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Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 26 lipca 2024

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1

LANZET, RAN. "A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL". Review of Symbolic Logic 10, nr 3 (8.05.2017): 549–82. http://dx.doi.org/10.1017/s1755020317000053.

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AbstractThis paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

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2

Friedman, Harvey M., i Andrej Ščedrov. "On the quantificational logic of intuitionistic set theory". Mathematical Proceedings of the Cambridge Philosophical Society 99, nr 1 (styczeń 1986): 5–10. http://dx.doi.org/10.1017/s0305004100063854.

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Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).

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3

Ruitenburg, Wim. "Basic Predicate Calculus". Notre Dame Journal of Formal Logic 39, nr 1 (styczeń 1998): 18–46. http://dx.doi.org/10.1305/ndjfl/1039293019.

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4

BEN-YAMI, HANOCH. "THE QUANTIFIED ARGUMENT CALCULUS". Review of Symbolic Logic 7, nr 1 (22.01.2014): 120–46. http://dx.doi.org/10.1017/s1755020313000373.

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AbstractI develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language.

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5

Liusti, Siti Ainim. "ANALISIS KALIMAT BERDASARKAN POLA KALIMAT DASAR DAN KALKULUS PREDIKAT". Adabiyyāt: Jurnal Bahasa dan Sastra 15, nr 2 (25.12.2016): 157. http://dx.doi.org/10.14421/ajbs.2016.15203.

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This study aims to analyze the sentences based on the basic sentence patterns of Indonesian and predicate calculus. These approaches put the predicate as a core component in the sentence structure. The object of study is focused on declarative sentences of Indonesian. The data analysis consists of several stages. Basic sentence patterns of Indonesian consist of identifying the type of sentences, identifying the elements forming sentences, and putting on elements which are based on basic sentence patterns of Indonesian. Predicate calculus consists of identifying atomic or compound propositions, determining the predicate and other components, defining a form of expression predicate calculus, and making a notation function. The results showed that the basic sentence pattern analysis only identifies the internal elements in a single sentence, while the predicate calculus can as well identifies the internal elements of a single or compound sentence.

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6

Andjelkovic, Danica. "Aristotle's syllogistic and modern logic". Theoria, Beograd 48, nr 3-4 (2005): 155–66. http://dx.doi.org/10.2298/theo0504155a.

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Different understandings of Aristotle's syllogistic as a logical theory are reviewed. Leibniz offered a mathematical interpretation of syllogistic. Boole expressed all syllogistic relations by means of algebraic formulas. Lukasiewicz built a system of syllogistic as a logical theory separate and different from the predicate calculus Comparing syllogistic with other formal systems, its definitional equivalence with Boolean algebra is proven. Many systems of syllogistic are built, and their differences are due to recognizing the bearer of existential sense of categorical propositions. It is shown that these systems can be embedded in the predicate calculus, which means that syllogistic is not a separate and different theory from the predicate calculus.

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7

Shalack, Vladimir. "On Some Applied First-Order Theories which Can Be Represented by Definitions". Bulletin of the Section of Logic 44, nr 1/2 (1.01.2015): 19–24. http://dx.doi.org/10.18778/0138-0680.44.1.2.03.

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In the paper we formulate a sufficient criterion in order for the first order theory with finite set of axioms to be represented by definitions in predicate calculus. We prove the corresponding theorem. According to this criterion such theories as the theory of equivalence relation, the theory of partial order and many theories based on the equality relation with finite set of functional and predicate symbols are represented by definitions in the first-order predicate calculus without equality.

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8

Konikowska, Beata, Andrzej Tarlecki i Andrzej Blikle. "A Three-Valued Logic for Software Specification and Validation. Tertium tamen datur". Fundamenta Informaticae 14, nr 4 (1.04.1991): 411–53. http://dx.doi.org/10.3233/fi-1991-14403.

