Showing papers presented at "Computer Science Logic in 1994"

A Mixed Linear and Non-Linear Logic: Proofs, Terms and Models (Extended Abstract)

Abstract: Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (‘of course’) modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satisfying certain extra conditions. Ordinary intuitionistic logic is then modelled in a cartesian closed category which arises as a full subcategory of the category of coalgebras for the comonad.

A Lambda-Calculus Structure Isomorphic to Gentzen-Style Sequent Calculus Structure

Abstract: We consider a λ-calculus for which applicative terms have no longer the form (...((u u1) u2)... un) but the form (u [u1;...;un]), for which [u1;...;un] is a list of terms. While the structure of the usual λ-calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this λ-calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cutelimination exactly matches with a β-reduction, a substitution propagation step or a concatenation computation step.

On the Interpretation of Type Theory in Locally Cartesian Closed Categories

Abstract: We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory.

Ramified Recurrence and Computational Complexity II: Substitution and Poly-Space

Abstract: We prove an applicative characterization of poly-space as the set of functions over \(\mathbb{W} = \{ 0,1\} *\)defined by ramified \(\mathbb{W}\)-recurrence with parameter substitution. Intuitively, parameter substitution allows re-use of space in ways disallowed by ramified recurrence without substitution: it permits capturing by recurrence the flow of computation backwards from accepting configurations, thereby enabling the simulation of parallel (alternating) computing. Conversely, parameter substitution can be simulated by a computation that can repeatedly bifurcate into subcomputations, i.e. by parallelism that can be captured in poly-space.

Logics For Context-Free Languages

Abstract: We define matchings, and show that they capture the essence of context-freeness. More precisely, we show that the class of context-free languages coincides with the class of those sets of strings which can be defined by sentences of the form ∃ bϕ, where ϕ is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.

Subtyping with Singleton Types

Abstract: We give syntax and a PER-model semantics for a typed λ-calculus with subtypes and singleton types. The calculus may be seen as a minimal calculus of subtyping with a simple form of dependent types. The aim is to study singleton types and to take a canny step towards more complex dependent subtyping systems. Singleton types have applications in the use of type systems for specification and program extraction: given a program P we can form the very tight specification {P} which is met uniquely by P. Singletons integrate abbreviational definitions into a type system: the hypothesis x: {M} asserts x=M. The addition of singleton types is a non-conservative extension of familiar subtyping theories. In our system, more terms are typable and previously typable terms have more (non-dependent) types.

A Subtyping for the Fisher-Honsell-Mitchell Lambda Calculus of Objects

Abstract: Labeled types and a new relation between types are added to the lambda calculus of objects as described in [6]. This relation is a trade-off between the possibility of having a restricted form of width subtyping and the features of the delegation-based language itself. The original type inference system allows both specialization of the type of an inherited method to the type of the inheriting object and static detection of errors, such as ‘ message-not-understood’. The resulting calculus is an extension of the original one. Type soundness follows from the subject reduction property.

An Intuitionistic Modal Logic with Applications to the Formal Verification of Hardware

Abstract: We investigate a novel intuitionistic modal logic, called Propositional Lax Logic, with promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness ‘up to’ behavioural constraints — a central notion in hardware verification — as a logical modality. The resulting logic is unorthodox in several respects. As a modal logic it is special since it features a single modal operator O that has a flavour both of possibility and of necessity. As for hardware verification it is special since it is an intuitionistic rather than classical logic which so far has been the basis of the great majority of approaches. Finally, its models are unusual since they feature worlds with inconsistent information and furthermore the only frame condition is that the O-frame be a subrelation of the ⊃-frame. We provide the motivation for Propositional Lax Logic and present several technical results. We investigate some of its proof-theoretic properties, and present a cut-elimination theorem for a standard Gentzen-style sequent presentation of the logic. We further show soundness and completeness for several classes of fallible two-frame Kripke models. In this framework we present a concrete and rather natural class of models from hardware verification such that the modality O models correctness up to timing constraints.

How to Lie Without Being (Easily) Convicted and the Length of Proofs in Propositional Calculus

Abstract: We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie. The second method is based on boolean valuations. For the first method, a log n + log log n -O(logloglog n) lower bound is given on the lengths of interactive proofs of certain permutation tautologies.

Decidability of Higher-Order Subtyping with Intersection Types

Abstract: The combination of higher-order subtyping with intersection types yields a typed model of object-oriented programming with multiple inheritance [11]. The target calculus, F ⋀ ω , a natural generalization of Girard's system Fω with intersection types and bounded polymorphism, is of independent interest, and is our subject of study.

