lim+, delta+, and Non-Permutability of beta-Steps

Using a human-oriented formal example proof of the (lim+) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a non-permutability of beta-steps and delta+-steps (according to Smullyan's classification), which is not visible with non-liberalized delta-rules and not serious with further liberalized delta-rules, such as the delta++-rule. Besides a careful presentation of the search for a proof of (lim+) with several pedagogical intentions, the main subject is to explain why the order of beta-steps plays such a practically important role in some calculi.Comment: ii + 36 page

Modal Hybrid Logic

This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory

Smart matching

One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behavior in interactive provers. The paper describes the superposition-based implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

Research Perspectives for Logic and Deduction

The article is meant to be kind of the author's manifesto for the role of logic and deduction within Intellectics. Based on a brief analysis of this role the paper presents a number of proposals for future scientic research along the various di-mensions in the space of logical explorations. These dimensions include the range of possible applications including modelling intelligent behavior, the grounding of logic in some semantic context, the choice of an appropriate logic from the great variety of alternatives, then the choice of an appropriate formal system for repre-senting the chosen logic, and nally the issue of developing the most ecient search strategies. Among the proposals is a conjecture concerning the treatment of cuts in proof search. Often a key advance is a matter of applying a small change to a single formula. Ray Kurzweil [Kur05, p.5]

Computational Aspects of Proofs in Modal Logic

Various modal logics seem well suited for developing models of knowledge, belief, time, change, causality, and other intensional concepts. Most such systems are related to the classical Lewis systems, and thereby have a substantial body of conventional proof theoretical results. However, most of the applied literature examines modal logics from a semantical point of view, rather than through proof theory. It appears arguments for validity are more clearly stated in terms of a semantical explanation, rather than a classical proof-theoretic one. We feel this is due to the inability of classical proof theories to adequately represent intensional aspects of modal semantics. This thesis develops proof theoretical methods which explicitly represent the underlying semantics of the modal formula in the proof. We initially develop a Gentzen style proof system which contains semantic information in the sequents. This system is, in turn, used to develop natural deduction proofs. Another semantic style proof representation, the modal expansion tree is developed. This structure can be used to derive either Gentzen style or Natural Deduction proofs. We then explore ways of automatically generating MET proofs, and prove sound and complete heuristics for that procedure. These results can be extended to most propositional system using a Kripke style semantics and a fist order theory of the possible worlds relation. Examples are presented for standard T, S4, and S5 systems, systems of knowledge and belief, and common knowledge. A computer program which implements the theory is briefly examined in the appendix

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten