POLYNOMIAL RING CALCULUS FOR MODAL LOGICS: A NEW SEMANTICS AND PROOF METHOD FOR MODALITIES

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra-Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.41150170Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fonds National de la Recherche LuxembourgFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2004/14107-2, 05/04123-3]CNPq [300702/2005-1

Education-oriented Proof Assistant Based on Calculational Logic: Proof Theory Algorithms and Assessment Experience

This work presents an interactive proof assistant, based on Dijkstra-Scholten logic, aimed at teaching logic and discrete mathematics in higher education. The assistant interface is web and easy to use, since inferences can be made just with the mouse. The educational experience is presented showing a correlation between the grades of the assessments in class and those made with the application web. Additionally, an algorithm proof theory for the Disjktra-Scholten system are made and the following algorithms are shown: 1) a versatile printing algorithm that allows the administrator to configure the symbols of a theory, by assigning the desired presentation with LaTeX; 2) An algorithm, based on Broda and Damas combinators, for generate monotonic or anti monotonic inferences in the Dijkstra-Scholten logic; 3) An algorithm to generate the proofs of dual theorems in Boolean Algebra theory

Isabelle/Isar --- eine vielseitige Umgebung für visuell lesbare, formale Beweis-Dokumente

The basic motivation of this work is to make formal theory developments with machine-checked proofs accessible to a broader audience. Our particular approach is centered around the Isar formal proof language that is intended to support adequate composition of proof documents that are suitable for human consumption. Such primary proofs written in Isar may be both checked by the machine and read by human-beings; final presentation merely involves trivial pretty printing of the sources. Sound logical foundations of Isar are achieved by interpretation within the generic Natural Deduction framework of Isabelle, reducing all high-level reasoning steps to primitive inferences. The resulting Isabelle/Isar system is generic with respect to object-logics and proof tools, just as pure Isabelle itself. The full Isar language emerges from a small core by means of several derived elements, which may be combined freely with existing ones. This results in a very rich space of expressions of formal reasoning, supporting many viable proof techniques. The general paradigms of Natural Deduction and Calculational Reasoning are both covered particularly well. Concrete examples from logic, mathematics, and computer-science demonstrate that the Isar concepts are indeed sufficiently versatile to cover a broad range of applications.[Abstract nur auf Englisch verfügbar.] The basic motivation of this work is to make formal theory developments with machine-checked proofs accessible to a broader audience. Our particular approach is centered around the Isar formal proof language that is intended to support adequate composition of proof documents that are suitable for human consumption. Such primary proofs written in Isar may be both checked by the machine and read by human-beings; final presentation merely involves trivial pretty printing of the sources. Sound logical foundations of Isar are achieved by interpretation within the generic Natural Deduction framework of Isabelle, reducing all high-level reasoning steps to primitive inferences. The resulting Isabelle/Isar system is generic with respect to object-logics and proof tools, just as pure Isabelle itself. The full Isar language emerges from a small core by means of several derived elements, which may be combined freely with existing ones. This results in a very rich space of expressions of formal reasoning, supporting many viable proof techniques. The general paradigms of Natural Deduction and Calculational Reasoning are both covered particularly well. Concrete examples from logic, mathematics, and computer-science demonstrate that the Isar concepts are indeed sufficiently versatile to cover a broad range of applications

