Set theory and logic-driven AI reasoning for decision-making systems

The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

World experience shows that intelligent information systems for decision support are an integral part of modern society functioning. The basis of many tasks that lies in the development of decision support systems is the presentation of knowledge of a particular subject field. Each decision-support system, depending on the application field, has its own characteristics that characterize the main objectives of this subject area, among which, for example, the presentation of knowledge of multi-level administration systems for decision support purposes. In modern conditions, mathematical modeling is the most effective for the formal representation of knowledge in decision support systems which can be used to represent declarative knowledge of land relations. Therefore, urgent scientific practical problem is relevant now that lies in the representing contradictory knowledge in multilevel administration systems. The scientific and practical task envisages the transformation of declarative knowledge of the subject field in the form of mathematical and informational models using elements of set theory. The task envisages the transformation of declarative knowledge in the form of mathematical models with using set theory. The research used methods: analysis, set theory, mathematical modeling, corpus linguistics, ontological engineering. The results of the study are important for organizing activities in a variety of industries using a large body of documents and laws. The purpose of this work is to develop models of knowledge representation on the use of set theory in ontology. The object of research is the process of decision support in multilevel systems. Subject of research is the mathematical models of knowledge representation in multilevel administration systems using set theory. Keywords: formalization, theory, predicate logic, corpus linguistics, ontology, set theory, model.

Logical and layout structures of documents

The aim of this study is to develop an automated rule-checking application that enables the rapid, transparent, and efficient verification of building design data, created through Building Information Modeling (BIM), in compliance with zoning regulations during the design and approval phases of construction projects in Turkiye. In this context, rules expressed in regulatory texts were transformed into a format adaptable to software using propositional logic, and decision structures were created. Distinctive parameters based on the rules to be checked were assigned to the BIM environment. Model data were exported in the shareable Industry Foundation Classes (IFC) format, and a Python-based software was developed to test the automated rule-checking method on four different sample projects, generating result reports. The findings show that regulatory provisions can be converted into logical structures and compared with Industry Foundation Classes data, making it possible to digitally assess a project’s compliance with regulations. This approach could contribute to enabling public authorities to implement a more effective and transparent control mechanism in the design and approval processes. The originality of the study lies in the successful application of classical propositional logic—despite its lack of complex logical operators—to complex regulatory rules. Moreover, the set of validated propositions not only defines the outcome of the rule check but also provides comprehensive contextual information related to the rule, allowing for a more in-depth evaluation of the results.

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

A formalism for declarative specification and programming is introduced that forms a logical and methodological framework for program and system specification and construction. It combines axiomatic techniques based on logical concepts for specifying properties and the possibility to introduce names for objects. In particular it comprises within one formalism, the possibilities of formulating specifications and defining algorithms. The logical formalism is based more or less on typed predicate logic. The development rules are particular proof rules together with the inference rules of predicate logic. As a special aspect, the authors consider logical formulas which explicitly specify typed identifiers as elements of signatures. >

This paper introduces the use of propositional logic calculus in the elaboration of educational quizzes to assess the level understanding of students in a specific theme of their courses. The technique introduced in this paper goes beyond multiple-choice quizzes. The technique requires several steps like a) to give a phrase, b) to re-order words of the given phrase in order to form a propositional logic formula, c) to make use of background knowledge for performing substitutions, d) to answer questions from one of the person in the team, e) to change synonyms/antonyms (if this is feasible), f) to perform actions in order to give value to both or at least one operand of the logic formula and g) to conclude the final answer of the logic formula (true or false) depending of the logic values of the operands in the logic formula. As a working example, the author shows a quiz for universal history, however, the same technique could be used to assess students in different courses.

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows.Pr(x), alias Bew(x), is the standard provability predicate of PA. For any formula F of PA, Pr[F] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F, i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[F], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[F] = Pr(‘F’), the sentence that expresses the PA-provability of F.Let υ1, υ2,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P* of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P.

We pursue the idea that predicate logic is a “fibred algebra” while propositional logic is a single algebra; in the context of intuitionism, this algebraic understanding of predicate logic goes back to Lawvere, in particular his concept of hyperdoctrine. Here, we aim at demonstrating that the notion of monad-relativised hyperdoctrines, which are what we call fibred algebras, yields algebraisations of a wide variety of predicate logics. More specifically, we discuss a typed, first-order version of the non-commutative Full Lambek calculus, which has extensively been studied in the past few decades, functioning as a unifying language for different sorts of logical systems (classical, intuitionistic, linear, fuzzy, relevant, etc.). Through the concept of Full Lambek hyperdoctrines, we establish both generic and set-theoretical completeness results for any extension of the base system; the latter arises from a dual adjunction, and is relevant to the tripos-to-topos construction and quantale-valued sets. Furthermore, we give a hyperdoctrinal account of Girard’s and Gödel’s translation.KeywordsPropositional LogicType TheoryCategorical SemanticResiduated LatticeUniversal AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Propositional and predicate logics of incomplete information

From the Publisher: This text makes in-depth explorations of a broad range of theoretical topics in computer science. It plunges into the applications of the abstract concepts in order to confront and address the skepticism of readers, and instill in them an appreciation for the usefulness of theory. A two-part presentation integrates logic and formal language—both with applications. Chapter topics cover mathematical preliminaries, propositional logic, proving things: why and how, predicate logic, proving with predicates, program verification, logic programming, language models for computer science, language models, finite automata and their languages, regular expressions, Lex: a tool for building lexical scanners, context-free grammars, pushdown automata and parsing, and turing machines. For future computer scientists.

