Showing papers presented at "Computer Science Logic in 2001"

Local Reasoning about Programs that Alter Data Structures

Abstract: We describe an extension of Hoare's logic for reasoning about programs that alter data structures. We consider a low-level storage model based on a heap with associated lookup, update, allocation and deallocation operations, and unrestricted address arithmetic. The assertion language is based on a possible worlds model of the logic of bunched implications, and includes spatial conjunction and implication connectives alongside those of classical logic. Heap operations are axiomatized using what we call the "small axioms", each of which mentions only those cells accessed by a particular command. Through these and a number of examples we show that the formalism supports local reasoning: A specification and proof can concentrate on only those cells in memory that a program accesses. This paper builds on earlier work by Burstall, Reynolds, Ishtiaq and O'Hearn on reasoning about data structures.

Locus Solum: From the Rules of Logic to the Logic of Rules

Abstract: Logic is no longer about a preexisting external reality, but about its own protocols, its own geometry. Typically the negation is not about saying "NOT", but about the mirror, the duality "I" vs. "the world"... The new approach encompasses the old one, typically if "I" win, "the world" loses, i.e., wins "NOT". When logical artifacts are identified with their own rules of production, LOCATIVE phenomenons arise. In particular, one realises that usual logic (including linear logic) is SPIRITUAL, i.e., up to isomorphism. But there is a deeper locative level, with indeed a more regular structure. Typically the usual (additive) conjunction has the value of categorical product in usual logic, and enjoys commutativity, associativity, etc. up to isomorphism. In ludics, what corresponds is a plain intersection G ∩ H, which is really associative, commutative, etc. (no isomorphisms); it contains the usual conjunction as a delocalised case ϕ(G) ∩ ψ(H). Incidentally this shows that the categorical view of logic - if very useful - is wrong... Nature abhors an isomorphism! LUDICS is a monist approach to logic-without this nonsense distinction syntax/semantics/meta - just plain logical artifacts, period.

Non-commutativity and MELL in the Calculus of Structures

Abstract: We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a self-dual noncommutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.

Categorical and Kripke Semantics for Constructive S4 Modal Logic

Abstract: We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

The Decidability of Model Checking Mobile Ambients

Abstract: The ambient calculus is a formalism for describing the mobility of both software and hardware. The ambient logic is a modal logic designed to specify properties of distributed and mobile computations programmed in the ambient calculus. In this paper we investigate the border between decidable and undecidable cases of model checking mobile ambients for some fragments of the ambient calculus and the ambient logic. Recently, Cardelli and Gordon presented a model-checking algorithm for a fragment of the calculus (without name restriction and without replication) against a fragment of the logic (without composition adjunct) and asked the question, whether this algorithm could be extended to include either replication in the calculus or composition adjunct in the logic. Here we answer this question negatively: it is not possible to extend the algorithm, because each of these extensions leads to undecidability of the problem. On the other hand, we extend the algorithm to the calculus with name restriction and logic with new constructs for reasoning about restricted names.

Modal Logic and the Two-Variable Fragment

Abstract: We introduce a modal language L which is obtained from standard modal logic by adding the difference operator and modal operators interpreted by boolean combinations and the converse of accessibility relations. It is proved that L has the same expressive power as the two-variable fragment FO^2 of first-order logic but speaks less succinctly about relational structures: if the number of relations is bounded, then L-satisfiability is ExpTime-complete but FO^2 satisfiability is NExpTime-complete. We indicate that the relation between L and FO^2 provides a general framework for comparing modal and temporal languages with first-order languages.

Beyond Regularity: Equational Tree Automata for Associative and Commutative Theories

Abstract: A new tree automata framework, called equational tree automata, is presented. In the newly introduced setting, congruence closures of recognizable tree languages are recognizable. Furthermore, we prove that in certain useful cases, recognizable tree languages are closed under union and intersection. To compare with early related work, e.g. [7], we discuss the relationship between linear bounded automata and equational tree automata. As a consequence, we obtain some (un)decidability results. We further present a hierarchy of 4 classes of tree languages.

Uniform Derivation of Decision Procedures by Superposition

Abstract: We show how a well-known superposition-based inference system for first-order equational logic can be used almost directly as a decision procedure for various theories including lists, arrays, extensional arrays and combinations of them. We also give a superposition-based decision procedure for homomorphism.

Applications of Alfred Tarski's Ideas in Database Theory

Abstract: Many ideas of Alfred Tarski - one of the founders of modern logic - find application in database theory. We survey some of them with no attempt at comprehensiveness. Topics discussed include the genericity of database queries; the relational algebra, the Tarskian definition of truth for the relational calculus, and cylindric algebras; relation algebras and computationally complete query languages; real polynomial constraint databases; and geometrical query languages.

