Bundy, Alan

Desimone, R.

Guinchiglia, F.

Hesketh, J.

Madden, P.

Sannella, D.

Smaill, A.

Stevens, A.

van Harmelen, F.

Wallen, L. A.

English

We outline the progress we have made in connection with the Alvey Grant "Proving Properties of Logic Programs" (SERC GR/D/44270 and Alvey IKBS 137). This grant runs for three years from 1st November 1985. The grant holders are Professor Alan Bundy and Dr Don Sannella and it employs or has employed Dr Fausto Giunchiglia, Frank van Harmelen, Jane Hesketh, Dr Alan Smaill and Dr Lincoln Wallen as Research Associates. Pete Madden and Andrew Stevens are attached Ph.D. students

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     Edinburgh Research Explorer                                      Proving properties of logic programs: A Progress ReportCitation for published version:Bundy, A, Wallen, LA, Sannella, D, Desimone, R, Guinchiglia, F, van Harmelen, F, Hesketh, J, Madden, P,Smaill, A & Stevens, A 1988, 'Proving properties of logic programs: A Progress Report' Alvey TechnicalConference 1998, vol UK IT88.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Author final version (often known as postprint)Published In:Alvey Technical Conference 1998General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015PROVING PROPERTIES OF LOGIC PROGRAMS:A PROGRESS REPORTA Bundy, D Sannella, R Desimone,F Giunchiglia, F van Harmelen,J Hesketh, P Madden, A Smaill,A Stevens & L WallenDAI RESEARCH PAPER NO.361Paper submitted to 1988 Alvey ConferenceCopyri9ht (c) A Bundy, D Sannella, R Desimone, F Giunchiglia,f van Harrnelen, J Hesketh, P Madden, A S~aill,A Stevens & L Wallen.Alan Bundy, Don Sannella,Roberto Desimone, Fausto Giunchiglia, Frank van Harmelen,Jane Hesketh, Pete Madden, Alan Smaill,Andrew Stevens, and Lincoln WallenDepartment of Artificial IntelligenceUniversity of EdinburghJanuary 20, 1988AbstractWe outline the progress we have made in connection with the Alvey Grant "Prov-ing Properties of Logic Programs" (SERC GRjDj44270 and Alvey IKBS 137). Thisgrant runs for three years from 1st November 1985. The grant holders are Profes-sor Alan Bundy and Dr Don Sannella and it employs or has employed Dr FaustoGiunchiglia, Frank van Harmelen, Jane Hesketh, Dr Alan Smaill and Dr LincolnWallen as Research Associates. Pete Madden and Andrew Stevens are attachedPh.D. students.Introduction1It is feared that current software techniques will not be adequate to meet the demandsfor quantity and complexity being placed on them. One of the fields whose developmentcould playa role in tackling this problem is that of automatic programming, ie theautomation of (part of) the processes of the generation of programs from specifications;of the verification that a program meets its specification; and of the transformation ofinefficient programs into more efficient programs with the same specification. So softwarereliability could be improved, provided that it is easier to write bug-free specificationsthan bug-free programs.Three recent developments are of help in this area..logic and functional programming languages, such as Prolog and Miranda;.the realisation that constructive logic provides a bridge between logical and com-putational domains (thus relating proofs closely to programs);.proof guidance techniques from automatic theorem proving.1Proving Properties of Logic Programs -A Progress Report 2So using logic programming and constructive logic, the task of generating programs canbe treated as the task of proving a theorem, and automatic proof guidance techniquescan be used.Logic ProgrammingIn logic programming languages the specification ofa program and source (inefficient) andtarget (efficient) programs can all be written in the same language, namely a [ormallogic.So the three processes of synthesis, verification and transformation of programs can betreated in a uniform framework, raising very similar problems. For logic programming, therelation specification/program or program/program according to which transformationsare carried out may vary according to which of these problems we are interested in. Itis also possible simply to prove that a target program has some useful property, as in[Boyer & Moore 73] and [Boyer & Moore 79].The advantage of logic programming in this situation over the case where the specifica-tion language is different from the target program language is that in the second caseoptimisation and synthesis cannot be treated uniformly, and in practice the resultingcombination of both processes leads to an increase of complexity.When the specification and implementation languages are the same, it should be easierto write specifications than programs, since