Showing papers presented at "Computer Science Logic in 1992"

The Semantics of the C Programming Language

Abstract: We present formal operational semantics for the C programming language. Our starting point is the ANSI standard for C as described in [KR]. Knowledge of C is not necessary (though it may be helpful) for comprehension, since we explain all relevant aspects of C as we proceed. Our operational semantics is based on evolving algebras. An exposition on evolving algebras can be found in the tutorial [Gu]. In order to make this paper self-contained, we recall the notion of a (sequential) evolving algebra in Sect. 0.1. Our primary concern here is with semantics, not syntax. Consequently, we assume that all syntactic information regarding a given program is available to us at the beginning of the computation (via static functions). We intended to cover all constructs of the C programming language, but not the C standard library functions (e.g. fprintf(), fscanf()). It is not di cult to extend our description of C to include any desired library function or functions. Evolving algebra semantic speci cations may be provided on several abstraction levels for the same language. Having several such algebras is useful, for one can examine the semantics of a particular feature of a programming language at the desired level of abstraction, with unnecessary details omitted. It also makes comprehension easier. We present a series of four evolving algebras, each a re nement of the previous one. The nal algebra describes the C programming language in full detail. Our four algebras focus on the following topics respectively:

Notes on Sconing and Relators

Abstract: This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a category-theoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined.

Solving 3-Satisfiability in Less Then 1, 579n Steps

Abstract: In this paper we describe and analyse an improved algorithm for solving the 3-Satisfiability problem. If F is a boolean formula in conjunctive normal form with n variables and r clauses, then we will show that this algorithm solves the Satisfiability problem for formulas with at most three literals per clause in time less than O(1,579n).

Linear λ-calculus and categorical models revisited

Abstract: Girard's Intuitionistic Linear Logic [7] is a renement of Intuitionistic Logic, where formulae must be used exactly once. In other words, the familiar Weakening and Contraction rules of Gentzen's sequent calculus [17] are removed. To regain the expressive power of Intuitionistic Logic, these rules are returned, but in a controlled manner. A logical operator, `!', is introduced which allows a formula to be used as many times as required (including zero). In this paper we shall consider multiplicative exponential linear logic (MELL), i.e. the fragment which has multiplicative conjunction or tensor, , linear implication, , and the logical operator \exponential", !. We recall the rules for MELL in a sequent calculus system in Fig. 1. We use capital Greek letters ;  for sequences of formulae and A; B for single formulae. The Exchange rule simply allows the permutation of assumptions. The `! rules' have been given names by other authors. !L1 is called Weakening , !L2 Contraction, !L3 Dereliction and (!R) Promotion . (We shall use these terms throughout this paper.) In the Promotion rule, ! means that every formula in the set  is modal, in other words, if  is the set fA1; A2; . . . Ang, then ! denotes the set f!A1; !A2; . . .!Ang.

The Basic Logic of Proofs

Abstract: Propositional Provability Logic was axiomatized in [5]. This logic describes the behaviour of the arithmetical operator “y is provable”. The aim of the current paper is to provide propositional axiomatizations of the predicate “x is a proof of y”by means of modal logic, with the intention of meeting some of the needs of computer science.

A Self-Interpreter of Lambda Calculus Having a Normal Form

Abstract: We formalize a technique introduced by Bohm and Piperno to solve systems of recursive equations in lambda calculus without the use of the fixed point combinator and using only normal forms. To this aim we introduce the notion of a canonical algebraic term rewriting system, and we show that any such system can be interpreted in the lambda calculus by the Bohm — Piperno technique in such a way that strong normalization is preserved. This allows us to improve some recent results of Mogensen concerning efficient godelizations ⌈⌉:Λ→Λ of lambda calculus. In particular we prove that under a suitable godelization there exist two lambda terms E (self-interpreter) and R (reductor), both having a normal form, such that for every (closed or open) lambda term M E⌈M⌉→M and if M has a normal form N, then R⌈M⌉→⌈N⌉.

On the Completeness of Narrowing as the Operational Semantics of Functional Logic Programming

Abstract: This paper is a continuation of [10]. It presents soundness and completeness results for a higher-order (HO) functional logic language which has a domain-based declarative semantics and uses conditional narrowing (for applicative, constructor based rewriting systems) as operational semantics. HO-unification is avoided by for bidding λ-abstractions in the language. However, narrowing must include a mechanism for binding HO logic variables to simple functional patterns built by partial application. A deeper investigation of lazy strategiees and infinite narrowing derivations is foreseen.