Well-Founded Recursive Relations

Abstract: We give a short constructive proof of the fact that certain binary relations > are well-founded, given a lifting ≫ a la Ferreira-Zantema and a well-founded relation ▹. This construction generalizes several variants of the recursive path ordering on terms and of the Knuth-Bendix ordering. It also applies to other domains, of graphs, of infinite terms, of word and tree automata notably. We then extend this construction further; the resulting family of well-founded relations generalizes Jouannaud and Rubio's higher-order recursive path orderings.

An Effective Extension of the Wagner Hierarchy to Blind Counter Automata

Abstract: The extension of the Wagner hierarchy to blind counter automata accepting infinite words with a Muller acceptance condition is effective. We determine precisely this hierarchy.

Markov's Principle for Propositional Type Theory

Abstract: In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov's principle from constructive recursive mathematics. Markov's principle is especially useful for proving termination of specific computations. Allowing a limited form of classical reasoning we get more powerful resulting system which remains constructive and valid in the standard constructive semantics of a type theory. We also show that this principle can be formulated and used in a propositional fragment of a type theory.

Intersection Logic

Abstract: The intersection type assignment system IT uses the formulas of the negative fragment of the predicate calculus (LJ) as types for the λ-terms. However, the deductions of IT only correspond to the proper sub-set of the derivations of LJ, obtained by imposinga metatheoretic condition about the use of the conjunction of LJ. This paper proposes a logical foundation for IT. This is done by introducing a logic IL. Intuitively, a derivation of IL is a set of derivations in LJ such that the derivations in the set can be thought of as writable in parallel. This way of looking at LJ, by means of IL, allows to transform the metatheoretic condition, mentioned above, into a purely structural property of IL. The relation between IL and LJ surely has a first main benefit: the strong normalization of LJ directly implies the same property on IL, which translates in a very simple proof of the strong normalizability of the λ-terms typable with IT.

Recursion for Higher-Order Encodings

Abstract: This paper describes a calculus of partial recursive functions that range over arbitrary andp ossibly higher-order objects in LF [HHP93]. Its most novel features include recursion under λ-binders and matching against dynamically introduced parameters.

Normalized Types

Abstract: We present a new method to specify a certain class of quotient in intentional type theory, and in the calculus of inductive constructions in particular. We define the notion of "normalized types". The main idea is to associate a normalization function to a type, instead of the usual relation. This function allows to compute on a particular element for each equivalence class, avoiding the difficult task of computing on equivalence classes themselves. We restrict ourselves to quotients that allow the construction of such a function, i.e. quotient having a canonical member for each equivalence class. This method is described as an extension of the calculus of constructions allowing normalized types. We prove that this calculus has the properties of strong normalization, subject reduction, decidability of typing. In order to show the example of the definition of Z by a normalized type, we finally present a pseudo Coq session.

An Improved Extensionality Criterion for Higher-Order Logic Programs

Abstract: Extensionality means, very roughly, that the semantics of a logic program can be explained in terms of the set-theoretic extensions of the relations involved. This allows one to reason about the program by ordinary extensional logic. First-order logic programming is extensional. Due to syntactic equality tests in the unification procedure, higher-order logic programming is generally not extensional. Extensionality is a highly undecidable property. We give a decidable extensionality criterion for simply typed logic programs, improving both on Wadge's definitional programs from [9] and on our good programs from [2].

The Anatomy of Innocence

Abstract: We reveal a symmetric structure in the ho/n games model of innocent strategies, introducing rigid strategies, a concept dual to bracketed strategies. We prove a direct definability theorem of general innocent strategies with respect to a simply typed language of extended Bohm trees, which gives an operational meaning to rigidity in call-byname. A corresponding factorization of innocent strategies into rigid ones with some form of conditional as an oracle is constructed.

Monotone Inductive and Coinductive Constructors of Rank 2

Abstract: A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and final coalgebras in a functor category and the Curry-Howard-correspondence. The definition of the system as a λ-calculus requires an appropriate definition of monotonicity to overcome subtle problems, most notably to ensure that the (co-)inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes.

Labelled Natural Deduction for Interval Logics

Abstract: We develop a Labelled Natural Deduction framework for a certain class of interval logics. With emphasis on Signed Interval Logic we consider normalization properties and show that normal derivations satisfy a subformula property. We have encoded our framework in the generic theorem proving system Isabelle. The labelled formalism turns out very convenient for conducting proofs and seems much closer to informal "pen and paper" reasoning than other proof systems. We give an example which supports this claim. We also sketch how the results are applicable to (non-signed) interval logic and Duration Calculus.

Semantic Characterisations of Second-Order Computability over the Real Numbers

Abstract: We propose semantic characterisations of second-order computability over the reals based on Σ-definability theory. Notions of computability for operators and real-valued functionals defined on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and real-valued functionals with validity of finite Σ-formulas.

