# Showing papers in "Mathematical Structures in Computer Science in 2001"

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TL;DR: This work reconsiders the foundations of modal logic, following Martin-Löf's methodology of distinguishing judgments from propositions, and gives a new presentation of lax logic, finding that the lax modality is already expressible using possibility and necessity.

Abstract: We reconsider the foundations of modal logic, following Martin-Lof's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's monadic metalanguage.

348 citations

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TL;DR: An-fang was near a city, the only living city with a pre-atomic name, and the headquarters of the People Programmer was at An-Fang, and there the mistake happened: A ruby trembled and a diamond noted the error.

Abstract: Go back to An-fang, the Peace Square at An-Fang, the Beginning Place at An-Fang, where all things start (…) An-Fang was near a city, the only living city with a pre-atomic name (…) The headquarters of the People Programmer was at An-Fang, and there the mistake happened: A ruby trembled. Two tourmaline nets failed to rectify the laser beam. A diamond noted the error. Both the error and the correction went into the general computer. Cordwainer Smith The Dead Lady of Clown Town, 1964.

210 citations

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TL;DR: It is proved, via a categorical structure theorem, that the categorical semantics is equivalent to a CPS semantics in the style of Hofmann and Streicher, and that the call-by-name λμ-calculus forms an internal language for the dual co-control categories.

Abstract: We give a categorical semantics to the call-by-name and call-by-value versions of Parigot's λμ-calculus with disjunction types. We introduce the class of control categories, which combine a cartesian-closed structure with a premonoidal structure in the sense of Power and Robinson. We prove, via a categorical structure theorem, that the categorical semantics is equivalent to a CPS semantics in the style of Hofmann and Streicher. We show that the call-by-name λμ-calculus forms an internal language for control categories, and that the call-by-value λμ-calculus forms an internal language for the dual co-control categories. As a corollary, we obtain a syntactic duality result: there exist syntactic translations between call-by-name and call-by-value that are mutually inverse and preserve the operational semantics. This answers a question of Streicher and Reus.

168 citations

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TL;DR: It is shown that injective matching provides additional expressiveness in two respects: for generating graph languages by grammars without non-terminals and for computing graph functions by convergent graph transformation systems.

Abstract: In this paper we investigate and compare four variants of the double-pushout approach to graph transformation. As well as the traditional approach with arbitrary matching and injective right-hand morphisms, we consider three variations by employing injective matching and/or arbitrary right-hand morphisms in rules. We show that injective matching provides additional expressiveness in two respects: for generating graph languages by grammars without non-terminals and for computing graph functions by convergent graph transformation systems. Then we clarify for each of the three variations whether the well-known commutativity, parallelism and concurrency theorems are still valid and – where this is not the case – give modified results. In particular, for the most general approach with injective matching and arbitrary right-hand morphisms, we establish sequential and parallel commutativity by appropriately strengthening sequential and parallel independence.

147 citations

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TL;DR: It is shown that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory, and a step-by-step simulation of higher-order resolution is given, showing that the well-studied improvements of proof search for first- order logic could be reused at no cost for higher- order automated deduction.

Abstract: We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.

81 citations

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TL;DR: The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening.

Abstract: Since Mellies showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented.

49 citations

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TL;DR: Horn linear logic is introduced as a comprehensive logical system capable of handling the typical AI problem of making a plan of the actions to be performed by a robot so that he could get into a set of final situations, if he started with a certain initial situation.

Abstract: We introduce Horn linear logic as a comprehensive logical system capable of handling the typical AI problem of making a plan of the actions to be performed by a robot so that he could get into a set of final situations, if he started with a certain initial situation Contrary to undecidability of propositional Horn linear logic, the planning problem is proved to be decidable for a reasonably wide class of natural robot systems
The planning problem is proved to be EXPTIME -complete for the robot systems that allow actions with non-deterministic effects Fixing a finite signature, that is a finite set of predicates and their finite domains, we get a polynomial time procedure of making plans for the robot system over this signature
The planning complexity is reduced to PSPACE for the robot systems with only pure deterministic actions
As honest numerical parameters in our algorithms we invoke the length of description of a planning task ‘from W to Z˜’ and the Kolmogorov descriptive complexity of Ax T, a set of possible actions

34 citations

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TL;DR: This special issue of Mathematical Structures in Computer Science is dedicated to the theory and applications of explicit substitutions, which have attracted a growing community of researchers in the last decade, especially in the study of explicit substitution as a means of bridging the gap between theory and practice in the implementation of programming languages.