Communicating Evolving Algebras

Abstract: We develop the first steps of a theory of concurrency within the framework of evolving algebras of Gurevich, with the aim of investigating its suitability for the role of a general framework for modeling concurrent computation. As a basic tool we introduce a ‘modal’ logic of transition rules and runs, which is, in the context of evolving algebras, just a definitional extension of ordinary first order logic. A notion of independence of rules and runs enables us to introduce a notion of (and notation for) concurrent runs, on which a logical theory of (‘true’) concurrency may be based. The notion of concurrent run also has (but does not depend on) an interleaving interpretation. Some basic constructs (concurrent composition, addition of guards and updates) and some derived constructs (internal and external choice, sequential composition) on evolving algebras are introduced and investigated. The power of the framework is demonstrated by developing simple and transparent evolving algebra models for the Chemical Abstract Machine of Berry and Boudol and for the π-calculus of Milner. Their respective notions of parallelism map directly and faithfully to native concurrency of evolving algebras.

Model Building by Resolution

Abstract: Resolution calculi are best known as basis for algorithms testing the unsatisfiability of sets of clauses. Only recently more attention is paid to the fact that various resolution refinements may also benefitly be employed as decision procedures for a wide range of decidable classes of clause sets. In this proof theoretic approach to the decision problem one usually tries to test for satisfiability by termination of complete resolution procedures. Building on such types of decidability results we show that, for certain classes of clause sets, we can extract (representations of) models from the set of resolvents generated by hyperresolution. The process of model construction proceeds in two steps: First hyperresolution is employed to arrive at a finite set of atoms that represents a description of an Herbrand-model. In a second step we extract from this set of atoms a full representation of a model with finite domain. We emphasize that no backtracking is needed in our model constructing algorithm.

Logical Characterization of Bounded Query Classes II: Polynomial-Time Oracle Machines

Abstract: We have shown that the logics (±HP)*[FOS] and (±HP)1[FOS] are of the same expressibility, and both capture P∥ NP . This result gives us the weakest possible hint that it might be wiser to try and show that (±STC*[FOS] collapses to (±STC)1[FOS] as opposed to trying to show that STC1[FOS] is closed under complementation: an attempt to use the methods of [Imm88] to achieve this latter result has failed (see [BCD89]).

On Asymptotic Probabilities of Monadic Second Order Properties

Abstract: We propose a new, general and easy method for proving nonexistence of asymptotic probabilities of monadic second-order sentences in classes of finite structures where first-order extension axioms hold almost surely.

Linear Time Algorithms and NP-Complete Problems

Abstract: We define and study a machine model (random access machine with powerful input/output instructions) and show that for this model the classes, DLINEAR and NLINEAR, of problems computable in deterministic (resp. nondeterministic) linear time are robust and powerful. In particular DLINEAR includes most of the concrete problems commonly regarded as computable in linear time (such as graph problems: topological sorting, strong connectivity...) and most combinatorial NP-complete problems belong to NLINEAR. The interest of NLINEAR class is enhanced by the following fact: some natural NP-complete problems, for example “reduction of incompletely specified automata” (in short: RISA), are NLINEAR-complete (consequently, NLINEAR ≠ DLINEAR iff RISA ∉ DLINEAR). That notion probably strengthens NP-completeness since we argue that propositional satisfiability is not NLINEAR-complete.

Algorithmic Structuring of Cut-free Proofs

Abstract: The problem of algorithmic structuring of proofs in the sequent calculi LK and LKB (LK where blocks of quantifiers can be introduced in one step) is investigated, where a distinction is made between linear proofs and proofs in tree form In this framework, structuring coincides with the introduction of cuts into a proof The algorithmic solvability of this problem can be reduced to the question of k/l-compressibility: “Given a proof of Π → Λ of length k, and l≤k: Is there is a proof of Π → Λ of length ≤l?” When restricted to proofs with universal or existential cuts, this problem is shown to be (1) undecidable for linear or tree-like LK-proofs (corresponds to the undecidability of second order unification), (2) undecidable for linear LKB-proofs (corresponds to the undecidability of semi-unification), and (3) decidable for tree-like LKB-proofs (corresponds to a decidable subproblem of semi-unification)

Optimization Problems: Expressibility, Approximation Properties and Expected Asymptotic Growth of Optimal Solutions

