A compact representation of proofs

Different automated theorem provers reason in various deductive systems and, thus, produce proof objects which are in general not compatible. To understand and analyze these objects, one needs to study the corresponding proof theory, and then study the language used to represent proofs, on a prover by prover basis. In this work we present an implementation that takes SMT and Connection proof objects from two different provers and imports them both as expansion trees. By representing the proofs in the same framework, all the algorithms and tools available for expansion trees (compression, visualization, sequent calculus proof construction, proof checking, etc.) can be employed uniformly. The expansion proofs can also be used as a validation tool for the proof objects produced.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, as opposed to the more traditional approaches that deal with tree-like objects such as formulas or sequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, is just a special, conservative fragment of this more general and, in a sense, more basic logic -- the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof complexities offered by cirquent calculus, and more. Among the main purposes of this paper is to provide an introductory-style starting point for what, as the author wishes to hope, might have a chance to become a new line of research in proof theory -- a proof theory based on circuits instead of formulas.

If one travels across time, continents and cultures, one needs to attend to the interests and attitudes of one’s audience. If one does not do so, one risks being thought strange. Of course, the same may hold true if one travels just from New York City to Texas (a mere 1600 miles or so). Many years ago I traveled from New York City, where I was teaching at the time, to Houston, Texas, to give a talk at one of the law schools in Houston. At the time I was much preoccupied with the work that had been done by people in artificial intelligence. I was interested in such stuff because I wondered about the possible applications of that research to the study of evidentiary processes in litigation and adjudication. In my talk in Houston, I tried to explain my tentative conclusion that artificial reasoning methods (including statistical methods) could not supplant ordinary methods of reasoning. After the talk, it soon became apparent that some or much of the audience viewed me as closely akin a man from Mars.1 In this paper, I will use some words—words such as ‘ontology’—that may seem strange or repellent to some or many readers. So I thought I would begin by talking briefly about some of my motivations for talking in ways that may seem strange or unpleasant to some or many readers. Several decades ago I revised the first volume of John Henry Wigmore’s multivolume treatise on the law of evidence.2 Being a reviser is both harder and easier than being an author of one’s own book. It is easier because one can, if one wishes, take the position of a critic and commentator rather than that of author. And in dealing with some or many of the theoretical portions of Wigmore’s magnum opus, I often did exactly that. However, after finishing my revision, I agreed to write a successor to my revision. Now, after a delay of many years, I am doing that. The job of this successor volume, like part of its precursor, is to examine ‘theoretical considerations’ that bear on the law of evidence and proof. This means that I can no longer just be a commentator. Now I must present my own ‘theory of proof’. In looking at what others had done in trying to develop a theory of the law of evidence and proof, I saw many impressive accomplishments. I also saw a variety of approaches. On the one hand, some of the authors seemed to focus on the abstract logic of uncertain reasoning; and their theory of the law of evidence and proof (a theory that they often called a theory of ‘relevance’) effectively amounted to their view of the nature of logical thinking about uncertain factual propositions.3 It seemed to me then—and it seems to me now—that there is something wrong with this approach. Above all, I wondered and I wonder still how it can be confidently said that the methods of reasoning and demonstration in a particular legal system (such as the American one) rest on and express such logic.

Automatic reasoning tools play an important role when developing provably correct software. Both main approaches, program verification and program synthesis employ automated reasoning tools, more specifically, automated theorem provers. Besides classical logic, non-classical logics are particularly relevant in this field. This chapter presents calculi to automate theorem proving in classical and some important non-classical logics, namely first-order intuitionistic and first-order modal logics. These calculi are based on the connection method, which permits a goal-oriented and, hence, a more efficient proof search. The connection calculi for these non-classical logics extend the calculus for classical logic in an elegant and uniform way by adding so-called prefixes to atomic formulae. The leanCoP theorem prover is a very compact PROLOG implementation of the connection calculus for classical logics. We present details of the implementation and describe some basic techniques to improve its efficiency. leanCoP is adapted to non-classical logics by integrating a prefix unification algorithm, which depends on the specific logic. This results in leading theorem provers for the aforementioned non-classical logics.

This paper surveys the work of the last two decades on the complexity of proofs in classical propositional logic, and its connection with open problems in theoretical computer science. The paper ends with a short list of open problems.