Abstract: This special issue of Mathematical Structures in Computer Science is dedicated to the theory and applications of explicit substitutions, which have attracted a growing community of researchers in the last decade, especially in the study of explicit substitutions as a means of bridging the gap between theory and practice in the implementation of programming languages, as well as theorem provers and proof checkers.Such implementations typically rely on formal calculi defined using implicit substitution operations that are left at the meta-level, so that they need to turn these meta-level operations into efficient executable code, and this is often fairly intricate and distant from the formal calculi. This causes a significant gap between theory and practice.Explicit substitutions considerably reduce this gap by bringing the meta-level operations down to the object-level calculus – where they are represented explicitly – allowing us in this way to give formal and robust models for the techniques actually used in implementations, and providing at the same time a more flexible tool for controlling the intermediate steps of evaluation.All the papers in this issue were invited on the basis of their quality and relevance to the domain, and subjected to the refereeing process of MSCS. Most of them are substantially expanded and revised versions of work originally presented at Westapp'98 and Westapp'99, the first and second ‘Workshop on Explicit Substitutions: Theory and Applications to Programs and Proofs’, which were held in conjunction with RTA'98 in Tsukuba, Japan, and with Floc'99 in Trento, Italy, respectively.As guest editor, I would like to express my warm thanks both to the authors, for their high-quality contributions to this special issue, and to the referees, whose scientific role was essential in improving the presentation of these contributions.

33 citations

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TL;DR: It turns out that associated with the Russell–Prawitz representation is a lax modal operator, which the paper compares with other possible representations of intuitionistic logic, and that any laxModal operator can be used to give a translation of intuitionists logic into itself that generalises both the double negation interpretation.

Abstract: In his 1903, Principles of Mathematics, Bertrand Russell mentioned possible definitions of conjunction, disjunction, negation and existential quantification in terms of implication and universal quantification, exploiting impredicative universal quantifiers over all propositions. In his 1965 Ph.D. thesis Dag Prawitz showed that these definitions hold in intuitionistic second order logic. More recently, these definitions have been used to represent logic in various impredicative type theories. This treatment of logic is distinct from the more standard Curry–Howard representation of logic in a dependent type theory.The main aim of this paper is to compare, in a purely logical, non type-theoretic setting, this Russell–Prawitz representation of intuitionistic logic with other possible representations. It turns out that associated with the Russell–Prawitz representation is a lax modal operator, which we call the Russell–Prawitz modality, and that any lax modal operator can be used to give a translation of intuitionistic logic into itself that generalises both the double negation interpretation, double negation being a paradigm example of a lax modality, and the Russell–Prawitz representation.

28 citations

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TL;DR: The system enjoys the decidability of typability, soundness of typed reduction with respect to the typing rules, the Church–Rosser and strong normalization properties and it is a conservative extension over the simply typed λ-calculus.

Abstract: This paper presents an extension of the simply typed λ-calculus that allows iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax. This calculus aims at being the kernel for a type theory in which the user will be able to formalize logics or formal systems using the LF methodology, while taking advantage of new induction and recursion principles, extending the principles available in a calculus such as the Calculus of Inductive Constructions. The key idea of our system is the use of modal logic S4. We present here the system, its typing rules and reduction rules. The system enjoys the decidability of typability, soundness of typed reduction with respect to the typing rules, the Church–Rosser and strong normalization properties and it is a conservative extension over the simply typed λ-calculus. These properties entail the preservation of the adequacy of encodings.