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Different calculi of partial or three-valued predicates have been used and studied by several authors in the context of software specification, development and validation. This paper offers a critical survey on the development of three-valued logics based on such calculi. In the first part of the paper we review two three-valued predicate calculi, based on, respectively, McCarthy’s and Kleene’s propositional connectives and quantifiers, and point out that in a three-valued logic one should distinguish between two notions of validity: strong validity (always true) and weak validity (never false). We define in model-theoretic terms a number of consequence relations for three-valued logics. Each of them is determined by the choice of the underlying predicate calculus and of the weak or strong validity of axioms and of theorems. We discuss mutual relationships between consequence relations defined in such a way and study some of their basic properties. The second part of the paper is devoted to the development of a formal deductive system of inference rules for a three-valued logic. We use the method of semantic tableaux (slightly modified to deal with three-valued formulas) to develop a Gentzen-style system of inference rules for deriving valid sequents, from which we then derive a sound and complete system of natural deduction rules. We have chosen to study the consequence relation determined by the predicate calculus with McCarthy’s propositional connectives and Kleene’s quantifiers and by the strong interpretation of both axioms and theorems. Although we find this choice appropriate for applications in the area of software specification, verification and development, we regard this logic merely as an example and use it to present some general techniques of developing a sequent calculus and a natural deduction system for a three-valued logic. We also discuss the extension of this logic by a non-monotone is-true predicate.

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9

Monahan, Brian. "Predicate calculus and program semantics". Science of Computer Programming 17, nr 1-3 (grudzień 1991): 259–62. http://dx.doi.org/10.1016/0167-6423(91)90048-3.

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10

Börger, Egon. "Predicate calculus and program semantics". Science of Computer Programming 23, nr 1 (październik 1994): 91–101. http://dx.doi.org/10.1016/0167-6423(94)90002-7.

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11

McQuade, Thomas J. "From Syllogism to Predicate Calculus". Teaching Philosophy 17, nr 4 (1994): 293–309. http://dx.doi.org/10.5840/teachphil199417448.

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12

Lin, Hui-Min. "Predicate ?-Calculus for Mobile Ambients". Journal of Computer Science and Technology 20, nr 1 (styczeń 2005): 95–104. http://dx.doi.org/10.1007/s11390-005-0011-7.

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13

Liusti, Siti Ainim. "ANALISIS KALKULUS PREDIKAT PADA TERJEMAHAN SURAH AL-SAFFAT". Adabiyyāt: Jurnal Bahasa dan Sastra 12, nr 1 (30.07.2013): 190. http://dx.doi.org/10.14421/ajbs.2013.12109.

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This study based on Chomsky’s generative grammar that about surface and deep structure. It is supported by Bierwish theory about projection rules, and limits choice theory by Katz and Bierwish. Predicate calculus is a study of sentence structure by relation to the componential meaning of the lexicon. Predicate calculus rests on the deep structure to form a sentence. Deep structure is able to describe such as subject, predicate, object, and others. Verb as a predicate is a core of sentence. The purposes of this research is to formulate the types of verbs and describing it based on predicate calculus. The object of this research is some sentences of the letter translation al-S{a>ffa>t from the Holy Qur’an. There are four types of verbs are : (1) V + NP, it is commonly like V kopula + NP as variation of verb. (2) V + NP + PP, the variation of verb can be V kopula+ NP + PP. (3) V + PP, or verbs can be V kopula + PP. (4) V, without other arguments followed. It can be VP just as a verb. In addition, it is common that the PP may consist of (a) Adj + Prep, and (b) Prep + NP.

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14

Bull, R. A. "MIPC as the formalisation of an intuitionist concept of modality". Journal of Symbolic Logic 31, nr 4 (styczeń 1997): 609–16. http://dx.doi.org/10.2307/2269696.

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In the course of a recent paper on modal' extensions of the intuitionist propositional calculus, [1], I made some suggestions as to the relationships between the system MIPC, the intuitionist predicate calculus, and the question of producing a genuine intuitionist concept of modality. This paper may be regarded as a clarification of those rather inaccurate ideas in the light of Kripke's outstanding analysis of the intuitionist predicate calculus, [2]. (I use Kripke's notation and terminology here without explanation — this work is intended to be read in conjunction with [2].) In particular, I shall adapt his interpretation of his modelling to give an account of MIPC in terms of differing mathematical intuitions.

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15

NEGRI, SARA, i JAN VON PLATO. "Proof systems for lattice theory". Mathematical Structures in Computer Science 14, nr 4 (sierpień 2004): 507–26. http://dx.doi.org/10.1017/s0960129504004244.

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A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent.An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.

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16

Kosovskaya, Tatiana M., i Nikolai N. Kosovskii. "Extraction of common properties of objects for creation of a logic ontology". Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 18, nr 1 (2022): 37–51. http://dx.doi.org/10.21638/11701/spbu10.2022.103.

