A Tableau Calculus with Automaton-Labelled Formulae for Regular Grammar Logics

Introduction to Elementary LogicrnIntroduction to Elementary Logic, which serves as an introduction to the discipline of logic and explores both its traditional and modern aspects, consists of two main sections. The first section addresses the conceptual background of traditional logic, while the second section aims to acquaint readers with logic on both a conceptual and practical level through modern syntactic and semantic methods. The first section, titled “Traditional Logic”, provides readers with a solid conceptual foundation for the discipline of logic. This section compiles the Aristotelian system of logic and the unique contributions of the Islamic logical tradition. By focusing on the terms tasawwur and tasdiq, the section examines the fundamental elements that constitute conceptual and propositional knowledge, analyzing the building blocks of cognition. Thus, the section highlights the primary function of logic: to organize thought and ensure consistency, or in other words, to establish the logical basis for correct reasoning. Additionally, traditional (classical) topics in logic, such as the five universals, categories, and types of propositions, are enriched with examples to enhance the section’s content. The second section, titled “Propositional Logic”, complements the knowledge of traditional logic with modern approaches to propositional logic. The section begins with technical subjects such as the formalization of propositions, logical connectives, and well-formed formulas, continuing with a semantic presentation of propositional logic. This provides a comprehensive analysis of both the formal (syntactic) and semantic dimensions of modern logic. Within the section, logical connections between propositions are explained clearly and accessibly through examples using truth tables, analytic tableaux, and natural deduction methods. Introduction to Elementary Logic is the product of a concise yet comprehensive approach to the field of logic. It aims to serve as a fundamental reference for both newcomers to the discipline and those who wish to learn modern methods in the field of logic.rnKeywords: Traditional logic, propositional logic, conception, assent, inference

This paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Löwenheim-Skolem theorem.The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

graphical sentential logic, the system of alpha graphs, is much more than an historical curiosity. Although a large alpha graph is difficult to manipulate hand, the alpha graphs are admirably suited to automation. In particular, a simple decision procedure for the alpha graphs is tailor-made for the programming language LISP. This decision procedure is described in section I below and is implemented by a LISP program CSProve in section II. Section III gives examples of CSProve as it ran on a desktop computer and indicates how the program is easily adapted to parallel processing. The decision procedure differs from tableau methods and the widely used Robinson resolution method in at least two interesting ways: unlike resolution it does not require that a formula be put into a normal form before the method can be applied, and unlike both tableau methods and resolution it does not require any branching tree constructions. If the method were formulated as a proof procedure in the usual notation it would have only four onepremise inference rules of the form:

In this paper we generalize the existing tableau methods for modal logics. First of all, while usual modal tableaux are based on trees, our basic structures are rooted directed acyclic graphs (RDAG). This allows natural tableau rules for some modal logics that are difficult to capture in the usual way (such as those having an accessibility relation that is dense or confluent). Second, tableau rules rewrite patterns, which are (schemas of) parts of a RDAG. A particular case of these rules are the single-step rules recently proposed by Massacci. This allows in particular tableau rule presentations for K5, KD5, K45, KD45, and S5 that respect the subformula property. Third, we divide modal tableau rules into propagation rules and structural rules. Structural rules construct new edges and nodes (without adding formulas to nodes), while propagation rules add formulas to nodes. This distinction allows to prove completeness in a modular way.

Building Decision Procedures for Modal Logics from Propositional Decision Procedures: The Case Study of Modal K(m)

In this paper we investigate tableau proof procedures for reasoning about actions and plans. Our framework is a multimodal language close to that of propositional dynamic logic, wherein we solve the frame problem by introducing the notion of dependence as a weak causal connection between actions and atoms. The tableau procedure is sound and complete for an important fragment of our language, within which all standard problems of reasoning about actions can be expressed, in particular planning tasks. Moreover, our tableaux are analytic and provide thus a decision procedure.

Dynamic linear time temporal logic

We consider the problem of binary image generation with given properties. This problem arises in a number of practical applications, including generation of artificial porous medium for an electrode of lithium-ion batteries, for composed materials, etc. A generated image represents a porous medium and, as such, it is subject to two sets of constraints: topological constraints on the structure and process constraints on the physical process over this structure. To perform image generation we need to define a mapping from a porous medium to its physical process parameters. For a given geometry of a porous medium, this mapping can be done by solving a partial differential equation (PDE). However, embedding a PDE solver into the search procedure is computationally expensive. We use a binarized neural network to approximate a PDE solver. This allows us to encode the entire problem as a logical formula. Our main contribution is that, for the first time, we show that this problem can be tackled using decision procedures. Our experiments show that our model is able to produce random constrained images that satisfy both topological and process constraints.