A Generalization of the Büchi-Elgot-Trakhtenbrot Theorem

Abstract: We consider the power of nondeterministic finite automata with generalized acceptance criteria and the corresponding logics. In particular, we examine the expressive power of monadic second-order logic enriched with monadic second-order generalized quantifiers for algebraic word-problems. Extending a well-known result by Buchi, Elgot, and Trakhtenbrot, we show that considering monoidal quantifiers, the obtained logic captures the class of regular languages. We also consider monadic second-order groupoidal quantifiers and show that these are powerful enough to define every language in LOGCFL.

Complete Categorical Equational Deduction

Abstract: A categorical four-rule deduction system for equational logics is presented. We show that under reasonable finiteness requirements this system is complete with respect to equational satisfaction abstracted as injectivity. The generality of the presented framework allows one to derive conditional equations as well at no extra cost. In fact, our deduction system is also complete for conditional equations, a new result at the author's knowledge.

Constrained Hyper Tableaux

Abstract: Hyper tableau reasoning is a version ofclausal form tableau reasoning where all negative literals in a clause are resolved away in a single inference step. Constrained hyper tableaux are a generalization ofh yper tableaux, where branch closing substitutions, from the point ofview ofmo del generation, give rise to constraints on satisfying assignments for the branch. These variable constraints eliminate the need for the awkward 'purifying substitutions' of hyper tableaux. The paper presents a non-destructive and proof confluent calculus for constrained hyper tableaux, together with a soundness and completeness proof, with completeness based on a new way to generate models from open tableaux. It is pointed out that the variable constraint approach applies to free variable tableau reasoning in general.

An Abstract Look at Realizability

Abstract: This paper is about the combinatorial properties necessary for the construction of realizability models with certain type-theoretic properties. We take as our basic construction a form of tagging in which elements of sets are equipped with tags, and functions must operate constructively on tags. To complete the construction we allow a form of closure under quotients by equivalence relations. In this paper we analyse first the condition for a natural monoidal structure to be product structure, and then investigate necessary conditions for the realizability model to be locally cartesian closed and to have a subobject classifier.

Inflationary Fixed Points in Modal Logic

Abstract: We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal µ-calculus extends basic modal logic with an operator for least fixed points. Least and inflationary fixed point operators have been studied and compared in other contexts, particularly in finite model theory, where it is known that the logics IFP and LFP that result from adding such fixed point operators to first order logic have equal expressive power. As we show, the situation in modal logic is quite different, as the modal iteration calculus (MIC) we introduce has much greater expressive power than the µ-calculus. Greater expressive power comes at a cost: the calculus is algorithmically much less manageable.

A Logic for Abstract State Machines

Abstract: We introduce a logic for sequential, non distributed Abstract State Machines. Unlike other logics for ASMs which are based on dynamic logic, our logic is based on atomic propositions for the function updates of transition rules. We do not assume that the transition rules of ASMs are in normal form, for example, that they concern distinct cases. Instead we allow structuring concepts of ASM rules including sequential composition and possibly recursive submachine calls. We show that several axioms that have been proposed for reasoning about ASMs are derivable in our system and that the logic is complete for hierarchical (non-recursive) ASMs.

A Principle of Induction

Abstract: We introduce an induction principle on complete partial orders and consider its applications to program verification and analysis on the real line. The highlight of this technique is that it allows one to make inductive arguments over continuous as well as discrete forms of data without ever having to distinguish between the two.

Stratified Context Unification Is in PSPACE

Abstract: Context unification is a variant of second order unification and also a generalization of string unification. Currently it is not known whether context unification is decidable. A decidable specialization of context unification is stratified context unification, which is equivalent to satisfiability of one-step rewrite constraints. This paper contains an optimization of the decision algorithm, which shows that stratified context unification can be done in polynomial space.

Actual Arithmetic and Feasibility

Abstract: This paper presents a methodology for reasoning about the computational complexity of functional programs We introduce a first order arithmetic AT0 which is a syntactic restriction of Peano arithmetic We establish that the set of functions which are provably total in AT0, is exactly the set of polynomial time functionsThe cut-elimination process is polynomial time computable Compared to others feasible arithmetics, AT0 is conceptually simpler The main feature of AT0 concerns the treatment of the quantification The range of quantifiers is restricted to the set of actual terms which is the set of constructor terms with variables The inductive formulas are restricted to conjunctions of atomic formulas

An Algebraic Foundation for Higraphs

Abstract: Higraphs, which are structures extending graphs by permitting a hierarchy of nodes, underlie a number of diagrammatic formalisms popular in computing. We provide an algebraic account of higraphs (and of a mild extension), with our main focus being on the mathematical structures underlying common operations, such as those required for understanding the semantics of higraphs and Statecharts, and for implementing sound software tools which support them.