abstract: In this thesis, we study linear type-theories and their semantics We present a general framework for such type-theories, and prove certain decidability properties of its equality We also present intuitionistic linear logic and Milner’s action calculi as instances of the framework, and use our results to show decidability of their respective equality judgements Firstly, we motivate our development by giving a split-context logic and typetheory, called dual intuitionistic linear logic (DILL), which is equivalent at the level of term equality to the familiar type-theory derived from intuitionistic linear logic (ILL) We give a semantics for the type-theory DILL, and prove soundness and completeness for it; we can then deduce these results for the type-theory ILL by virtue of our translation Secondly, we generalise DILL to obtain a general logic, type-theory and semantics based on an arbitrary set of operators, or general natural deduction rules We again prove soundness and completeness results, augmented in this case by an initiality result We introduce Milner’s action calculi, and present example instances of our framework which are isomorphic to them We extend this isomorphism to three significant higher-order variants of the action calculi, having functional properties, and compare the induced semantics for these action calculi with those given previously Thirdly, motivated by these functional extensions, we define higher-order instances of our general framework, which are equipped with functional structure, proceeding as before to give logic, type-theory and semantics We show that the logic and type-theory DILL arise as a higher-order instance of our general framework We then define the higher-order extension of any instance of our framework, and use a Yoneda lemma argument to show that the obvious embedding from an instance into its higher-order extension is conservative This has the corollary that the embeddings from the action calculi into the higher-order action calculi are all conservative, extending a result of Milner Finally, we introduce relations, a syntax derived from proof-nets, for our general framework, and use them to show that certain instances of our framework, including some higher-order instances, have decidable equality judgements This immediately shows that the equalities of DILL, ILL, the action calculi and the higher-order action calculi are decidable

Quantifiers, Anaphora, and Intensionality

Abstract: The relationship between Lexical-Functional Grammar (LFG) functional structures (f-structures) for sentences and their semantic interpretations can be formalized in linear logic in a way that correctly explains the observed interactions between quantifier scope ambiguity, bound anaphora and intensionality. Our linear-logic formalization of the compositional properties of quantifying expressions in natural language obviates the need for special mechanisms, such as Cooper storage, in representing the scoping possibilities of quantifying expressions. Instead, the semantic contribution of a quantifier is recorded as a linear-logic formula whose use in a proof will establish the scope of the quantifier. Different proofs can lead to different scopes. In each complete proof, the properties of linear logic ensure that quantifiers are properly scoped. The interactions between quantified NPs and intensional verbs such as ’’seek‘‘ are also accounted for in this deductive setting. A single specification in linear logic of the argument requirements of intensional verbs is sufficient to derive the correct reading predictions for intensional-verb clauses both with nonquantified and with quantified direct objects. In particular, both de dicto and de re readings are derived for quantified objects. The effects of type-raising or quantifying-in rules in other frameworks just follow here as linear-logic theorems. While our approach resembles current categorial approaches in important ways (Moortgat, 1988, 1992a; Carpenter, 1993; Morrill, 1994) it differs from them in allowing the greater compositional flexibility of categorial semantics (van Benthem, 1991) while maintaining a precise connection to syntax. As a result, we are able to provide derivations for certain readings of sentences with intensional verbs and complex direct objects whose derivation in purely categorial accounts of the syntax-semantics interface appears to require otherwise unnecessary semantic decompositions of lexical entries.

A Proof Theoretical Approach to Communication

Abstract: The paper investigates a concurrent computation model, chi calculus, in which communications resemble cut eliminations for classical proofs. The algebraic properties of the model are studied. Its relationship to sequential computation is illustrated by showing that it incorporates the operational semantics of the call-by-name lambda calculus. Practically the model has pi calculus as a submodel.

Believe it or not, AJM's games model is a model of classical linear logic

Abstract: A general category of games is constructed. A subcategory of saturated strategies, closed under all possible codings in copy games, is shown to model reduction in classical linear logic.

Lambda Calculus and Intuitionistic Linear Logic

Abstract: The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.

A logic for Petri nets

Abstract: This paper presents a translation from Petri nets to linear logic with the objective of enhancing the analysis technics of Petri nets. The net markings are denoted by formulas generated by a connective, the transitions are denoted by implicatives formulas between marking formulas and the firing operation is expressed by a sequent (i.e. a reasoning scheme). Starting from this translation, a logical calculus on transition sequences is defined, which allows to compose transitions in a sequential way as well as in a concurrent one.

