Goal-directed proof theory

This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape

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

Deep Inference and Symmetry in Classical Proofs

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

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

Decidability for Residuated Lattices and Substructural Logics

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical methods for establishing decidability as well as tractability of decision procedures. On the undecidability front, we develop a notion of algebraic machines for which the theory of residuated frames will allow us to encode decision problems within the theories of residuated lattices and their substructural analogues. We prove the undecidability of the word problem for a broad class of simple extensions for both commutative and non-commutative residuated lattices. Furthermore, through a deduction theorem we establish the undecidability of the equational theory for a broad class of simple extensions. Translated in terms of substructural logics, we prove that the undecidability of both provability and deducibility for a multitude of extensions of FLe by simple rules

LDS - Labelled Deductive Systems: Volume 1 - Foundations

Traditional logics manipulate formulas. The message of this book is to manipulate pairs; formulas and labels. The labels annotate the formulas. This sounds very simple but it turned out to be a big step, which makes a serious difference, like the difference between using one hand only or allowing for the coordinated use of two hands. Of course the idea has to be made precise, and its advantages and limitations clearly demonstrated. `Precise' means a good mathematical definition and `advantages demonstrated' means case studies and applications in pure logic and in AI. To achieve that we need to address the following: \begin{enumerate} \item Define the notion of {\em LDS}, its proof theory and semantics and relate it to traditional logics. \item Explain what form the traditional concepts of cut elimination, deduction theorem, negation, inconsistency, update, etc.\ take in {\em LDS}. \item Formulate major known logics in {\em LDS}. For example, modal and temporal logics, substructural logics, default, nonmonotonic logics, etc. \item Show new results and solve long-standing problems using {\em LDS}. \item Demonstrate practical applications. \end{enumerate} This is what I am trying to do in this book. Part I of the book is an intuitive presentation of {\em LDS} in the context of traditional current views of monotonic and nonmonotonic logics. It is less oriented towards the pure logician and more towards the practical consumer of logic. It has two tasks, addressed in two chapters. These are: \begin{itemlist}{Chapter 1:} \item [Chapter1:] Formally motivate {\em LDS} by starting from the traditional notion of `What is a logical system' and slowly adding features to it until it becomes essentially an {\em LDS}. \item [Chapter 2:] Intuitively motivate {\em LDS} by showing many examples where labels are used, as well as some case studies of familiar logics (e.g.\ modal logic) formulated as an {\em LDS}. \end{itemlist} The second part of the book presents the formal theory of {\em LDS} for the formal logician. I have tried to avoid the style of definition-lemma-theorem and put in some explanations. What is basically needed here is the formulation of the mathematical machinery capable of doing the following. \begin{itemize} \item Define {\em LDS} algebra, proof theory and semantics. \item Show how an arbitrary (or fairly general) logic, presented traditionally, say as a Hilbert system or as a Gentzen system, can be turned into an {\em LDS} formulation. \item Show how to obtain a traditional formulations (e.g.\ Hilbert) for an arbitrary {\em LDS} presented logic. \item Define and study major logical concepts intrinsic to {\em LDS} formalisms. \item Give detailed study of the {\em LDS} formulation of some major known logics (e.g.\ modal logics, resource logics) and demonstrate its advantages. \item Translate {\em LDS} into classical logic (reduce the `new' to the `old'), and explain {\em LDS} in the context of classical logic (two sorted logic, metalevel aspects, etc). \end{itemize} \begin{itemlist}{Chapter 1:} \item [Chapter 3:] Give fairly general definitions of some basic concepts of {\em LDS} theory, mainly to cater for the needs of the practical consumer of logic who may wish to apply it, with a detailed study of the metabox system. The presentation of Chapter 3 is a bit tricky. It may be too formal for the intuitive reader, but not sufficiently clear and elegant for the mathematical logician. I would be very grateful for comments from the readers for the next draft. \item [Chapter 4:] Presents the basic notions of algebraic {\em LDS}. The reader may wonder how come we introduce algebraic {\em LDS} in chapter 3 and then again in chapter 4. Our aim in chapter 3 is to give a general definition and formal machinery for the applied consumer of logic. Chapter 4 on the other hand studies {\em LDS} as formal logics. It turns out that to formulate an arbitrary logic as an {\em LDS} one needs some specific labelling algebras and these need to be studied in detail (chapter 4). For general applications it is more convenient to have general labelling algebras and possibly mathematically redundant formulations (chapter 3). In a sense chapter 4 continues the topic of the second section of chapter 3. \item [Chapter 5:] Present the full theory of {\em LDS} where labels can be databases from possibly another {\em LDS}. It also presents Fibred Semantics for {\em LDS}. \item [Chapter 6:] Presents a theory of quantifers for {\em LDS}. The material for this chapter is still under research. \item [Chapter 7:] Studies structured consequence relations. These are logical system swhere the structure is not described through labels but through some geometry like lists, multisets, trees, etc. Thus the label of a wff $A$ is implicit, given by the place of $A$ in the structure. \item [Chapter 8:] Deals with metalevel features of {\em LDS} and its translation into two sorted classical logic. \end{itemlist} Parts 3 and 4 of the book deals in detail with some specific families of logics. Chapters 9--11 essentailly deal with substructural logics and their variants. \begin{itemlist}{Chapter10:} \item [Chapter 9:] Studies resource and substructural logics in general. \item [Chapter 10:] Develops detailed proof theory for some systems as well as studying particular features such as negation. \item [Chapter 11:] Deals with many valued logics. \item [Chapter 12:] Studies the Curry Howard formula as type view and how it compres with labelling. \item [Chapter 13:] Deals with modal and temporal logics. \end{itemlist} Part 5 of the book deals with {\em LDS} metatheory. \begin{itemlist}{Chapter15:} \item [Chapter 14:] Deals with labelled tableaux. \item [Chapter 15:] Deals with combining logics. \item [Chapter 16:] Deals with abduction. \end{itemlist