As we sketched in [Godo and Hajek, 1996], most of fuzzy inference in Zadeh’s style is well formalized in the many-sorted many-valued Pavelka-style fuzzy predicate logic — and this can be done in several ways. The most important fact is that patterns of fuzzy inference can be presented as sound deduction rules, with all necessary assumptions expressed by some formulas of predicate logic (with the usual quantifiers ∀, ∃) in the premisse part of the rule. Here we complement the presentation of [Godo and Hajek, 1996] by a somewhat unexpected formalization of Zadeh’s Generalized Modus Ponens (GMP, cf. e.g. Zadeh [1988; 1990; 1994]). We shall be sketchy; a full detailed presentation is to be found in [Godo and Hajek, to appear]. For the reader’s convenience we repeat the definition of the many-sorted RQL (rational quantification logic, cf. [Hajek, 1995; Hajek, 1997]).

Until today, many layout recognition/analysis methods have been proposed, but the guideline for knowledge representation applicable to documents which should be analyzed newly is not always discussed directly. This paper addresses such a subject. Generally, the documents can be categorized into the appropriate document types on a basis of the features of layout structures. Then, the processing mechanisms are assessed with a view to establishing the criteria for selecting the knowledge representation means appropriate to the document types. First, we define the physical layout structure and logical layout structure in addition to the traditional concepts of layout structure and logical structure. Second, we define the document type on the basis of the relationship between the physical layout structure and logical layout structure. Third, we make the knowledge representation means and the processing mechanisms clear under the document types. Finally, we show a criterion or guideline for knowledge representation means and processing mechanisms with respect to the logical layout structure and physical layout structure. For this discussion, our basic view is derived from our document understanding methods which we have developed for several different documents.

DIMACS Series in Discrete Mathematicsand Theoretical Computer ScienceVolume00, 19xxA Branch and Cut Algorithm for MAX-SAT and WeightedMAX-SATSteveJoy, John Mitchell, and Brian BorchersAbstract.We describ e a branch and cut algorithm for b oth MAX-SAT andweighted MAX-SAT. This algorithm uses the GSAT pro cedure as a primalheuristic. At eachnode we solve a linear programming (LP) relaxation of theproblem. Twostyles of separating cuts are added: resolution cuts and o ddcycle inequalities.We compare our algorithm to an extension of the Davis Putnam Love-land (EDPL) algorithm. Our algorithm is more e ective than EDPL on someproblems, notably MAX-2-SAT. EDPL is more e ective on some other classesof problems.1. Intro ductionThe satis ability problem (SAT) is a problem in prop ositional logic. A logicformula consists of the conjunction of clauses. Each clause consists of a disjunctionof literals. Each literal is a variable or its negation. The SAT problem seeks to ndan assignment to the variables which satis es the logic formula, or an indicationthat no such assignment exists. The satis ability problem is NP-complete [5].There are a numb er of exact algorithms for the satis ability problem. Theseinclude Davis-Putnam-Loveland [4,23], resolution [26], and integer programmingapproaches [1,16, 18, 20,21]. A numb er of heuristics that use randomizationalso exist; the rst randomized lo cal search algorithm for satis abilitywas due toGu [9, 10, 11, 12]. Other algorithms include the GSAT heuristic [29, 30] and theGRASP heuristic [25]. For surveys of algorithms for SAT problems see [13,14].In this pap er weinvestigate the related MAX-SAT problem. Given a collectionof clauses, we seek a variable assignment that maximizes the numb er of satis edclauses. The weighted MAX-SAT problem assigns a wt to each clause, and seeksan assignment that maximizes the sum of the weights of the satis ed clauses. Bothof these problems are NP-hard. It is p ossible to approximate MAX-SAT within afactor of 1.325 in p olynomial time [6].Most SAT heuristics have b een extended to MAX-SAT. Several heuristics forMAX-SAT are summarized in Hansen and Jaumard [15]. The GSAT heuristic hasalso b een extended to weighted MAX-SAT[22].1991Mathematics Subject Classi cation.03B05, 49M35, 65K05, 90C10.Research supp orted by ONR Grant N00014{94{1{0391 to Rensselaer Polytechnic Institute.c0000 American Mathematical So ciety1052-1798/00 $1.00 + $.25 p er page1

This chapter proposes a methodology to implement the uncertain linguistic summarizer posed in Liu’s uncertain logic to measure the performance of investment funds in the Colombian capital market. The algorithm extracts a truth value for a set of linguistic summaries, written as propositions in predicate logic, where the terms for the quantifier, subject, and predicate are unsharp. The linguistic summarizer proves to be autonomous, successful, efficient, and close to human language. Furthermore, the implementation has a general scope and could become a data mining tool under uncertainty. The propositions found characterize with plenty of sense the investment funds data. Finally, a Corollary that allows accelerating the obtention of the summaries is presented.