Stratified Default Theories

Abstract: Default logic is a nonstandard formal system, especially suitable for knowledge representation and commonsense reasoning. In this paper we study a class of propositional default theories for which computation of extensions simplifies. We introduce the notions of stratification and strong stratification. We investigate properties of stratified default theories. We show how to determine whether a given default theory is stratified or strongly stratified and how to find the finest partition into strata. We present algorithms for computing extensions for stratified default theories and analyze their complexity.

The Girard Translation Extended with Recursion

Abstract: This paper extends Curry-Howard interpretations of Intuitionistic Logic and Intuitionistic Linear Logic rules for recursion. The resulting term languages, the λrec-calculus and the linear λrec-calculus respectively, are given sound categorical interpretations. The embedding of proofs of Intuitionistic Logic into proofs of Intuitionistic Linear Logic given by the Girard Translation is extended with the rules for recursion such that an embedding of terms of the λrec-calculus into terms of the linear λrec-calculus is induced via the extended Curry-Howard isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations.

Resolution Games and Non-Liftable Resolution Orderings

Abstract: We prove the completeness of the combination of ordered resolution and factoring for a large class of non-liftable orderings, without the need for any additional rules, as for example saturation. This is possible because of a new proof method which avoids making use of the standard ordered lifting theorem. This new proof method is based on a new technique, which we call the resolution game.

A Sound Metalogical Semantics for Input/Output Effects

Abstract: We study the longstanding problem of semantics for input/output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove it is a congruence using Howe's method.

An Algebraic View of Structural Induction

Abstract: We propose a uniform, category-theoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T: \(\mathbb{B} \to \mathbb{B}\)on a bicartesian/distributive category \(\mathbb{B}\). Regarding a predicate logic as a fibration p: \(\mathbb{P} \to \mathbb{B}\)over \(\mathbb{B}\), we consider a logical predicate lifting of T to the total category \(\mathbb{P}\). Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that the ‘truth’ predicate functor ⊤: \(\mathbb{B} \to \mathbb{P}\)preserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the ‘stability’ property of the induction principle.

Monadic Logical Definability of NP-Complete Problems

Abstract: It is well known that monadic second-order logic with linear order captures exactly regular languages. On the other hand, if addition is allowed, then J.F.Lynch has proved that existential monadic secondorder logic captures at least all the languages in NTIME(n), and then expresses some NP-complete languages (e.g. knapsack problem).

Logic Programming in Tau Categories

Abstract: Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally non-ground character, and the uniform way in which such languages have been extended to typed domains subject to constraints, suggest that a categorical treatment of constraint domains, of programming syntax and of semantics may be closer in spirit to declarative programming than conventional set theoretic semantics.

Algorithmic Aspects of Propositional Tense Logics

Abstract: It is well-known that tense logics can be treated as logics of computations if we understand "moments of time" as "states of a computing system". In this paper we are concerned with tense logics in the traditional Priorean language having two non-classical connectives: G ("it will always be the case that . . . "), H ("it was always the case that . . . "), and their duals: F = -~G-~, P = ~ H . First let us recall the definition of the minimal lense logic (denoted by K t 1). A x i o m s : Classical tautologies and the formulas G(p --+ q) ~ (Gp ~ Gq), H(p --+ q) ~ (Hp ~ gq), F g p ~ p, PGp ~ p. I n f e r e n c e ru les : Modus Ponens; Generalization (t9 ~ FG 9 and ~9 ==~ ~Hg) ; Substitution (of variables by formulas). In general , by a lense logic we mean an extension of K t by some new axioms; if the number of new axioms is finite, the logic is called a tense calculus. E.g. the calculus K 4 t is obtained by adding Gp --+ GGp (this is written as follows: K 4 t = K t + Gp --~ GGp). Semantics of tense logics is given by Kripke frames. The latter is a nonempty set (of "moments of time" or of "states") with a binary relation ("earlier than") . We get a Kripke model on a frame (W, R) if for every moment w E W and for every formula 9 we say whether 9 is true at w (w ~ 9) or not, and also if the following holds: w ~ 9 A r iff w ~ 9 and w ~ r etc. (similarly for the other Boolean connectives); w ~ G 9 iff Vv(wRv ~ v ~ 9); w ~ H 9 iffVv(vRw ~ v ~ 9). A formula is called true in a Kripke model if it is true at every moment; it is called valid in a frame if it is true in every model on this frame. It is well-known that K t is exactly the set of formulas valid in every frame, K 4 t is the set of formulas valid in every transitive frame. For proofs and motivations of tense logics the reader may consult [5]. The main topic of our paper is undecidability. In Section 2.2 we construct undecidable tense calculi which are axiomatized by formulas of a very simple kind

First-Order Spectra with One Binary Predicate

Abstract: The spectrum, Sp(ϕ), of a sentence ϕ is the set of cardinalities of finite structures which satisfy ϕ. We prove that any set of integers which is in Func 1 ∞ i.e. in the class of spectra of first-order sentences of type containing only unary function symbols is also in BIN1 i.e. in the class of spectra of first-order sentences of type involving only a single binary relation.

