View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by New University of Lisbon's Repository Jos´eJ´ulioAlves Alferes Semantics of Logic Programs with Explicit Negation Disserta¸c˜aoapresentada para a obten¸c˜aodo Grau de Doutor em Engenharia Inform´atica, especialidade InteligˆenciaArtificial, pela Uni- versidade Nova de Lisboa, Faculdade de Ciˆenciase Tecnologia. Lisboa (1993) Acknowledgements First of all, I would like to thank my parents, Ac´acioand Dion´ısia,for the incentive they gave me to pursue the goal of making the PhD, their encouragement, support and many other reasons. Also special thanks to Cristina for her support, and especially for bearing with me during the period while this thesis was being written. Special thanks to my supervisor, Lu´ısMoniz Pereira, for lots of reasons including but not limited to: his availability for discussions, even when he had lots of other urgent things to do; the valuable comments and advice he gave me about this work; for introducing me to the right researchers at the right time, giving me the chance of learning a lot with those researchers; for providing for the good working conditions I had during the preparation of the PhD thesis; for teaching me how to do scientific research in its many aspects, and for the pleasure of doing it together with him. I would also like to thank my colleagues Joaquim Nunes Apar´ıcioand Carlos Dam´asio,who collaborated with me in many tasks during the three years of preparation of this thesis. For many profitable discussions, thanks are due to, among others, my colleagues Salvador Abreu, Lu´ısCaires, Gabriel David, Lu´ısMonteiro, Ant´onioPorto, and Irene Rodrigues. On the international side, I would like to thank, for valuable comments and discussions, Marc Denecker, Michael Gelfond, Tony Kakas, Vladimir Lifschitz, Paolo Mancarella, David Peirce, Halina Przymusinska, Teodor Przymusinski, and Gerd Wagner. Special thanks to Phan Minh Dung. I’m indebted to Carlos Dam´asioand Lu´ısMonteiro, who kindly accepted to read a prelimi- nary version of this report and give me their comments, and to Salvador Abreu for helping me with the French “sommaire”. I would like to acknowledge JNICT (Junta Nacional de Investiga¸c˜aoCient´ıficae Tecnol´ogica) for giving me the PhD grant (no. 137/90-IA) that paid for my survival during these last three years. I thank Esprit BR projects Compulog (no. 3012) and Compulog 2 (no. 6810) for their support, and for giving me the chance of knowing many european researchers in the area of logic programming, with whom I had many aclarifying discussion. For giving me working conditions, I acknowledge the Departamento de Inform´aticaof Fac- uldade de Ciˆenciase Tecnologia of Universidade Nova de Lisboa, the Centro de Inteligˆencia Artificial (CRIA) of Uninova, and Centro de Inform´aticada Universidade Nova de Lisboa (CIUNL). I’m indebted to Anabela Rodrigues and Filipa Reis, for managing most of my bureaucracy, and arranging my travels, during the period of preparation of this thesis and even before. i ii Abstract After a historical introduction, the bulk of the thesis concerns the study of a declarative seman- tics for logic programs. The main original contributions are: ² WFSX (Well–Founded Semantics with eXplicit negation), a new semantics for logic pro- grams with explicit negation (i.e. extended logic programs), which compares favourably in its properties with other extant semantics. ² A generic characterization schema that facilitates comparisons among a diversity of se- mantics of extended logic programs, including WFSX. ² An autoepistemic and a default logic corresponding to WFSX, which solve existing prob- lems of the classical approaches to autoepistemic and default logics, and clarify the mean- ing of explicit negation in logic programs. ² A framework for defining a spectrum of semantics of extended logic programs based on the abduction of negative hypotheses. This framework allows for the characterization of different levels of scepticism/credulity, consensuality, and argumentation. One of the semantics of abduction coincides with WFSX. ² O–semantics, a semantics that uniquely adds more CWA hypotheses to WFSX. The tech- niques used for doing so are applicable as well to the well–founded semantics of normal logic programs. ² By introducing explicit negation into logic programs contradiction may appear. I present two approaches for dealing with contradiction, and show their equivalence. One of the approaches consists in avoiding contradiction, and is based on restrictions in the adoption of abductive hypotheses. The other approach consists in removing contradiction, and is based in a transformation of contradictory programs into noncontradictory ones, guided by the reasons for contradiction. iii iv