24 citations

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TL;DR: The method to associate calculi of proof terms and rewrite rules with cut elimination procedures for logical deduction systems (i.e., Gentzen-style sequent calculi) in the case of intuitionistic logic is introduced and various subject reduction, termination, and confluence properties are proved.

Abstract: We introduce a method to associate calculi of proof terms and rewrite rules with cut elimination procedures for logical deduction systems (i.e., Gentzen-style sequent calculi) in the case of intuitionistic logic. We illustrate this method using two different versions of the cut rule for a variant of the intuitionistic fragment of Kleene's logical deduction system G3.Our systems are in fact calculi of explicit substitution, where the cut rule introduces an explicit substitution and the left-→ rule introduces a binding of the result of a function application. Cut propagation steps of cut elimination correspond to propagation of explicit substitutions, and propagation of weakening (to eliminate it) corresponds to propagation of index-updating operations. We prove various subject reduction, termination, and confluence properties for our calculi.Our calculi improve on some earlier calculi for logical deduction systems in a number of ways. By using de Bruijn indices, our calculi qualify as first-order term rewriting systems (TRS's), allowing us to use correctly certain results for TRS's about termination. Unlike in some other calculi, each of our calculi has only one cut rule and we do not need unusual features of sequents.We show that the substitution and index-updating mechanisms of our calculi work the same way as the substitution and index-updating mechanisms of Kamareddine and Rios' λs and λt, two well-known systems of explicit substitution for the standard λ-calculus. By a change in the format of sequents, we obtain similar results for a known λ-calculus with variables and explicit substitutions, Rose's λbxgc.

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TL;DR: Ochmański's theorem on recognizable languages in free partially commutative monoids is obtained as a consequence of defining the class of divisibility monoids that arise as quotients of the free monoid Σ* modulo certain equations of the form ab = cd.

Abstract: We define the class of divisibility monoids that arise as quotients of the free monoid Σa modulo certain equations of the form ab = cd. These form a much larger class than free partially commutative monoids, and we show, under certain assumptions, that the recognizable languages in these divisibility monoids coincide with c-rational languages. The proofs rely on Ramsey's theorem, distributive lattice theory and on Hashigushi's rank function generalized to these monoids. We obtain Ochmanski's theorem on recognizable languages in free partially commutative monoids as a consequence.

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TL;DR: To complete the study of Fes, the property of subject reduction is shown to hold by extending type assignments of the typing rules to allow non-pure types (types with possible occurrences of the type substitution operator).

Abstract: We study perpetuality in the calculus of explicit substitutions λx. A reduction is called perpetual if it preserves the possibility of infinite reduction sequences. We then take a look at applications of this study: an inductive characterization of the λx-strongly normalizing terms, two perpetual reduction strategies for λx and finally a proof of strong normalization of a polymorphic lambda calculus with explicit substitutions Fes. To complete the study of Fes, the property of subject reduction is shown to hold by extending type assignments of the typing rules to allow non-pure types (types with possible occurrences of the type substitution operator).

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Open University

^{1}TL;DR: Certain ‘Finite Structure Conditions’ on a geometric theory are shown to be sufficient for its classifying topos to be a presheaf topos, and to hold for the theory of strongly algebraic information systems and some variants, as well as for some other theories already known to be classified by presheAF toposes.

Abstract: Certain ‘Finite Structure Conditions’ on a geometric theory are shown to be sufficient for its classifying topos to be a presheaf topos. The conditions assert that every homomorphism from a finite structure of the theory to a model factors via a finite model, and they hold in cases where the finitely presentable models are all finite.
The conditions are shown to hold for the theory of strongly algebraic (or SFP) information systems and some variants, as well as for some other theories already known to be classified by presheaf toposes.
The work adheres to geometric constructivism throughout, and in consequence provides ‘topical’ categories of domains (internal in the category of toposes and geometric morphisms) with an analogue of Plotkin's double characterization of strongly algebraic domains, by sets of minimal upper bounds and by sequences of finite posets.

