Journal•ISSN: 0168-0072

# Annals of Pure and Applied Logic

Elsevier BV

About: Annals of Pure and Applied Logic is an academic journal published by Elsevier BV. The journal publishes majorly in the area(s): Mathematics & Countable set. It has an ISSN identifier of 0168-0072. Over the lifetime, 2491 publications have been published receiving 50771 citations.

Topics: Mathematics, Countable set, Axiom, Intuitionistic logic, Decidability

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TL;DR: A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language.

Abstract: Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 (1991) 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed.

687 citations

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612 citations

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TL;DR: Abramsky as discussed by the authors introduced a notion of universes of discourse for various computational situations, and a standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms.

Abstract: Abramsky, S., Domain theory in logical form, Annals of Pure and Applied Logic 51 (1991) 1–77. • Domain theory, the mathematical theory of computation introduced by Scott as a foundation for detonational semantics • The theory of concurrency and systems behaviour developed by Milner, Hennesy based on operational semantics. • Logics of programs Stone duality provides a junction between semantics (spaces of points=detonations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric , which can be computationally interpreted as the logic of observable properties—i.e., properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme. (1) A metalanguage is introduced, comprising • types = universes of discourse for various computational situations; • terms = programs = syntactic intensions for models or points. (2) A standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms. (3) The metalanguage is also given a logical interpretation, in which types are interpreted as propositional theories and terms are interpreted via a program logic, which axiomatises the properties they satisfy. (4) The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. (5) This opens the way to a whole range of applications. Given a denotational description of a computational situation in our metalanguage, we can turn the handle to obtain a logic for that situation.

496 citations

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TL;DR: A theorem is presented characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context, Dedekind–MacNeille completion of a partial fuzzy order and results provide foundations for formal concept analysis of vague data.

Abstract: The theory of concept lattices (i.e. hierarchical structures of concepts in the sense of Port-Royal school) is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions “object x has attribute y ”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better.

436 citations

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TL;DR: In this paper, the fall of Hilbert's program is discussed, and the first Incompleteness Theorem is shown to be equivalent to the cut-free rules of the Calculus of Sequents.

Abstract: Introduction: Elementary Proof Theory. The Fall of Hilbert's Program. Hilbert's Program. Recursive Functions. The First Incompleteness Theorem. The Second Incompleteness Theorem. Exercises. Annex: Intuitionism. Part I: Sigma 0 1 Proof Theory. The Calculus of Sequents. Definitions. Completeness of the Sequent Calculus. The Cut-Elimination Theorem. The Subformula Property. Intuitionistic Sequent Calculus. Herbrand's Theorem. Generalization. Annex: Natural Deduction. The Church-Rosser Property. Strong Normalization. The Semantics of Sequent Calculus. Completeness of the Cut-Free Rules. Three-Valued Models. Three-Valued Logic. Annex: Takeuti's Conjecture. Limitations of Takeuti's Conjecture. Three-Valued Equivalence. Cut-Free Analysis. Three-Valued Semantics and Generalized Logics. Applications of the ``Hauptsatz''. The Interpolation Lemma. The Reflection Schema of PA. Elementary Consistency Proofs. 1-Consistency. Annex: The Hauptsatz in a Concrete Case. Normalization in HA. Normalization for NL 2 J. Part II: Pi 1 1 Proof Theory. Pi 1 1 Formulas and Well-Foundedness. The Projective Hierarchy. Well-Founded Trees. Well-Orders. Equivalents of (Sigma 0 1 -CA * ). Recursive Well-Orders. Hyperarithmetical Sets. Annex: Kleene's 0. Hierarchies Indexed by 0. Paths Through 0. The Classification Problem. The omega-Rule. omega-Logic. The Cut-Elimination Theorem. Bounds for Cut-Elimination. Equivalents for (Sigma 0 1 -CA * ). Annex: The Calculus Lomega 1 omega. Cut-Elimination in Lomega 1 omega. The Ordinal epsilon o and Arithmetic. Ordinal Analysis of PA. Extensions to other Systems. Ordinals and Theories. Annex: Godel's System T. Functional Interpretation. Spector's Interpretation. No Conterexample Interpretation. An Application. Bibliography. Analytical Index.

401 citations