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

A Semantic Framework for Proof Evidence

International audienceTheorem provers produce evidence of proof in many different formats, such as proof scripts, natural deductions, resolution refutations, Herbrand expansions, and equational rewritings. In implemented provers, numerous variants of such formats are actually used: consider, for example, such variants of or restrictions to resolution refu-tations as binary resolution, hyper-resolution, ordered-resolution, paramodulation, etc. We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence. This framework allows both producers of proof certificates and the checkers of those certificates to have a clear formal definition of the semantics of a wide variety of proof evidence. Employing the FPC framework will allow one to separate a proof from its provenance and to allow anyone to construct their own proof checker for a given style of proof evidence. The foundation on which FPC relies is that of proof theory, particularly recent work into focused proof systems: such proof systems provide protocols by which a checker extracts information from the certificate (mediated by the so called clerks and experts) as well as performs various deterministic and non-deterministic computations. While we shall limit ourselves to first-order logic in this paper, we shall not limit ourselves in many other ways. The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

A defence of predicativism as a philosophy of mathematics

A specification of a mathematical object is impredicative if it essentially involves quantification over a domain which includes the object being specified (or sets which contain that object, or similar). The basic worry is that we have no non-circular way of understanding such a specification. Predicativism is the view that mathematics should be limited to the study of objects which can be specified predicatively. There are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. The other is the positive project, begun by Weyl in Das Kontinuum (1918), to reconstruct as much as possible of classical mathematics on the basis of a predicatively acceptable set theory, which accepts only countably infinite objects. This is a revisionary project, and certain parts of mathematics will not be saved. Chapter 2 contains an account of the historical background to the predicativist project. The rigorization of analysis led to Dedekind's and Cantor's theories of the real numbers, which relied on the new notion of abitrary infinite sets; this became a central part of modern classical set theory. Criticism began with Kronecker; continued in the debate about the acceptability of Zermelo's Axiom of Choice; and was somewhat clarified by Poincaré and Russell. In the light of this, chapter 3 examines the formulation of, and motivations behind the predicativist position. Chapter 4 begins the critical task by detailing the epistemological problems with the classical account of the continuum. Explanations of classicism which appeal to second-order logic, set theory, and primitive intuition are examined and are found wanting. Chapter 5 aims to dispell the worry that predicativism might collapses into mathematical intuitionism. I assess some of the arguments for intuitionism, especially the Dummettian argument from indefinite extensibility. I argue that the natural numbers are not indefinitely extensible, and that, although the continuum is, we can nonetheless make some sense of classical quantification over it. We need not reject the Law of Excluded Middle. Chapter 6 begins the positive work by outlining a predicatively acceptable account of mathematical objects which justifies the Vicious Circle Principle. Chapter 7 explores the appropriate shape of formalized predicative mathematics, and the question of just how much mathematics is predicatively acceptable. My conclusion is that all of the mathematics which we need can be predicativistically justified, and that such mathematics is particularly transparent to reason. This calls into question one currently prevalent view of the nature of mathematics, on which mathematics is justified by quasi-empirical means.Supported by the Arts and Humanities Research Council [grant number 111315]

Arithmetic and Modularity in Declarative Languages for Knowledge Representation

The past decade has witnessed the development of many important declarative languages for knowledge representation and reasoning such as answer set programming (ASP) languages and languages that extend first-order logic. Also, since these languages depend on background solvers, the recent advancements in the efficiency of solvers has positively affected the usability of such languages. This thesis studies extensions of knowledge representation (KR) languages with arithmetical operators and methods to combine different KR languages. With respect to arithmetic in declarative KR languages, we show that existing KR languages suffer from a huge disparity between their expressiveness and their computational power. Therefore, we develop an ideal KR language that captures the complexity class NP for arithmetical search problems and guarantees universality and efficiency for solving such problems. Moreover, we introduce a framework to language-independently combine modules from different KR languages. We study complexity and expressiveness of our framework and develop algorithms to solve modular systems. We define two semantics for modular systems based on (1) a model-theoretical view and (2) an operational view on modular systems. We prove that our two semantics coincide and also develop mechanisms to approximate answers to modular systems using the operational view. We augment our algorithm these approximation mechanisms to speed up the process of solving modular system. We further generalize our modular framework with supported model semantics that disallows self-justifying models. We show that supported model semantics generalizes our two previous model-theoretical and operational semantics. We compare and contrast the expressiveness of our framework under supported model semantics with another framework for interlinking knowledge bases, i.e., multi-context systems, and prove that supported model semantics generalizes and unifies different semantics of multi-context systems. Motivated by the wide expressiveness of supported models, we also define a new supported equilibrium semantics for multi-context systems and show that supported equilibrium semantics generalizes previous semantics for multi-context systems. Furthermore, we also define supported semantics for propositional programs and show that supported model semnatics generalizes the acclaimed stable model semantics and extends the two celebrated properties of rationality and minimality of intended models beyond the scope of logic programs

Universal (Meta-)Logical Reasoning: Recent Successes

Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

The complexity of theorem proving in circumscription and minimal entailment

We provide the first comprehensive proof-complexity analysis of different proof systems for propositional circumscription. In particular, we investigate two sequent-style calculi: MLK defined by Olivetti [28] and CIRC introduced by Bonatti and Olivetti [8], and the tableaux calculus NTAB suggested by Niemelä [26]. In our analysis we obtain exponential lower bounds for the proof size in NTAB and CIRC and show a polynomial simulation of CIRC by MLK. This yields a chain NTAB < CIRC < MLK of proof systems for circumscription of strictly increasing strength with respect to lengths of proofs