Automated theorem proving is a method to establish or disprove logical theorems. While these can be theorems in the classical mathematical sense, we are more concerned with logical encodings of properties of algorithms, hardware and software. Especially in the area of hardware verification, propositional logic is used widely in industry. Satisfiability Module Theories (SMT) is a set of logics which extend propositional logic with theories relevant for specific application domains. In particular, software verification has received much attention, and efficient algorithms have been devised for reasoning over arithmetic and data types. Built-in support for theories by decision procedures is often significantly more efficient than reductions to propositional logic (SAT). Most efficient SAT solvers are based on the DPLL architecture, which is also the basis for most efficient SMT solvers. The main shortcoming of both kinds of logics is the weak support for non-ground reasoning, which noticeably limits the applicability to real world systems. The Model Evolution Calculus (ME) was devised as a lifting of the DPLL architecture from the propositional setting to full first-order logic. In previous work, we created the solver Darwin as an implementation of ME, and showed how to adapt improvements from the DPLL setting. The first half of this thesis is concerned with ME and Darwin. First, we lift a further crucial ingredient of SAT and SMT solvers, lemma-learning, to Darwin and evaluate its benefits. Then, we show how to use Darwin for finite model finding, and how this application benefits from lemma-learning.

A symbolic algorithm for the case-split rule in solving word constraints with extensions

There are three main families of inference algorithms in first-order logic: direct inference and its application to deductive databases and production systems; backward inference procedures and logic programming systems; theorem proving systems based on the resolution method. When solving specific problems, the most effective algorithms are those that allow you to cover all the facts and axioms and must be taken into account in the process of inference. An example is considered in which it is necessary to prove the guilt of a person in murder. On the basis of statements, a knowledge base is formed from expressions, with the help of which an expression of first-order logic is compiled and proved using direct logical inference. The proof of the reasoning obtained in direct inference using the proof tree is given. However, direct inference provides for the implementation of all admissible stages of logical inference based on all known facts. The article also considers a method based on the resolution when implementing the reverse inference, taking into account the expression obtained in the direct inference. This expression is converted into a conjunctive normal formula using the laws of Boolean algebra and is proved by the elimination of events using the conjunction operation.

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

Description logics provide powerful languages for representing and reasoning about knowledge of static application domains. The main strength of description logics is that they offer considerable expressive power going far beyond propositional logic, while reasoning is still decidable. There is a demand to bring the power and character of description logics into the description and reasoning of dynamic application domains which are characterized by actions. In this paper, based on a combination of the propositional dynamic logic PDL, a family of description logics and an action formalism constructed over description logics, we propose a family of dynamic description logics DDL(X @) for representing and reasoning about actions, where X represents well-studied description logics ranging from the to the , and X @ denotes the extension of X with the @ constructor. The representation power of DDL(X @) is reflected in four aspects. Firstly, the static knowledge of application domains is represented as RBoxes and acyclic TBoxes of the description logic X. Secondly, the states of the world and the pre-conditions of atomic actions are described by ABox assertions of the description logic X @, and the post-conditions of atomic actions are described by primitive literals of X @. Thirdly, starting with atomic actions and ABox assertions of X @, complex actions are constructed with regular program constructors of PDL, so that various control structures on actions such as the "Sequence", "Choice", "Any-Order", "Iterate", "If-Then-Else", "Repeat-While" and "Repeat-Until" can be represented. Finally, both atomic actions and complex actions are used as modal operators for the construction of formulas, so that many properties on actions can be explicitly stated by formulas. A tableau-algorithm is provided for deciding the satisfiability of DDL(X @)-formulas; based on this algorithm, reasoning tasks such as the realizability, executability and projection of actions can be effectively carried out. As a result, DDL(X @) not only offers considerable expressive power going beyond many action formalisms which are propositional, but also provides decidable reasoning services for actions described by it.

We study a framework for the specification of architecture styles as families of architectures involving a common set of types of components and coordination mechanisms. The framework combines two logics: 1 interaction logics for the specification of architectures as generic coordination schemes involving a configuration of interactions between typed components; 2 configuration logics for the specification of architecture styles as sets of interaction configurations. The presented results build on previous work on architecture modelling in BIP. We show how propositional interaction logic can be extended into a corresponding configuration logic by adding new operators on sets of interaction configurations. We provide a complete axiomatisation of the propositional configuration logic, as well as a decision procedure for checking that an architecture satisfies given logical specifications. To allow genericity of specifications, we study first-order and second-order extensions of the propositional logic. We provide examples illustrating the application of the results to the characterization of architecture styles. Finally, we provide an experimental evaluation using the Maude rewriting system to implement the decision procedure for the propositional logic.

A new decision procedure for Propositional Projection Temporal Logic (PPTL) is proposed which is an improvement to the decision procedure given in [4]. The main contribution of the paper is as follows: (1) the relationship between paths in the NFG of a formula R and its models is established and proved; (2) a new Labeled NFG (LNFG) with a set of labels (propositions) is defined; (3) given a formula R, an LNFG of R can be generated by the new decision algorithm, and all models of R can be found; (4) based on the new decision procedure, an improved model checking algorithm is presented and implemented.KeywordsPropositional Projection Temporal LogicSatisfiabilityDecision ProcedureModel CheckingVerification