Abstract: Although the notion of NP-completeness was defined in terms of decision problems, the prime motivation for its study and development was the apparent intractability of a large family of combinatorial optimization problems. NP-completeness of a decision problem rules out the possibility of finding an optimal solution of the corresponding optimization problem in polynomial time unless P = NP. It does not exclude, however, the possibility that there are efficient algorithms which produce approximate solutions. In fact, for many optimization problems with NP-complete decision problems, there are simple and efficient algorithms that produce solutions differing from optimal solutions by at most a constant factor. For some problems, there even exist so-called polynomial-time approximation schemes (PTAS), which produce approximate solutions to any desired degree of accuracy. For other problems, notably the Traveling Salesperson Problem, there do not exist efficient approximations unless P = NP (see [7]). Until now the "structural" reasons for the different approximation properties of NP optimization problems have not been sufficiently understood. Papadimitriou and Yannakakis [17] provided a new perspective by relating the approximation properties of optimization problems to their logical representation. Exploiting Fagin's characterization of NP by existential second order logic [6], they defined two classes of optimization problems, MAX SNP and MAX NP, and showed that all problems in these classes are approximable in polynomial time up to a constant factor. They also identified a host of problems which are complete for MAX SNP with respect to reductions that preserve polynomial-time approximation schemes. Very recently, the classes MAX SNP and MAX NP have received a lot of attention due to results by Arora et al. [2] showing that problems which are hard for MAX SNP cannot have a PTAS, unless P = NP. We present the syntactic criterion of Papadimitriou and Yannakakis in the more general form and notation provided by Kolaitis and Thakur [11].

A Universal Turing Machine

Abstract: The aim of this paper is to give an example of a universal Turing machine, which is somewhat small. To get a small universal Turing machine a common constructions would go through simulating tag system (see Minsky 1967). The universal machine here simulate two-symbol Turing machines directly.

Logical Definability of NP-Optimization Problems with Monadic Auxiliary Predicates

Abstract: Given a first-order formula ϕ with predicate symbols e1el, so,,sr, an NP-optimisation problem on -structures can be defined as follows: for every -structure G, a sequence of relations on G is a feasible solution iff satisfies ϕ, and the value of such a solution is defined to be ¦S0¦ In a strong sense, every polynomially bounded NP-optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1, ,sr are restricted to be monadic The result is proved by an Ehrenfeucht-Fraisse game and remains true in several more general situations

Universes in the Theory of Types and Names

Abstract: In this paper we recall the basic ideas of the theories of types and names, which were introduced by Jager and which are closely related to Feferman's systems of explicit mathematics. We start off from the elementary theory of types and names with the datatype of the natural numbers, which is equivalent to EM0 or to EM0,↾ depending on the form of induction taken. The elementary theory is extended by universes, i.e. types enjoying special closure properties, to a theory UTN. We show how new type constructions become possible in presence of the universes and prove a lower bound for its proof theoretic strength. We sketch the proof of the upper bound of the theory, which shows that UTN does not exceed the limits of predicativity.

Comparative Transition System Semantics

Abstract: Employing the notion of a transition system, programs, conceived as binary (transition) relations on states, are related to processes, viewed as dynamic states. The comparative study is carried out syntactically over rules for transitions, and semantically in terms of bisimulation equivalence. A certain form of transitions is studied, and a “logical” approach to the notion of a bisimulation is taken that are somewhat non-standard (but, it is hoped, illuminating). Sequential composition, non-deterministic choice, iteration, and interleaving are analyzed alongside a notion of data. Atomization and synchronization are also considered.

Negation-Complete Logic Programs

Abstract: We give a short, direct proof that a logic program is negation-complete if, and only if, it has the cut-property. The property negation-complete refers to three-valued models, the cut-property is defined in terms of ESLDNF-computations only.

Reasoning with Higher Order Partial Functions

Abstract: In this paper we introduce the logic PHOL, which embodies higher-order functions through a simply-typed λ-calculus and deals with partial objects by using partially ordered domains and three truth values. We define a refutationally complete tableaux method for PHOL and we show how to derive a sound and complete cut free sequent calculus through a systematic analysis of the rules for tableaux construction.

Logical semantics of modularization

Abstract: An algebra of theories, signatures, renamings and the operations import and export is investigated. A normal form theorem for terms of this algebra is proved. Another algebraic approach and the relation with a fragment of second order logic are also considered.

Recursive Inseparability in Linear Logic

Abstract: We first give our version of the register machines to be simulated by proofs in propositional linear logic. Then we look further into the structure of the computations and show how to extract ”finite counter models” from this structure. In that way we get a version of Trakhtenbrots theorem without going through a completeness theorem for propositional linear logic. Lastly we show that the interpolant I in propositional linear logic of a provable formula A ⊸ B cannot be totally recursive in A and B.