Proof search issues in some non-classical logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed.This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin’s cutfree LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1–1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called ‘permutation-free’ calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a ‘permutation-free’ calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to besound and complete with respect to a proof-theoretic semantics and (weak) cutelimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter

Proof Search Issues in Some Non-Classical Logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli (citeandreoli-92) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut-free LJT (citeherb-95, here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1--1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called `permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding emphet al (citeheu-sey-zim-96). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a `permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut-elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Proof search issues in some non-classical logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4

Constructivisation through Induction and Conservation

The topic of this thesis lies in the intersection between proof theory and alge- braic logic. The main object of discussion, constructive reasoning, was intro- duced at the beginning of the 20th century by Brouwer, who followed Kant’s explanation of human intuition of spacial forms and time points: these are constructed step by step in a finite process by certain rules, mimicking con- structions with straightedge and compass and the construction of natural numbers, respectively. The aim of the present thesis is to show how classical reasoning, which admits some forms of indirect reasoning, can be made more constructive. The central tool that we are using are induction principles, methods that cap- ture infinite collections of objects by considering their process of generation instead of the whole class. We start by studying the interplay between cer- tain structures that satisfy induction and the calculi for some non-classical logics. We then use inductive methods to prove a few conservation theorems, which contribute to answering the question of which parts of classical logic and mathematics can be made constructive.Tämän opinnäytetyön aiheena on todistusteorian ja algebrallisen logiikan leikkauspiste. Keskustelun pääaiheen, rakentavan päättelyn, esitteli 1900-luvun alussa Brouwer, joka seurasi Kantin selitystä ihmisen intuitiosta tilamuodoista ja aikapisteistä: nämä rakennetaan askel askeleelta äärellisessä prosessissa tiettyjen sääntöjen mukaan, jotka jäljittelevät suoran ja kompassin konstruktioita ja luonnollisten lukujen konstruktiota. Tämän opinnäytetyön tavoitteena on osoittaa, kuinka klassista päättelyä, joka mahdollistaa tietyt epäsuoran päättelyn muodot, voidaan tehdä rakentavammaksi. Keskeinen työkalu, jota käytämme, ovat induktioperiaatteet, menetelmät, jotka keräävät äärettömiä objektikokoelmia ottamalla huomioon niiden luomisprosessin koko luokan sijaan. Aloitamme tutkimalla vuorovaikutusta tiettyjen induktiota tyydyttävien rakenteiden ja joidenkin ei-klassisten logiikan laskelmien välillä. Todistamme sitten induktiivisten menetelmien avulla muutamia säilymislauseita, jotka auttavat vastaamaan kysymykseen siitä, mitkä klassisen logiikan ja matematiikan osat voidaan tehdä rakentaviksi

Supporting Format Migration with Ontology Model Comparison

Die ausschließliche Bewahrung der reinen Bitfolge eines digitalen Dokuments führt nicht dazu, dass zu einem späteren Zeitpunkt auch Information aus dem Dokument extrahiert werden kann. Wenn keine Formatinformation verfügbar ist, muss der Inhalt des Dokuments stattdessen als verloren angesehen werden. Eine Lösung für das Problem ist die (wiederholte) Konvertierung in immer neuere Formate. Entsprechende Verfahren, diese Konvertierungen zu ermöglichen und zu automatisieren sind Teil der aktuellen Forschung. Diese Arbeit geht von der Annahme aus, dass digitale Dokumente als formale Ontologien repräsentiert werden können, was es wiederum ermöglicht, existierende Verfahren aus dem Ontology Matching zu verwenden, um Dokumentenformate aufeinander abzubilden. Bis auf wenige Ausnahmen sind existierende Verfahren beschränkt: Sie bilden Klassen auf Klassen, Rollen auf Rollen und Individuen auf Individuen ab. Solche einfachen Abbildungen sind für komplexe Dokumentenformate unzureichend. In dieser Arbeit wird zum einen eine Methode entwickelt, um einfache Abbildungen heuristisch zu komplexeren Regeln zu verfeinern. Das neue Verfahren basiert auf Tableau-Verfahren für Beschreibungslogiken und verwendet eine modelbasierte Repräsentation von komplexen Korrespondenzen. Ein zweiter Teil verwendet die modelbasierte Darstellung, um Vorschläge für bestmögliche Abbildungen zwischen Dokumenten zu finden. Die hier entwickelte Methode verwendet die semantischen Information sowohl aus dem Tableau- wie auch aus dem Verfeinerungsverfahren. Das Ergebnis ist eine neuartige Methode zur halbautomatischen Ableitung komplexer Abbildungen zwischen beschreibungslogischen Ontologien. Das Verfahren ist zugeschnitten aber nicht beschränkt auf das Feld der Formatmigration.Being able to read successfully the bits and bytes stored inside a digital archive does not necessarily mean we are able to extract meaningful information from an archived digital document. If information about the format of a stored document is not available, the contents of the document are essentially lost. One solution to the problem is format conversion, but due to the amount of documents and formats involved, manual conversion of archived documents is usually impractical. There is thus an open research question to discover suitable technologies to transform existing documents into new document formats and to determine the constraints within which these technologies can be applied successfully. In the present work, it is assumed that stored documents are represented as formal description logic ontologies. This makes it possible to view the translation of document formats as an application of ontology matching, an area for which many methods and algorithms have been developed over the recent years. With very few exceptions, however, current ontology matchers are limited to element-level correspondences matching concepts against concepts, roles against roles, and individuals against individuals. Such simple correspondences are insufficient to describe mappings between complex digital documents. This thesis presents a method to refine simple correspondences into more complex ones in a heuristic fashion utilizing a modified form of description logic tableau reasoning. The refinement process uses a model-based representation of correspondences. Building on the formal semantics, the process also includes methods to avoid the generation of inconsistent or incoherent correspondences. In a second part, this thesis also makes use of the model-based representation to determine the best set of correspondences between two ontologies. The developed similarity measures make use of semantic information from both description logic tableau reasoning as well as from the refinement process. The result is a new method to semi-automatically derive complex correspondences between description logic ontologies tailored but not limited to the context of format migration

Model Checking and Model-Based Testing : Improving Their Feasibility by Lazy Techniques, Parallelization, and Other Optimizations

This thesis focuses on the lightweight formal method of model-based testing for checking safety properties, and derives a new and more feasible approach. For liveness properties, dynamic testing is impossible, so feasibility is increased by specializing on an important class of properties, livelock freedom, and deriving a more feasible model checking algorithm for it. All mentioned improvements are substantiated by experiments

Almost duplication-free tableau calculi for propositional Lax logics

In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1 , where the calculus for PLL 1 is duplication--free while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in relation to the problem of Formal Hardware Verification. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Namely, we enlarge the language containing the usual sings T and F with the new sign F c . PLL and PLL 1 logics are characterized by a Kripke--semantics which is a "weak" version of the semantics for ordinary intuitionistic modal logics. In this paper we establish the soundness and completeness theorems for these calculi