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

Picturing resources in concurrency

Inspired by the pioneering work of Petri and the rise of diagrammatic formalisms to reason about networks of open systems, we introduce the resource calculus---a graphical language for distributed systems. Like process algebras, the resource calculus is modular, with primitive connectors from which all diagrams can be built. We characterise its equational theory by proving a full completeness result for an interpretation in the symmetric monoidal category of additive relations---a result that constitutes the main contribution of this thesis. Additive relations are frequently exploited by model-checking algorithms for Petri nets. In this thesis, we recognise them as a fundamental algebraic structure of concurrency and use them as an axiomatic framework. Surprisingly, the resource calculus has the same syntax as that of interacting Hopf algebras, a diagrammatic formalism for linear (time-invariant dynamical) systems. Indeed, the approach stems from the simple but fruitful realisation that, by replacing values in a field with values in the semiring of non-negative integers, concurrent behaviour patterns emerge. This change of model reflects the interpretation of diagrams as systems manipulating limited and discrete resources instead of continuous signals. We also extend the resource calculus in two orthogonal directions. First, by adding an affine primitive to express access to a constant quantity of resources. The extended calculus is remarkably expressive and allows the formulation of non-additive patterns of behaviour, like mutual exclusion. Once more, we characterise it---this time as the equational theory of the symmetric monoidal category of polyhedral relations, discrete analogues of polyhedra in convex geometry. Secondly, we add a synchronous register to model stateful systems. The stateful resource calculus is expressive enough to faithfully capture the behaviour of Petri nets while being strictly more expressive. It is also shown to axiomatise a category of open Petri nets, in the style of the connector algebras of nets with boundaries first studied by Bruni, Melgratti, Montanari and Sobociński

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Foundations of Software Science and Computation Structures

586 p.This book constitutes the proceedings of the 21st International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2018, which took place in Thessaloniki, Greece, in April 2018, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2018. The 31 papers presented in this volume were carefully reviewed and selected from 103 submissions. The papers are organized in topical sections named: semantics; linearity; concurrency; lambda-calculi and types; category theory and quantum control; quantitative models; logics and equational theories; and graphs and automata

The Theory of Interacting Deductions and its Application to Operational Semantics

This thesis concerns the problem of complexity in operational semantics definitions. The appeal of modern operational semantics is the simplicity of their metatheories, which can be regarded as theories of deduction about certain shapes of operational judgments. However, when applied to real programming languages they produce bulky definitions that are cumbersome to reason about. The theory of interacting deductions is a richer metatheory which simplifies operational judgments and admits new proof techniques. An interacting deduction is a pair (F, I), where F is a forest of inference trees and I is a set of interaction links (a symmetric set of pairs of formula occurrences of F), which has been built from interacting inference rules (sequences of standard inference rules, or rule atoms). This setting allows one to decompose operational judgments. For instance, for a simple imperative language, one rule atom might concern a program transition, and another a store transition. Program judgments only interact with store judgments when necessary: so stores do not have to be propagated by every inference rule. A deduction in such a semantics would have two inference trees: one for programs and one for stores. This introduces a natural notion of modularity in proofs about semantics. The proof fragmentation theorem shows that one need only consider the rule atoms relevant to the property being proved. To illustrate, I give the semantics for a simple process calculus, compare it with standard semantics and prove three simple properties: nondivergence, store correctness and an equivalence between the two semantics. Typically evaluation semantics provide simpler definitions and proofs than transition semantics. However, it turns out that evaluation semantics cannot be easily expressed using interacting deductions: they require a notion of sequentiality. The sequential deductions contain this extra structure. I compare the utility of evaluation and transition semantics in the interacting case by proving a simple translation correctness example. This proof in turn depends on proof-theoretic concerns which can be abstracted using dangling interactions. This gives rise to the techniques of breaking and assembling interaction links. Again I get the proof fragmentation theorem, and also the proof assembly theorem, which allow respectively both the isolation and composition of modules in proofs about semantics. For illustration, I prove a simple type-checking result (in evaluation semantics) and another nondivergence result (in transition semantics). I apply these results to a bigger language, CSP, to show how the results scale up. Introducing a special scoping side-condition permits a number of linguistic extensions including nested parallelism, mutual exclusion, dynamic process creation and recursive procedures. Then, as an experiment I apply the theory of interacting deductions to present and prove sound a compositional proof system for the partial correctness of CSP programs. Finally, I show that a deduction corresponds to CCS-like process evaluation, justifying philosophically my use of the theory to give operational semantics. A simple corollary is the undecidability of interacting-deducibility. Practically, the result also indicates how one can build prototype interpreters for definitions