Subrecursion as a Basis for a Feasible Programming Language

Abstract: We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of a pairing function because in this way we can explore the amazing coding powers of S-expressions of LISP within the domain of natural numbers. We introduce three Grzegorczyk-like hierarchies based on pairing and characterize them both in terms of Grzegorczyk hierarchy and computational complexity.

A Homomorphism Concepts for omega-Regularity

Abstract: The Myhill-Nerode Theorem (that for any regular language, there is a canonical recognizing device) is of paramount importance for the computational handling of many formalisms about finite words.

Powerdomains, Powerstructures and Fairness

Abstract: We introduce the framework of powerstructures for comparing models of non-determinism and concurrency, and we show that in this context the Plotkin powerdomain plot(D) [6] naturally occurs as a quotient of a refined and generalized player model ipf (D), following Moschovakis [2, 3]. On the other hand, Plotkin's domains for countable non-determinism plotω(D) [7] are not comparable with these structures, as they cannot be realized concretely in the powerset of D.

Is First Order Contained in an Initial Segment of PTIME

Abstract: By “initial segments of P” we mean classes DTime(n k ). The question of whether for any fixed signature the first-order definable predicates in finite models of this signature are all in an initial segment of P is shown to be related to other intriguing open problems in complexity theory and logic, like P=PSpace.

Lambda Representation of Operations Between Fifferent Term Algebras

Abstract: There is a natural isomorphism identifying second order types of the simple typed λ calculus with free homogeneous term algebras Let τA and τB be types representing algebras A and B respectively Any closed term of the type τA → τB represents a computable function between algebras A and B The problem investigated in the paper is to find and characterize the set of all λ definable functions between structures A and B The problem is presented in a more general setting If algebrasA1,, A n ,B are represented respectively by second order types \(\tau ^{A_l } ,,\tau ^{A_n } \), τB then \(\tau ^{A_l } \)→ ((\(\tau ^{A_n } \)→ τB) is a type of functions from the product A1×xAn into algebra B Any closed term of this type is a representation of algorithm which transforms the tuple of terms of types \(\tau ^{A_l } ,,\tau ^{A_n } \) respectively into a term of type τB, which represents an object in algebra B (see [BoB85]) The problem investigated in the paper is to find an effective computational characteristic of the λ definable functions between arbitrary free algebras and the expressiveness of such transformations As an example we will consider λ definability between well known free structures such as: numbers, words and trees The result obtained in the paper is an extension of the results concerning λ definability in various free structures described in [Sch75] [Sta79] [Lei89] [Zai87] [Zai90] and [Zai91]

A homomorphism concept for ω-regularity

Abstract: The Myhill-Nerode Theorem (that for any regular language, there is a canonical recognizing device) is of paramount importance for the computational handling of many formalisms about finite words

Reasoning and Rewriting with Set-Relations I: Ground Completeness

Abstract: The paper investigates reasoning with set-relations: intersection, inclusion and identity of 1-element sets. A language is introduced which, interpreted in a multi-algebraic semantics, allows one to specify such relations. An inference system is given and shown sound and refutationally ground-complete for a particular proof strategy which selects only maximal literals from the premise clauses. Each of the introduced set-relations satisfies only two among the three properties of the equivalence relations — we study rewriting with such non-equivalence relations and point out differences from the equational case. As a corollary of the main ground-completeness theorem we obtain ground-completeness of the introduced rewriting technique.

Semi-Unification and Generalizations of a Particularly Simple Form

Abstract: This paper describes a criterion for the existence of generalizations of a particularly simple form given complex terms in short proofs within schematic theories: The soundness of replacing single quantifiers, which bind variables in schema instances, by blocks of quantifiers of the same type. The criterion is shown to be necessary in general and sufficient for languages consisting of monadic function symbols and constants. The proof is mainly based on the existence of most general solutions for solvable semi-unification problems.

Convergence and 0–1 laws for L∞,ωk under arbitrary measures

Abstract: We prove some general results about the existence of 0–1 and convergence laws for L ∞,ω k and L ∞,ω k on classes of finite structures equipped with a sequence of arbitrary probability measures {μ n }, as well as a few results for particular classes First, two new proofs of the characterization theorem of Kolaitis and Vardi [9] are given Then this theorem is generalized to obtain a characterization of the existence of L ∞,ω ω convergence laws on a class with arbitrary measure We use this theorem to obtain some results about the nonexistence of L ∞,ω ω convergence laws for particular classes of structures We also disprove a conjecture of Tyszkiewicz [16] relating the existence of L ∞,ω ω and MSO 0–1 laws on classes of structures with arbitrary measures