Soundness and Completeness Proofs by Coinductive Methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion.     

Proofs, Programs, Processes

The objective of this paper is to provide a theoretical foundation for program extraction from inductive and coinductive proofs geared to practical applications. The novelties consist in the addition of inductive and coinductive definitions to a realizability interpretation for first-order proofs, a soundness proof for this system, and applications to the synthesis of non-trivial provably correct programs in the area of exact real number computation. We show that realizers, although per se untyped, can be assigned polymorphic recursive types and hence represent valid programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. the binary signed digit representation.     

Extension table built-ins for prolog

The ET* algorithm is a complete evaluation strategy for Datalog programs, which are logic programs without function symbols. The ET* algorithm uses extension tables and depth-first iterative deepening to provide the evaluation of pure function-free logic programs as declarative specifications. Extension tables are a memo facility that the algorithm uses both to cut infinite derivation paths for complete evaluation and to optimise the evaluation of logic programs. The original implementation of the ET* algorithm incorporated extension tables as part of the Prolog database using the built-in predicates assert and retract. The advantage of implementing the extension table using the Prolog database is the portability of the ET* algorithm. There are several disadvantages, however, with this approach. One disadvantage is the cost associated with the built-in predicates assert and retract, which are known to be expensive operations in most current Prolog systems. Another disadvantage is the differences across implementations in the semantics that these built-ins provide for dynamic predicates. This paper presents an efficient implementation of extension tables as a global data structure in Prolog, which includes a set of built-in primitives for manipulating the extension table. The ET* algorithm is updated to reflect the utilisation of the global extension table data structure. The implementations of the ET* algorithm are compared using time and space performance on a variety of benchmark programs.     

程序语言中的共归纳数据类型及其应用

归纳数据类型利用代数方法从构造的角度归纳地描述数据类型的有限语法结构,但在描述动态行为方面存在一定的不足。作为归纳数据类型的范畴对偶概念,共归纳数据类型利用共代数方法从观察的角度共归纳地描述了数据类型的动态行为。首先,从范畴论和代数的角度给出程序语言中的归纳数据类型定义,并分析了相应的递归操作;接着,利用共代数给出共归纳数据类型的范畴论定义,并根据共归纳数据类型的终结性分析了相应的共递归操作;最后,指出如何利用无双代数及分配律将归纳与共归纳数据类型有机地融合起来,探讨数据类型的语法构造与动态行为关系。     

Distributive laws for the coinductive solution of recursive equations

This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws.     

计算机科学中的范畴数据类型的研究综述

范畴数据类型是指以范畴论为数学理论基础研究数据类型的描述、计算、语义和应用。早期的范畴数据类型研究以归纳数据类型为主,采用代数从归纳的角度研究有限数据类型的构造语义和递归性质。近年来,归纳数据类型的对偶概念——共归纳数据类型逐渐引起计算机科学工作者的关注与研究,他们采用共代数从观察的角度研究无限数据类型的行为语义和共递归性质。利用范畴论可以为数据类型研究提供统一的数学理论基础,并将代数和共代数中的各种重要研究成果有机地融合在一起,如语法构造与动态行为、递归与共递归、同余与互模拟等。目前,范畴数据类型已经在程序语言、计算描述、理论证明器和并行计算等领域得到广泛的应用。对范畴数据类型的基本概念、数学理论基础、逻辑基础及应用等方面的最新研究成果进行介绍,以引起国内外相关研究领域的学者对计算机科学中的范畴数据类型理论的关注。     

On computability by logic programs

The problem of computational completeness of Horn clause logic programs is revisited. The standard results on representability of all computable predicates by Horn clause logic programs are not related to the real universe on which logic programs operate. SLD-resolution, which is the main mechanism to execute logic programs, may give answer substitutions with variables. As the main result we prove that computability by Horn clause logic programs is equivalent to standard computability over the Herbrand universe with variables. The semantics we use isS-semantics introduced by Falaschi et al. [3]. As an application of the main result we prove the existence of a metainterpreter for a sublanguage of real Prolog, written in the language of Horn clauses with the S-semantics. We also show that the traditional semantics of Prolog do not reflect its computational behavior from the viewpoint of recursion theory.This article is a revised version of [13]. The work was essentially done during author's visit to ECRC.     