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The paper describes an approach to the formation of ontology based on descriptions of objects in terms of the predicate calculus language. With this approach, an object is represented as a set of its elements, on which a set of predicates is defined that defines the properties of these elements and the relationship between them. A description of an object is a conjunction of literals that are true on elements of an object. In the present work, ontology is understood as an oriented graph with descriptions of subsets as nodes and such that the elements of a set at the end of an oriented edge have the properties of the elements of the set at the beginning of this edge. Three settings of an ontology construction problem are considered: 1) all predicates are binary and subsets of the original set of objects are given; 2) all predicates are binary and it is required to find subsets of the original set; 3) among the predicates there are many-valued ones and subsets of the original set of objects are given. The main tool for construction such a description is to extract an elementary conjunction of literals of predicate formulas that is isomorphic to subformulas of some formulas. The definition of an isomorphism of elementary conjunctions of atomic predicate formulas is given. The method of ontology construction is formulated. An illustrative example is provided.

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17

S. Veloso, Paulo A., i Sheila R. M. Veloso. "A Graph Calculus for Predicate Logic". Electronic Proceedings in Theoretical Computer Science 113 (28.03.2013): 153–68. http://dx.doi.org/10.4204/eptcs.113.15.

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18

Ben-Yami, Hanoch. "Attributive adjectives and the predicate calculus". Philosophical Studies 83, nr 3 (wrzesień 1996): 277–89. http://dx.doi.org/10.1007/bf00364609.

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19

Interdisciplinary Seminar on Peirce. "“Betagraphic”: An Alternative Formulation of Predicate Calculus". Transactions of the Charles S. Peirce Society 51, nr 2 (2015): 137. http://dx.doi.org/10.2979/trancharpeirsoc.51.2.137.

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20

Boute, Raymond. "Functional declarative language design and predicate calculus". ACM Transactions on Programming Languages and Systems 27, nr 5 (wrzesień 2005): 988–1047. http://dx.doi.org/10.1145/1086642.1086647.

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21

Yankov, V. A. "Dialogue interpretation of the classical predicate calculus". Izvestiya: Mathematics 61, nr 1 (28.02.1997): 225–33. http://dx.doi.org/10.1070/im1997v061n01abeh000112.

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22

Weiß, Benjamin. "Predicate abstraction in a program logic calculus". Science of Computer Programming 76, nr 10 (październik 2011): 861–76. http://dx.doi.org/10.1016/j.scico.2010.06.008.

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23

Paul, Etienne. "Equational methods in first order predicate calculus". Journal of Symbolic Computation 1, nr 1 (marzec 1985): 7–29. http://dx.doi.org/10.1016/s0747-7171(85)80026-2.

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24

Bijlsma, Lex, i Rob Nederpelt. "Dijkstra-Scholten predicate calculus: concepts and misconceptions". Acta Informatica 35, nr 12 (1.12.1998): 1007–36. http://dx.doi.org/10.1007/s002360050150.

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25

Drăgulici, Daniel, i George Georgescu. "Algebraic Logic for Rational Pavelka Predicate Calculus". MLQ 47, nr 3 (sierpień 2001): 315–26. http://dx.doi.org/10.1002/1521-3870(200108)47:3<315::aid-malq315>3.0.co;2-0.

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26

Huet, Gérard. "Residual theory in λ-calculus: a formal development". Journal of Functional Programming 4, nr 3 (lipiec 1994): 371–94. http://dx.doi.org/10.1017/s0956796800001106.

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AbstractWe present the complete development, in Gallina, of the residual theory of β-reduction in pure λ-calculus. The main result is the Prism Theorem, and its corollary Lévy's Cube Lemma, a strong form of the parallel-moves lemma, itself a key step towards the confluence theorem and its usual corollaries (Church-Rosser, uniqueness of normal forms). Gallina is the specification language of the Coq Proof Assistant (Dowek et al., 1991; Huet 1992b). It is a specific concrete syntax for its abstract framework, the Calculus of Inductive Constructions (Paulin-Mohring, 1993). It may be thought of as a smooth mixture of higher-order predicate calculus with recursive definitions, inductively defined data types and inductive predicate definitions reminiscent of logic programming. The development presented here was fully checked in the current distribution version Coq V5.8. We just state the lemmas in the order in which they are proved, omitting the proof justifications. The full transcript is available as a standard library in the distribution of Coq.