programs are written with efficiency consid-erations in mind whereas such considerations are not involved in writing specifications.Constructive LogicRecently increasing attention has been paid to the use of constructive logic as providinga link between logical formalism and the computational content embodied in statementsin such constructive logics ([Martin-LoC 79, Constable et al 86]).In particular, we can suppose that we are interested in some-relation <I>(x, y) that we wantto hold between input x and output y. Then simply by finding a constructive proof ofthe statement'v'x3ycI>(x, y)we also find, by a routine computation, an algorithm (the "extract term" T) with theproperty that there is a proof thatVxcI>(x,r(x)).So such a T computes the output from the input in such a way as to provably satisfy thegiven specification. In this way, the problem of program synthesis and verification canbe treated using the better understood techniques of theorem proving.It is essential that the logic used should be constructive; for example, when a theoremis proved by considering a number of cases, it is constructively necessary to be able to3Proving Properties of Logic Programs -A Progress Reporttell the cases apart: if we are in such a position, the resulting algorithm will be able toappeal to some test as to which case is relevant. In classical logic, there may be no suchtest, so a proof that uses (classical) axioms like<I> v -, <I>in a proof by cases, where there is no decision procedure for 4>, will fail to yield analgorithm in the way described above.An implementation of this form of program generation and verification has been imple-mented at Cornell ([Constable et al 86]) based on Martin-Lof's constructive type theory.This is an interactive system, with little to help the user in the search for proofs withinthe system; this project aims to use ideas on search control, in particular meta-levelinference, to facilitate this process.As is the case for logic programming, this framework can be used in different ways. Forexample, for straight verification we can conjecture a target program T and show that itsatisfies a specification by provingVx<I>(x,r(x)).Meta-Level Inference1.3For several years the Mathematical Reasoning Group in the Department of ArtificialIntelligence under the direction of Prof A Bundy has investigated the use of meta-levelinference as a tool for controlling implementations of mathematical activities, such assolving equations or proving theorems.In this method, the control and inference for some particular theory, say algebraic equa-tions (called the object theory), is represented in a separate mathematical theory (calledthe meta-theory) where the formulae, axioms, and inference patterns of the object theoryare represented as objects, and so can be reasoned about. The axioms of the meta-theorydescribe properties of the object theory and so in the meta-theory we can formally rea-son about solutions to object-level problems and different methods for obtaining thesesolutions. This method was successfully applied to the area of solving equations in thePRESS program ([Bundy & Welham 79, Sterling et al 82]).Later the same technique was applied to the domain of logic programs in the IMPRESSprogram ([Sterling & Bundy 82]); this was capable of verifying Prolog programs, using aproof plan to guide the verification process.The presence of an explicit meta-level theory also makes it possible to apply deductionand learning notions to the meta-theory itself. The LP program ([Silver 84]) was capableof generalising from a sample solution a plan that could apply in greater generality. Someta-level inference not only helps to guide the object-level inference processes, thusmaking more complex tasks tractable, it also opens the way to automatic learning of thismeta-level knowledge.Proving Properties of Logic Programs 4A Progress ReportGoals2The overall goal is to incorporate techniques from meta-level reasoning into a theoremproving program so as to tackle the problems of synthesis and verification of programs,building both on work in logic programming (eg [Hogger 81]) and in the relation of pro-grams to proofs via constructive logic (eg [Constable 82]). Using the notion of proof plan,the lower-level tactics used to guide object-level inference can be combined so as to guidethe search for proofs of certain forms. It is possible to prove properties of these proofplans, provided that they are expressed in an appropriate declarative language.Other goals included the automatic transformation of programs treated via the transfor-mation of proofs, and the exploration of automatic learning of proof plans from