Local possibilistic logic

Abstract: Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a t-norm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natural interpretation of logical connectives as operations on fuzzy sets. Due to the quantal structure of information states, we obtain a system which shares several features with (exponential-free) intuitionistic linear logic. Soundness and completeness are proved, parametrically on the choice of the t-norm operation.

Partial Order and SOS Semantics for Linear Constraint Programs

Abstract: In this paper we consider linear constraint programming (Icp), a non-monotonic extension of concurrent constraint programming (ccp) which allows to remove information. The entailment relation of a linear constraint system, in terms of which linear constraint programs are defined, is based on the main underlying idea of linear logic: hypotheses in a logical derivation represent physical resources which are consumed, once used in the entailment relation.

Subnets of proof-nets in multiplicative linear logic with MIX

Abstract: This paper studies the properties of the subnets of a proof-net for first-order Multiplicative Linear Logic without propositional constants ( MLL − ), extended with the rule of Mix: from [vdash ]Γ and [vdash ]Δ infer [vdash ]Γ, Δ. Asperti's correctness criterion and its interpretation in terms of concurrent processes are extended to the first-order case. The notions of kingdom and empire of a formula are extended from MLL − to MLL − + MIX . A new proof of the sequentialization theorem is given. As a corollary, a system of proof-nets is given for De Paiva and Hyland's Full Intuitionistic Linear Logic with Mix; this result gives a general method for translating Abramsky-style term assignments into proof-nets, and vice versa .

Timed Petri Nets and Temporal Linear Logic

Abstract: It is well known that Petri nets constitute the algebraic structure of quantales, which can be models of linear logic. As a timed extension to quantales, timed R-monoids are defined, which are constructed from timed Petri nets. Next, temporal linear logic is introduced, which has timed Petri nets as its models, i.e., whose formulas can be interpreted as sets of timed markings of a timed Petri net. Soundness of the logic with respect to timed Petri net interpretation is shown. Finally, examples show how to express properties of timed Petri nets by temporal linear logic.

From Action Calculi to Linear Logic

Abstract: Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a type-theoretic account of action calculi using the propositions-as-types paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a model-embedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi.

Connection-based proof construction in linear logic

Abstract: We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrix-based proof search procedure for this fragment and a procedure which translates the machine-found proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found.

A Formulation of Linear Logic Based on Dependency-Relations

Abstract: In this paper we describe a solution to the problem of proving cut-elimination for FILL, a variant of exponential-free and multiplicative Linear Logic originally introduced by Hyland and de Paiva. In the work of Hyland and de Paiva, a term assignment system is used to describe the intuitionistic character of FILL and a proof of cut-elimination is barely sketched. In the present paper, as well as correcting a small mistake in their work and extending the system to deal with exponentals, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut-elimination theorem. The formal system is based on a dependency-relation between formulae occurrences within a given proof and seems of independent interest. The procedure for cut-elimination applies to (multiplicative and exponential) Classical Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut-elimination proofs, is a little involved and we have not seen it published anywhere.

Connection-Based Proof Construction in Linear Logic

Abstract: We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrix-based proof search procedure for this fragment and a procedure which translates the machine-found proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found.

Resource-Distribution via Boolean Constraint (Extended Abstract)

Abstract: Proof-search (the basis of logic programming) with multiplicative inference rules, such as linear logic's ⊗R and '8L, is problematic because of the required non-deterministic splitting of resources. Similarly, searching with additive rules such as &L and ⊕R requires a non-deterministic choice between two formulae. Many strategies which resolve such non-determinism, either locally or globally, are available.

Concurrent Constraint Programming and Non-commutative Logic

Abstract: This paper presents a connection between the intuitionistic fragment of a non-commutative version of linear logic introduced by the first author (NLI) and concurrent constraint programming (CC). We refine existing logical characterizations of operational aspects of CC, by providing a logical interpretation of finer observable properties of CC programs, namely stores, successes and suspensions.

Logical Consequence: A Turn in Style

Abstract: This talk summarizes some of the things that contemporary logic and, in particular, proof theory stemming from Gentzen have to say about the notion of consequence. It starts from very elementary facts, the understanding of which doesn’t require any technical knowledge, to reach the more specialized areas of substructural logics and categorial proof theory. There, one may turn to a style of proof-theoretical investigation whose goal is not just the elimination of cut. Some tentative philosophical suggestions are drawn from this summary.