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27

Konovalov, Aleksandr Yu. "Generalized Realizability and Basic Logic". ACM Transactions on Computational Logic 22, nr 4 (31.10.2021): 1–23. http://dx.doi.org/10.1145/3468856.

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Let V be a set of number-theoretical functions. We define a notion of absolute V -realizability for predicate formulas and sequents in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Basic Predicate Calculus is sound with respect to the semantics of absolute V -realizability if V satisfies some natural conditions.

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28

Xiao, Hua Bo, i Shu Li Huang. "Research in Test Case Method in Software Testing". Applied Mechanics and Materials 457-458 (październik 2013): 1163–66. http://dx.doi.org/10.4028/www.scientific.net/amm.457-458.1163.

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Based on existing software test case designing method, combining artificial intelligence domain knowledge and methods, we design a method of Predicate Calculus for Test case designing, using this method to provide detailed information to testers.

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29

Aghaei, Mojtaba, i Mohammad Ardeshir. "A Gentzen-style axiomatization for basic predicate calculus". Archive for Mathematical Logic 42, nr 3 (1.04.2003): 245–59. http://dx.doi.org/10.1007/s001530100132.

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30

McKinnon, K. I. M., i H. P. Williams. "Constructing integer programming models by the predicate calculus". Annals of Operations Research 21, nr 1 (grudzień 1989): 227–45. http://dx.doi.org/10.1007/bf02022101.

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31

Orevkov, V. P. "A new decidable Horn fragment of predicate calculus". Journal of Mathematical Sciences 134, nr 5 (maj 2006): 2403–10. http://dx.doi.org/10.1007/s10958-006-0117-7.

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32

Ostermann, Pascal. "Many-valued modal logics: Uses and predicate calculus". Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36, nr 4 (1990): 367–76. http://dx.doi.org/10.1002/malq.19900360411.

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33

Carbone, A. "The cost of a cycle is a square". Journal of Symbolic Logic 67, nr 1 (marzec 2002): 35–60. http://dx.doi.org/10.2178/jsl/1190150028.

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AbstractThe logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be non-elementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with (kn+1) lines. In particular, there is a polynomial time algorithm which eliminates cycles from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.

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34

ALIZADEH, MAJID, FARZANEH DERAKHSHAN i HIROAKIRA ONO. "UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS". Review of Symbolic Logic 7, nr 3 (27.05.2014): 455–83. http://dx.doi.org/10.1017/s175502031400015x.

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AbstractUniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4, which have Craig’s interpolation property but do not have uniform interpolation property. The situation is even worse for predicate logics, as classical predicate logic does not have uniform interpolation property as pointed out by L. Henkin.In this paper, uniform interpolation property of basic substructural logics is studied by applying the proof-theoretic method introduced by A. Pitts (Pitts, 1992). It is shown that uniform interpolation property holds even for their predicate extensions, as long as they can be formalized by sequent calculi without contraction rules. For instance, uniform interpolation property of full Lambek predicate calculus, i.e., the substructural logic without any structural rule, and of both linear and affine predicate logics without exponentials are proved.

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35

MARANGET, LUC. "FUNCTIONAL PEARL Functional satisfaction". Journal of Functional Programming 14, nr 6 (27.10.2004): 647–56. http://dx.doi.org/10.1017/s0956796804005155.

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This work presents simple decision procedures for the propositional calculus and for a simple predicate calculus. These decision procedures are based upon enumeration of the possible values of the variables in an expression. Yet, by taking advantage of the sequential semantics of boolean connectors, not all values are enumerated. In some cases, dramatic savings of machine time can be achieved. In particular, an equivalence checker for a small programming language appears to be usable in practice.

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36

McGee, Vann. "The complexity of the modal predicate logic of “true in every transitive model of ZF”". Journal of Symbolic Logic 62, nr 4 (grudzień 1997): 1371–78. http://dx.doi.org/10.2307/2275648.