sampleproofs.The system in use is a customised version of Cornell University's NuPRL system[Constable et al86], which is an implementation of Martin-Lors Type Theory. As alogical system, this is a rich language incorporating a higher order constructive logic; as aprogramming language it corresponas to a polymorphic higher-order functional languagesuch as ML or Miranda.The meta-level tactics (as in [Gordon et al79]) which implement the proof constructionoperations are written in MLj the typing of the ML language is used to ensure thecorrect~ess of these proof construction operators, making it impossible to produce anunsound proof.3 Progress and Current StatusThe original Cornell system was first ported to the SUN workstation and adapted to rununder Xwindows. It has been adapted in several ways to make it more suitable for itsprojected role.A library module system has been added, allowing not only hierarchical treatment ofsets of definitions and theorems built up inside the system, but also allowing "abstract"theories to be built up, where a number of objects and properties can be assumed, andthe development carried ahead under these assumptions, to be fully justified later. Thisallows the incorporation of a degree of top-down implementation (in the usual mannerof specification refinement) into the basic bottom-up approach, and makes the use of thesystem for life-size problems more practicable.The type system has been extended with the addition of a new induction principle (for"course-of-values" induction). Since inductive proofs playa central role in this work(showing that a specification can be satisfied by some recursively defined object corre-sponds to showing termination of an algorithm), the provision of this more flexible formallows the investigation of alternative inductive proofs and comparisons between theirassociated algorithms.Using the system, problems such as the synthesis of a variety of sorting algorithms andthe synthesis and verification of a unification algorithm have been carried out.Proving Properties of Logic Programs -A Progress Report 5These sorting algorithms are being used as a domain to examine how the transformationof proofs can be used to optimise program efficiency ([Madden 87]). A proof from which aprogram is derived using the programs-as-proofs principle will contain more informationthan the program derived from it. Using notions from analogy, proofs may be systemati-cally related in such a way that more efficient programs can be obtained by transformingthe associated proofs. The crucial element in the proof is which forms of induction areinvoked and how they are related.In order to exploit the implicit proof-plan found in Boyer and Moore's theorem prover, itis important to have an explicit representation in which proof plans can be specified andtactics combined. Work is currently in hand on elaborating such a specification language.The basic proof plan for inductive proofs has been elaborated in several formalisms tocompare their merits, and proofs of the correctness of such plans have been set out[Bundy 88].Ultimately plans should be capable of reacting to local failures by means of dynamicpatching. The plans could themselves be the object of systematic transformation. We arealso exploring the relation between these proof plans and the AI planning tradition. -The use of pre-condition analysis to learn proof plans has been investigated, resultingin extensions to the previous techniques ([Desimone 87]) and applied to NuPRL proofscorresponding to insert functions in lists.Currently investigation is also proceeding in the use of NuPRL itself as a possible specifica-tion language, using some form of reflection as in [Knoblock & Constable 86]. This wouldallow the processes of specification, synthesis and verification of tactics to be treated inthe same way as their object-level counterparts. Building on the work of Weyrauch andthe FOL group ([Weyhrauch 80]), this would allow formalised reasoning at both the meta-level and object-level to take place in a single system, providing a formal link betweenmeta-reasoning and object-level reasoning. This is one systematic way in which the userof the system can extend its metamathematical capability in the same way as is alreadypossible at the object-level using the definition and library mechanisms.Summary4We outline the work undertaken in connection with the Alvey "Proving Properties ofLogic Programs" Grant. The project aims to synthesise experience in automated theo-rem proving (in particular Boyer and Moore) and the proofs-as-programs school (usingconstructive logic, in particular the NuPRL system) to tackle the problems