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TL;DR: This work investigates how the topology of such nets relates to the number of exchange rules in corresponding proofs by studying orientable surfaces with boundary.

Abstract: Proofnets may be seen as orientable surfaces with boundary. We investigate how the topology of such nets relates to the number of exchange rules in corresponding proofs.

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TL;DR: A Gentzen sequent calculus for Lax Logic is presented, the proofs in which correspond naturally in a 1–1 way to the normal natural deductions for the logic.

Abstract: A Gentzen sequent calculus for Lax Logic is presented, the proofs in which correspond naturally in a 1–1 way to the normal natural deductions for the logic. The propositional fragment of this calculus is used as the basis for another calculus that uses a history mechanism in order to give a decision procedure for propositional Lax Logic.

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TL;DR: In this system, meta-variables, as well as substitutions, are first-class objects and it is shown that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence, and weak normalization.

Abstract: We present a dependent-type system for a λ-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence, and weak normalization.

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TL;DR: It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are, and a more complicated extension is proposed for which subject reduction does hold in general.

Abstract: We define an extension of pure type systems with explicit substitution. We show that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker, though still useful, subject reduction property is established. A more complicated extension is proposed for which subject reduction does hold in general.

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TL;DR: This special issue of MSCS reports on some recent advances in the area of intuitionistic modal type theories and their application to Computer Science.

Abstract: This special issue reports on some recent advances in the area of intuitionistic modal type theories and their application to Computer Science. It collects a selection of papers presented at the Logic in Computer Science (LICS'99) satellite workshop on Intuitionistic Modal Logics and Applications (IMLA'99) held at Trento, Italy in July 1999. All of the contributors to this one day workshop, which was widely attended, were invited to submit a full and revised version of their papers to this special issue of MSCS. The selection was based on a second round of peer reviewing.

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TL;DR: The theory presented here is a new, radical step in the research program that started with linear logic and aiming at an interactive, resource and space conscious account of reasoning and programming.

Abstract: The theory presented here is a new, radical step in the research program that started with linear logic and aiming at an interactive, resource and space conscious account of reasoning and programming.

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TL;DR: The notion of uniformly semiconstructive calculus is discussed, the information extraction mechanism is introduced and the mechanism is applied to two calculi extending Intuitionistic Arithmetic.

Abstract: In this abstract we will describe research in progress on the problem of extracting information from proofs. Here we will concentrate our attention on semiconstructive calculi, which is a kind of calculus that is of interest in the framework of program synthesis and formal verification. We will discuss the notion of uniformly semiconstructive calculus, introduce our information extraction mechanism and apply it to two calculi extending Intuitionistic Arithmetic.

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TL;DR: Two theorems are important for design of an interpreter of a ν act, which is a representation of mathematical action, that is, such formulae do not have an equivalent first-order formula.

Abstract: This paper concerns the elimination of higher type quantifiers and gives two theorems. The first theorem shows that quantifiers in formulae of a specific form can be eliminated. The second theorem shows that quantifiers in formulae of a similar form cannot be eliminated, that is, such formulae do not have an equivalent first-order formula. The proof is based on the Ehrenfeucht game. These theorems are important for design of an interpreter of a ν act, which is a representation of mathematical action. Moreover, even if the universe is assumed to be finite, these theorems hold.

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TL;DR: This paper considers the propositional modal mu-calculus, a logic proposed by Kozen in 1983, and exhibits a new formula requiring 3 alternations, independent of the technique of Bradfield, based on a new kind of game on infinite trees.

Abstract: We consider the propositional modal mu-calculus, a logic proposed by Kozen in 1983. In this logic two operators μ and v are present, which express the least and greatest fixpoints of monotone operators on sets. Bradfield in 1998 proved for any n the existence of a mu-calculus formula that requires n alternations of μ and v. In this paper we consider the particular case n = 3 and we exhibit a new formula requiring 3 alternations. Our proof is independent of the technique of Bradfield, and is based on a new kind of game on infinite trees.