Coinductive big-step operational semantics

Using a call-by-value functional language as an example, this article illustrates the use of coinductive definitions and proofs in big-step operational semantics, enabling it to describe diverging evaluations in addition to terminating evaluations. We formalize the connections between the coinductive big-step semantics and the standard small-step semantics, proving that both semantics are equivalent. We then study the use of coinductive big-step semantics in proofs of type soundness and proofs of semantic preservation for compilers. A methodological originality of this paper is that all results have been proved using the Coq proof assistant. We explain the proof-theoretic presentation of coinductive definitions and proofs offered by Coq, and show that it facilitates the discovery and the presentation of the results.     

SLT-Resolution for the Well-Founded Semantics

Global SLS-resolution and SLG-resolution are two representative mechanisms for top-down evaluation of the well-founded semantics of general logic programs. Global SLS-resolution is linear for query evaluation but suffers from infinite loops and redundant computations. In contrast, SLG-resolution resolves infinite loops and redundant computations by means of tabling, but it is not linear. The principal disadvantage of a nonlinear approach is that it cannot be implemented by using a simple, efficient stack-based memory structure nor can it be easily extended to handle some strictly sequential operators such as cuts in Prolog. In this paper, we present a linear tabling method, called SLT-resolution, for top-down evaluation of the well-founded semantics. SLT-resolution is a substantial extension of SLDNF-resolution with tabling. Its main features are the following. First, it resolves infinite loops and redundant computations while preserving the linearity. Second, it is terminating and is sound and complete w.r.t. the well-founded semantics for programs with the bounded-term-size property with nonfloundering queries. Its time complexity is comparable with SLG-resolution and polynomial for function-free logic programs. Third, because of its linearity for query evaluation, SLT-resolution bridges the gap between the well-founded semantics and standard Prolog implementation techniques. It can be implemented by an extension to any existing Prolog abstract machines such as WAM or ATOAM.     

Oracle Semantics for Prolog

This paper proposes to specify semantic definitions for logic programming languages such as Prolog in terms of an oracle which specifies the control strategy and identifies which clauses are to be applied to resolve a given goal. The approach is quite general. It can be applied to Prolog to specify both operational and declarative semantics as well as other logic programming languages. Previous semantic definitions for Prolog typically encode the sequential depth-first search of the language into various mathematical frameworks. Such semantics mimic a Prolog interpreter in the sense that following the "leftmost" infinite path in the computation tree excludes computation to the right of this path from being considered by the semantics. The basic idea in this paper is to abstract away from the sequential control of Prolog and to provide a declarative characterization of the clauses to apply to a given goal. The decision whether or not to apply a clause is viewed as a query to an oracle which is specified from within the semantics and reasoned about from outside. This approach results in simple and concise semantic definitions which are more useful for arguing the correctness of program transformations and providing the basis for abstract interpretations than previous proposals.     

Integration of prolog and databases in both evaluational and non-evaluational approaches

DB-Prolog is an extended version of Prolog which allows users to access relational databases in both evaluational and non-evaluational approaches. Using the former approach, DB-Prolog offers some built-in predicates for retrieving relational databases. Using the latter approach, DB-Prolog enables relational databases to store facts (ground unit clauses) which may have compound terms in their arguments. As a result, DB-Prolog can execute Prolog programs which contain both a large set of facts, and conventional data (both numerical and character) efficiently. These Prolog programs are required for the construction of practical expert systems. DB-Prolog implementation techniques are described, and some of its available functions are discussed.     