Sum´ario Depois de uma breve introdu¸c˜aohist´orica,o grosso da tese consiste no estudo de semˆanticas declarativas de programas em l´ogica. As principais contribui¸c˜oesoriginais s˜ao: ² A WFSX (semˆantica bem–fundada com nega¸c˜aoexpl´ıcita), uma nova semˆantica para programas em l´ogicacom nega¸c˜aoexpl´ıcita(i.e. programas em l´ogicaextendidos), que ´e melhor que outras semˆantica existentes, nas suas propriedades. ² Um esquema gen´ericode caracteriza¸c˜aoque facilita compara¸c˜oesentre uma diversidade de semˆanticas, incluindo a WFSX. ² Uma l´ogicaauto–epist´emicae uma l´ogicade regras por omiss˜aocorrespondentes `a WFSX, que por um lado resolvem problemas das abordagens cl´assicasa l´ogicasauto–epist´emicas e a l´ogicasde regras por omiss˜ao,e por outro clarificam o significado da nega¸c˜aoexpl´ıcita em programas em l´ogica. ² Um enquadramento de semˆanticas para programas em l´ogicaextendidos com base na abdu¸c˜aode hip´otesesnegativas. Este enquadramento permite a caracteriza¸c˜aode difer- entes graus de cepticismo/credulidade, consensualidade e argumenta¸c˜ao. Uma das semˆanticas de abdu¸c˜aocoincide com a WFSX. ² A “semˆantica O”, uma semˆantica que acrescenta `a WFSX hip´otesesn˜aocontradit´aveis. As t´ecnicasusadas para a defini¸c˜aodesta semˆantica s˜aotamb´emaplic´aveis `asemˆantica bem–fundada de programas normais. ² Com a introdu¸c˜aoda nega¸c˜aoexpl´ıcitap˜oe–sea quest˜aodo tratamento da contradi¸c˜ao. Introduzem–se duas abordagens, que se mostram equivalentes, para lidar com a con- tradi¸c˜ao.Uma consiste em evit´a–lae a outra em removˆe–la,e s˜aotratadas respectivamente atrav´esde restri¸c˜oesna adop¸c˜aode hip´otesesabdutivas, e da transforma¸c˜aode programas contradit´oriosem programas n˜aocontradit´orios,guiada pelas raz˜oesda contradi¸c˜ao. v vi Sommaire Apr`esune br`eve introduction historique, le gros de la th`eseconsiste en une ´etudede s´emantiques d´eclaratives de la programmation logique. Les principales contribution originales sont: ² La WFSX (s´emantique bien fond´eeavec de la n´egationexplicite), une nouvelle s´emantique pour les programmes logiques avec la n´egationexplicite (c.a.d. programmes logiques ´etendus),laquelle est meilleure que d’autres s´emantiques existantes, en virtu de ses pro- priet´es. ² Un sch´ema caract´erisant g´en´erique qui permet la comparison entre une vari´et´e de s´emantiques, y compris la WFSX. ² Une logique auto–´epist´emiqueet une logique de r´egles`ad´efautcorrespondantes `ala WFSX, qui d’un cot´er´esoudent quelques probl`emesdes abordages classiques aux logiques ´epist´emiqueset des logiques de r´egles`ad´efaut,et qui d’autre part clarifient la signification de la n´egationexplicite en programmation logique. ² Un cadre pour de s´emantiques de la programmation logique ´etendue,fond´esur l’abduction d’hypoth`esesn´egatives. Ce cadre va permettre la caract´erisationde plusieurs degr´esdes cepticisme/cr´edulit´e,de consensualit´eet d’argumentation. Une de ces s´emantiques de l’abduction coincide avec la WFSX. ² L’O–s´emantique, une s´emantique qui ajoute `a la WFSX des hypoth´eses non– contraditables. Les techniques emploiy´eespour la d´efinitionde cette s´emantique sont aussitˆotvalables pour la s´emantique bien fond´eede programmes normaux. ² Avec l’introduction de la n´egationexplicite il se pose la question du traitement de la con- tradiction. On introduit deux approches, qui se r´evelent tout `afait ´equivalens, pour bien faire face `ala contradiction. L’une l’entre elles consiste `al’´eviteret l’autre `ala d`efaire.Ils sont ateints, resp´ectivement, soit par des restrictions sur l’adoption des hypoth´esesabduc- tives, soit par une transformation de programmes contradictoires en d’autres programmes non contradictoires, qui est guid´eepar les causes mˆemesde la contradiction. vii viii Contents Acknowledgements i Abstract iii Sum´ario v Sommaire vii Preface 1 I Semantics of Logic Programs: A Brief Historical Overview 5 1 Normal logic programs 9 1.1 Language :::::::::::::::::::::::::::::::::::::::: 9 1.1.1 Interpretations and models :::::::::::::::::::::::::: 10 1.2 Semantics :::::::::::::::::::::::::::::::::::::::: 11 1.2.1 Stable model semantics :::::::::::::::::::::::::::: 14 1.2.2 Well–founded semantics :::::::::::::::::::::::::::: 15 2 Extended logic programs 17 2.1 Language :::::::::::::::::::::::::::::::::::::::: 19 2.2 Semantics :::::::::::::::::::::::::::::::::::::::: 20 II A New Semantics for Extended Logic Programs 23 3 Why a new semantics for extended programs? 25 4 The WFSX semantics 29 4.1 Interpretations and models :::::::::::::::::::::::::::::: 29 4.2 The definition of WFSX :::::::::::::::::::::::::::::::: 31 4.3 Existence of the semantics ::::::::::::::::::::::::::::::: 36 5 WFSX and autoepistemic logics 39 5.1 Generic semantics