of the auto-mated synthesis, verification and transformation of programs. The N uPRL system hasbeen adapted and altered to make it more suitable for this task, and the experience ofthe group in meta-level inference brought to bear on the problems raised. The notionof proof-plan has proved central in allowing AI techniques of learning, analogy, and ex-plicit meta-level inference to be applied. It seems that the constructive logic approachopens up many possible avenues of research, though the problem of automating significantapplications remains difficult.6Proving Properties of Logic Programs .A Progress ReportReferencesR.S. Boyer and J.S. Moore. Proving theorems about lispfunctions. In N. Nilsson, editor, Proceedings of the third IJ-GAl, pages 486-493, International Joint Conference on Ar-tificial Intelligence, August 1973. Also available from Edin-burgh as DCL memo No. 60.[Boyer &, Moore 73][Boyer & Moore 79] R.S. Boyer and J.S. Moore. A Computational Logic. Aca-demic Press, 1979. ACM monograph series.[Bundy & Welham 79] A. Bundy and B. Welham. Using meta-level descriptionsfor selective application of multiple rewrite rules in algebraicmanipulation. Working Paper 55, Dept. of Artificial Intel-ligence, Edinburgh, May 1979. Revised version in ArtificialIntelligence 16(2). Also Research Paper 121.[Bundy 88] A. Bundy. The Use of Explicit Plans to Guide InductiveProofs. Research Paper forthcoming, Dept. of Artificial In-telligence, Edinburgh, 1988. Short version submitted toCADE9.[Constable 82] R.L. Constable. Programs as Proofs. Technical Report TR82-532, Dept. of Computer Science, Cornell University,November 1982.[Constable et al 86] R.L. Constable, S.F. Allen, H.M. Bromley, et al. Implement-ing Mathematics with the Nuprl Proof Development System.Prentice Hall, 1986.R. V. Desimone. Explanation-based learning of proof plans.In Y. Kodratoff, editor, Machine and Human Learning, EllisHorwood, 1987. Also available as DAI Research Paper 304.Previous version in proceedings of EWSL-86.[Desimone 87][Gordon et al 79] M.J. Gordon, A.J. Milner, and C.P. Wadsworth. EdinburghLCF -A mechanised logic of computation. Volume 78 ofLecture Notes in Computer Science, Springer Verlag, 1979.[Hogger 81] C.J. Hogger. Derivation of logic programs.28(2):372-392, April 1981.JACM,[Knoblock & Constable 86] T. B. Knoblock and R.L. Constable. Formalized metareason-ing in type theory. In Proceedings of LICS, pages 237-248,IEEE, 1986.[Madden 87] P. Madden. The Application of Analogy to Program Opti-mization. Discussion paper 51, DAI, September 1987.7Proving Properties of Logic Programs -A Progress Report[Martin-LoC 79] Per Martin-Lo£. Constructive mathematics and computerprogramrriing. In 6th International Congress for Logic,Methodology and Philosophy of Science, pages 153-175, Han-nover, August 1979. Published by North Holland, Amster-dam. 1982.[Silver 84] B. Silver. Precondition analysis: learning control informa-tion. In Machine Learning p., Tioga Publishing Company,1984.[Sterling & Bundy 82] L. Sterling and A. Bundy. Meta-level inference and programverification. In D. W. Loveland, editor, 6th Conference onAutomated Deduction, pages 144-150, Springer Verlag, 1982.Lecture Notes in Computer Science No. 138. Also availablefrom Edinburgh as Research Paper 168.[Sterling et al 82] L. Sterling, A. Bundy, L. Byrd, R. O'Keefe, and B. Silver.Solving symbolic equations with press. In J. Calmet, editor,Computer Algebra) Lecture Notes in Computer Science No.1 , pages 109-116, Springer Verlag, 1982. Also availablefrom Edinburgh as Research Paper 171.[Weyhrauch 80] R. W. Weyhrauch. Prolegomena to a theory of mechanizedformal reasoning. Artificial Intelligence, 13:133-170, 1980.

A Computational Logic.

Constructive mathematics and computer programrriing.

Derivation of logic programs.

Edinburgh LCF -A mechanised logic of computation.

Formalized metareasoning in type theory.

Implementing Mathematics with the Nuprl Proof Development System.

Meta-level inference and program verification. In

Precondition analysis: learning control information.

Programs as Proofs.

Prolegomena to a theory of mechanized formal reasoning.

Proving theorems about lisp functions.

Solving symbolic equations with press.

The Application of Analogy to Program Optimization. Discussion paper 51,

The Use of Explicit Plans to Guide Inductive Proofs. Research Paper forthcoming,

Using meta-level descriptions for selective application of multiple rewrite rules in algebraic manipulation. Working Paper 55,

Proving properties of logic programs: A Progress Report

Proving properties of logic programs: A Progress Report

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