Operational and goal-independent denotational semantics for Prolog with cut

In this paper we propose an operational and a denotational semantics for Prolog. We deal with the control rules of Prolog and the cut operator. Our denotational semantics provides a goal-independent semantics. This means that the behaviour of a goal in a program is defined as the evaluation of the goal in the denotation (semantics) of the program. We show how our denotational semantics can be specialised into a computed answer semantics and into a call pattern semantics. Our work provides a basis for a precise abstract interpretation of Prolog programs.     

Bi-inductive Structural Semantics: (Extended Abstract)

We propose a simple order-theoretic generalization of set-theoretic inductive definitions. This generalization covers inductive, co-inductive and bi-inductive definitions and is preserved by abstraction. This allows the structural operational semantics to describe simultaneously the finite/terminating and infinite/diverging behaviors of programs. This is illustrated on the structural bifinitary small/big-step trace/relational/operational semantics of the call-by-value λ-calculus.     

A metalogic programming approach: language,semantics and applications

Abstract

This paper presents a logic programming language of novel conception, called Reflective Prolog, which allows declarative metaknowledge representation and metareasoning. The language is defined by augmenting pure Prolog (Horn clauses) with capabilities of self-reference and logical reflection. Self-reference is designed as a quotation device (a carefully defined naming relation) which allows the construction of metalevel terms that refer to object-level terms and atoms. Logical reflection is designed as an unquotation mechanism (a distinguished truth predicate) which relates names to what is named, thus extending the meaning of domain predicates. The reflection mechanism is embodied in an extended resolution procedure which automatically switches the context between levels. This implicit reflection relieves the programmer from having to explicitly deal with control aspects of the inference process. The declarative semantics of a Reflective Prolog definite program P is provided in terms of the least reflective Herbrand model of P, characterized by means of a suitable mapping defined over the Herbrand interpretations of P. The extended resolution is proved sound and complete with respect to the least reflective Herbrand model. By illustrating Reflective Prolog solutions to an organic set of problems, and by discussing the main differences with respect to other approaches to logic metaprogramming, we show that the proposed language deploys, within its field of action, greater expressive and inferential power than those available till now. The interpreter of the language has been fully implemented. Because of its enhanced power, logic semantics and working interpreter, Reflective Prolog is offered as a contribution toward making the declarative approach of logic programming applicable to the development of increasingly sophisticated knowledge-based systems.     

Diagram Chase in Relational System Development

We propose diagrammatic techniques for visualizing relational reasoning in formal methods like B or Z; in particular for induction and coinduction. These are similar to those for functional diagrams in category theory and inspired by rewriting theory. Diagrams are endowed with a simple algebraic semantics that imposes a convenient balance between expressive and algorithmic power. This makes the approach particularly suitable for mechanization and automation. Its usefulness for visual reasoning is illustrated by various examples.     

Complexity and undecidability results for logic programming

This paper surveys complexity, degree of uncomputability, and expressive power results for logic programming. Some major decision problem complexity results and other results for logic programming are also covered. It also proves several new results filling in previous gaps in the literature. The paper considers seven logic programming semantics: the van Emden-Kowalski semantics for definite (Horn) logic programs; the perfect model semantics for stratified and for locally stratified logic programs; and the two- and three-valued program completion semantics, the well-founded semantics, and the stable semantics, all for normal logic programs, under skeptical inference. The main results concern expressibility and query complexity/uncomputability in five contexts: for propositional logic programs, for first order logic programs with infinite Herbrand universes on their Herbrand universes (a closed domain assumption), for first order logic programs with infinite Herbrand universes on those universes extended with infinitely many new elements (an open domain assumption), and for logic programs without function or constant symbols evaluated over varying extensional databases (DATALOG-type results, data complexity results only) under both closed and open domain assumptions. Several of the open domain assumption results are new to this paper. Other results surveyed are (1) results about the family of all stable models of a program and (2) decision questions about when a logic program has nice properties with respect to a semantics (e.g., a unique stable model). One decision result, for well-founded semantics, is new to this paper.Work supported in part by NSF grant IRI-8905166.     

Abstract interpretation of resolution-based semantics

We extend the abstract interpretation point of view on context-free grammars by Cousot and Cousot to resolution-based logic programs and proof systems. Starting from a transition-based small-step operational semantics of Prolog programs (akin to the Warren Machine), we consider maximal finite derivations for the transition system from most general goals. This semantics is abstracted by instantiation to terms and furthermore to ground terms, following the so-called c- and s-semantics approach. Orthogonally, these sets of derivations can be abstracted to SLD-trees, call patterns and models, as well as interpreters providing effective implementations (such as Prolog). These semantics can be presented in bottom–up fixpoint form. This abstract interpretation-based construction leads to classical bottom–up semantics (such as the s-semantics of computed answers, the c-semantics of correct answers of Keith Clark, and the minimal-model semantics of logical consequences of Maarten van Emden and Robert Kowalski). The approach is general and can be applied to infinite and top–down semantics in a straightforward way.     

Algebraic computational models of OR-parallel execution of Prolog

A specification of the OR-parallel execution of Prolog programs, using CHOCS (calculus of higher order communicating systems) [24], is presented in the paper. A translation is defined from Prolog programs and goals to CHOCS processes: the execution of the CHOCS process corresponding to a goal mimics the OR-parallel execution of the original Prolog goal. In the translation, clauses and predicate definitions of a Prolog program correspond to processes. To model OR-parallelism, the processes , corresponding to clauses (having the same head predicate ) start their execution concurrently, but, in order to respect the depth-first search rule, each is guarded by the termination of the executions of processes 's, . The computational model is proved correct with respect to the semantics of Prolog, as given in [4, 5]. Our model, because of its algebraic specification, can be easily used to prove properties of the parallel execution of Prolog programs. Moreover, the model exploits the maximum degree of parallelism, by giving the Prolog solutions in parallel, without any order among them. However, this model, being close to the Prolog semantics definition, contains sources of inefficiency which make it unpractical as a guide for the implementation. To overcome these problems, a new computational model is defined. This model is obtained by modifications of the basic one and thus its correctness can be easily proved. Finally, we show how to obtain models of different real implementations of OR-parallel Prolog by slight modification of the new model. The relations among all these models, in terms of parallelism degree, are studied by using the concepts of bisimulation and simulation, developed for concurrent calculi. Received: 5 May 1995 / 28 May 1996     

Coinductive Verification of Program Optimizations Using Similarity Relations

Formal verification methods have gained increased importance due to their ability to guarantee system correctness and improve reliability. Nevertheless, the question how proofs are to be formalized in theorem provers is far from being trivial, yet very important as one needs to spend much more time on verification if the formalization was not cleverly chosen. In this paper, we develop and compare two different possibilities to express coinductive proofs in the theorem prover Isabelle/HOL. Coinduction is a proof method that allows for the verification of properties of also non-terminating state-transition systems. Since coinduction is not as widely used as other proof techniques as e.g. induction, there are much fewer “recipes” available how to formalize corresponding proofs and there are also fewer proof strategies implemented in theorem provers for coinduction. In this paper, we investigate formalizations for coinductive proofs of properties on state transition sequences. In particular, we compare two different possibilities for their formalization and show their equivalence. The first of these two formalizations captures the mathematical intuition, while the second can be used more easily in a theorem prover. We have formally verified the equivalence of these criteria in Isabelle/HOL, thus establishing a coalgebraic verification framework. To demonstrate that our verification framework is suitable for the verification of compiler optimizations, we have introduced three different, rather simple transformations that capture typical problems in the verification of optimizing compilers, even for non-terminating source programs.     

Assert,retract and external processes in prolog

As Prolog becomes more widely used, it becomes important to provide clear and consistent implementations of the assert and retract primitives. This paper introduces a bit-map indexing system with a consistent semantics. It also considers the implementation of the interface between Prolog and an external source of clauses, concentrating on the storage of clauses on secondary storage, but considering also the presentation of data to Prolog programs by non-Prolog processes.