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Robert Solovay [8] investigated the version of the modal sentential calculus one gets by taking “□ϕ” to mean “ϕ is true in every transitive model of Zermelo-Fraenkel set theory (ZF).” Defining an interpretation to be a function * taking formulas of the modal sentential calculus to sentences of the language of set theory that commutes with the Boolean connectives and sets (□ϕ)* equal to the statement that ϕ* is true in every transitive model of ZF, and stipulating that a modal formula ϕ is valid if and only if, for every interpretation *, ϕ* is true in every transitive model of ZF, Solovay obtained a complete and decidable set of axioms.In this paper, we stifle the hope that we might continue Solovay's program by getting an analogous set of axioms for the modal predicate calculus. The set of valid formulas of the modal predicate calculus is not axiomatizable; indeed, it is complete .We also look at a variant notion of validity according to which a formula ϕ counts as valid if and only if, for every interpretation *, ϕ* is true. For this alternative conception of validity, we shall obtain a lower bound of complexity: every set which is in the set of sentences of the language of set theory true in the constructible universe will be 1-reducible to the set of valid modal formulas.

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37

Marek, W. "Stable Theories in Autoepistemic Logic". Fundamenta Informaticae 12, nr 2 (1.04.1989): 243–54. http://dx.doi.org/10.3233/fi-1989-12209.

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We investigate the operator producing a stable theory out of its objective part (A stable theory is a set of beliefs of a rational agent). We characterize the objective parts of stable theories. Finally, we discuss the predicate calculus case.

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38

López Astorga, Miguel. "Exclusive Disjunctions With Three Disjuncts from First-Order Predicate Calculus". Open Insight 15, nr 34 (10.06.2024): 168–82. http://dx.doi.org/10.23924/oi.v15i34.629.

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First-order predicate logic seems to be incompatible with the way people understand embedded exclusive disjunctions with three disjuncts. In classical logic, an exclusive disjunction with three disjuncts holds when the three disjuncts hold. However, it is hard to note that for people. The theory of mental models can explain this fact. According to that theory, individuals tend to process embedded exclusive disjunctions with three disjuncts intuitively. Thus, they only consider possible situations in which just one of the disjuncts is the case. The present paper tries to explain this problem within first-order predicate logic. The main point is that, in the latter logic, in inferences having, as its first premise, an embedded exclusive disjunction with three disjuncts, and, as its second premise, the first disjunct of that very exclusive disjunction, it is possible to infer none of the other two disjuncts.

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39

Maffezioli, Paolo, i Eugenio Orlandelli. "Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate". Bulletin of the Section of Logic 48, nr 2 (30.06.2019): 137–58. http://dx.doi.org/10.18778/0138-0680.48.2.04.

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In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and (partly) overcome the failure of interpolation for the implication-free fragment.

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40

KOSTIĆ, JOVANA. "LOGIC FOR THE THEORY OF CONCEPTS". Arhe 27, nr 34 (17.03.2021): 85–102. http://dx.doi.org/10.19090/arhe.2020.34.85-102.

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In this paper, we follow Gödel’s remarks on an envisioned theory of concepts to determine which properties should a logical basis of such a theory have. The discussion is organized around the question of suitability of the classical predicate calculus for this role. Some reasons to think that classical logic is not an appropriate basis for the theory of concepts, will be presented. We consider, based on these reasons, which alternative logical system could fare better as a logical foundation of, in Gödel’s opinion, the most important theory in logic yet to be developed. This paper should, in particular, motivate the study of partial predicates in a certain system of three-valued logic, as a promising starting point for the foundation of the theory of concepts.

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41

Unno, Hiroshi, Tachio Terauchi, Yu Gu i Eric Koskinen. "Modular Primal-Dual Fixpoint Logic Solving for Temporal Verification". Proceedings of the ACM on Programming Languages 7, POPL (9.01.2023): 2111–40. http://dx.doi.org/10.1145/3571265.

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We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novel modular primal-dual method. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the original primal instance has a corresponding (co-)inductive predicate representing its complement in the dual instance obtained by taking the standard De Morgan’s dual of the primal instance, and (2) partial solutions for (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is also modular in that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems.

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42

Purdy, William C. "Fluted formulas and the limits of decidability". Journal of Symbolic Logic 61, nr 2 (czerwiec 1996): 608–20. http://dx.doi.org/10.2307/2275678.

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AbstractIn the predicate calculus, variables provide a flexible indexing service which selects the actual arguments to a predicate letter from among possible arguments that precede the predicate letter (in the parse of the formula). In the process of selection, the possible arguments can be permuted, repeated (used more than once), and skipped. If this service is withheld, so that arguments must be the immediately preceding ones, taken in the order in which they occur, the formula is said to be fluted. Quine showed that if a fluted formula contains only homogeneous conjunction (conjoins only subformulas of equal arity), then the satisfiability of the formula is decidable. It remained an open question whether the satisfiability of a fluted formula without this restriction is decidable. This paper answers that question.