This paper is the second part of a series exploring how, given a proof, we can inductively transform it into a proof that contains no irrelevancies and is as strong as possible. In the prequel paper, I defined a weaker and a stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. There, I showed how proofs in core logic and classical core logic can be transformed into perfect proofs. In this paper I study gaunt proofs. I show how proofs in core logic can be inductively transformed into gaunt core proofs, but that this property fails for the natural deduction system of classical core logic.

General and effective methods are required for providing good automation strategies to prove properties of sequent systems. Structural properties such as admissibility, invertibility, and permutability of rules are crucial in proof theory, and they can be used for proving other key properties such as cut-elimination. However, finding proofs for these properties requires inductive reasoning over the provability relation, which is often quite elaborated, exponentially exhaustive, and error prone. This paper aims at developing automatic techniques for proving structural properties of sequent systems. The proposed techniques are presented in the rewriting logic metalogical framework, and use rewrite- and narrowing-based reasoning. They have been fully mechanized in Maude and achieve a great degree of automation when used on several sequent systems, including intuitionistic and classical logics, linear logic, and normal modal logics.

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.

In the 1990s Yankov published two papers containing important contributions to proof theory and based on the application of a dialogical interpretation of proofs. In both cases the method is used for providing constructive proofs of important metalogical results concerning classical logic and fundamental mathematical theories. In the first paper it is shown that impredicative extensions of intuitionistic versions of arithmetic, analysis and set theory, enriched with suitable bar induction schemata, are sufficiently strong for proving the consistency of their classical counterparts. In the second paper the same method is applied to provide a constructive proof of the completeness theorem for classical logic. In both cases a version of a one-sided sequent calculus in Schütte-style is used and cut elimination is established in the second case. Although the obtained results are important, and the applied method is original and interesting, they have not received the attention they deserve from the wider community of researchers in proof theory. In this paper we briefly recall the content of both papers. We focus on essential features of Yankov’s approach and provide comparisons with other results of similar character.KeywordsProof theoryDialogical interpretation of proofsConsistency proofsBar inductionArithmeticAnalysisSet theory

This paper explores how, given a proof, we can systematically transform it into a proof that contains no irrelevancies and which is as strong as possible. I define a weaker and stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. Using classical core logic to study classical validities and core logic to study intuitionistic validities, I show that every core proof or classical core proof can be transformed into a perfect proof. In a sequel paper, I show how proofs in core logic can also be transformed into gaunt proofs and I observe that this property fails for classical core logic.

It is argued that the Curry-Howard correspondence for classical logic implies limitations for logical reasoning that can be learned and performed by large language models. The correspondence establishes an isomorphism between proofs in logic and programs in functional typed lambda calculus. While intuitionistic logic maps to a version of lambda calculus that can be carried out in a local way, i.e., considering local parts of the code in isolation, the version of lambda calculus that corresponds to classical logic requires non-local relations – and this non-locality cannot be learned by large language models due to their restriction to investigate a relative short sequence of tokens. A possible way to go beyond this limitation is sketched as well. Implications for other areas are investigated as well.

We consider some advanced topics in logic including fuzzy logic, temporal logic, intuitionist logic, undefined values in logic, logic and AI and theorem provers. Fuzzy logic is an extension of classical logic that acts as a mathematical model for vagueness. Temporal logic is concerned with the expression of properties that have time dependencies. Brouwer and others developed intuitionist logic as the logical foundation for intuitionism, which was a controversial theory of the foundations of mathematics based on a rejection of the law of the excluded middle and an insistence on constructive existence. We discuss several approaches that have been applied to dealing with undefined values that arise with partial functions including the logic of partial functions; Dijkstra’s approach with his cand and cor operators; and Parnas’s approach which preserves a classical two-valued logic.

In this paper a technique is presented which provides us with a means to develop resolution-based calculi for (first-order) modal and temporal logics. The approach is based on three parts: A special translation technique from modal and temporal logic formulae into classical predicate logic, a certain kind of saturation technique which is to be applied to given background theories, and an extraction of either suitable “simpler” background theories or logic-specific inference rules. The former is interesting in case existing classical logic theorem provers are to be utilized; the latter gains importance if one is prepared to extend theorem provers that are already at hand.

We propose a purely extensional semantics for higher-order logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixed-point of an immediate consequence operator. We also propose an SLD-resolution proof system which is proven sound and complete with respect to the minimum Herbrand model semantics. In other words, we provide a purely extensional theoretical framework for higher-order logic programming which generalizes the familiar theory of classical (first-order) logic programming.