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Шалак, В. И. "On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus". Logical Investigations 21, nr 2 (28.09.2015): 9–14. http://dx.doi.org/10.21146/2074-1472-2015-21-2-9-14.

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In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

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Satish, B. N. V., i G. Ganesan. "Approximations on intuitionistic fuzzy predicate calculus through rough computing". Journal of Intelligent & Fuzzy Systems 27, nr 4 (2014): 1873–79. http://dx.doi.org/10.3233/ifs-141153.

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Nissan, Ephraim. "Data analysis using a geometrical representation of predicate calculus". Information Sciences 41, nr 3 (kwiecień 1987): 187–258. http://dx.doi.org/10.1016/0020-0255(87)90010-7.

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Megill, Norman D. "A Finitely Axiomatized Formalization of Predicate Calculus with Equality". Notre Dame Journal of Formal Logic 36, nr 3 (lipiec 1995): 435–53. http://dx.doi.org/10.1305/ndjfl/1040149359.

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Koletsos, George. "Church-Rosser theorem for typed functional systems". Journal of Symbolic Logic 50, nr 3 (wrzesień 1985): 782–90. http://dx.doi.org/10.2307/2274330.

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Introduction. This paper contains a new proof of the Church-Rosser theorem for the typed λ-calculus, which also applies to systems with infinitely long terms.The ordinary proof of the Church-Rosser theorem for the general untyped calculus goes as follows (see [1]). If is the binary reduction relation between the terms we define the one-step reduction 1 in such a way that the following lemma is valid.Lemma. For all terms a and b we have: ab if and only if there is a sequence a = a0, …, an = b, n ≥ 0, such that aiiai + 1for 0 ≤ i < n.We then prove the Church-Rosser property for the relation 1 by induction on the length of the reductions. And by combining this result with the above lemma we obtain the Church-Rosser theorem for the relation .Unfortunately when we come to infinite terms the above lemma is not valid anymore. The difficulty is that, assuming the hypothesis for the infinitely many premises of the infinite rule, there may not exist an upper bound for the lengths n of the sequences ai = a0, …, an = bi (i < α); cf. the infinite rule (iv) in §6.A completely new idea in the case of the typed λ-calculus would be to exploit the type structure in the way Tait did in order to prove the normalization theorem. In this we succeed by defining a suitable predicate, the monovaluedness predicate, defined over the type structure and having some nice properties. The key notion permitting to define this predicate is the notion of I-form term (see below). This Tait-type proof has a merit, namely that it can be extended immediately to the case of infinite terms.

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RÖCKL, CHRISTINE, i DANIEL HIRSCHKOFF. "A fully adequate shallow embedding of the π-calculus in Isabelle/HOL with mechanized syntax analysis". Journal of Functional Programming 13, nr 2 (marzec 2003): 415–51. http://dx.doi.org/10.1017/s0956796802004653.

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This paper discusses an application of the higher-order abstract syntax technique to general-purpose theorem proving, yielding shallow embeddings of the binders of formalized languages. Higher-order abstract syntax has been applied with success in specialized logical frameworks which satisfy a closed-world assumption. As more general environments (like Isabelle/HOL or Coq) do not support this closed-world assumption, higher-order abstract syntax may yield exotic terms, that is, datatypes may produce more terms than there should actually be in the language. The work at hand demonstrates how such exotic terms can be eliminated by means of a two-level well-formedness predicate, further preparing the ground for an implementation of structural induction in terms of rule induction, and hence providing fully-fledged syntax analysis. In order to apply and justify well-formedness predicates, the paper develops a proof technique based on a combination of instantiations and reabstractions of higher-order terms. As an application, syntactic principles like the theory of contexts (as introduced by Honsell, Miculan, and Scagnetto) are derived, and adequacy of the predicates is shown, both within a formalization of the π-calculus in Isabelle/HOL.

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Dekkers, Wil, Martin Bunder i Henk Barendregt. "Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic". Journal of Symbolic Logic 63, nr 3 (wrzesień 1998): 869–90. http://dx.doi.org/10.2307/2586717.

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AbstractIllative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions.

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HUET, GÉRARD. "Preface". Mathematical Structures in Computer Science 21, nr 4 (1.07.2011): 671–77. http://dx.doi.org/10.1017/s0960129511000235.

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This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

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