Deduction theorem

About: Deduction theorem is a research topic. Over the lifetime, 392 publications have been published within this topic receiving 7910 citations.

Lambda-My-Calculus: An Algorithmic Interpretation of Classical Natural Deduction

Abstract: 1 INTRODUCTION This paper presents a way of extending the paradigm "proofs as programs" to classical proofs. The system we use is derived from the general Free Deduction system presented in [31. Usually when considering proofs as programs, one has only in mind some kind of intuitionistic proofs. There is an obvious reason for that restriction: only intuitionistic proofs are contructive, in the sense that from the proof of an existential statement, one can get a witness of this existential statement. But from the programming point of view, constructivity is only needed for E~-statements, for which classical and intuitionistic provability coincide. This means that, classical proofs are also candidates for being programs. In order to use them as programs, one has two tasks to achieve: (i) to find a system in which one can extract directly a program from a classical proof (and not by means of a translation to intuitionistic logic), and (ii) to understand the algorithmic meaning of classical constructions. The system we will consider is a natural deduction system with multiple conclusions, we will call it Classical Natural Deduction (the one with the absurdity rule being called Usual Natural Deduction). It is a particular subsystem of Free Deduction (FD) with inputs fixed to the left, chosen for its simplicity: it can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known. In this context, the contribution of classical constructs to programming appears clearly: they correspond to control operators added to functional languages, like call/ce in Scheme. In both contexts, the role of the classical constructs is the same: they allow to take shorter routes in the construction of a proof~program. The link between control operators and classical constructs has first been made by T. Griffin in [1], where he proposes to type the C operator of Felleisen, with the type-~'-,A-* A. The system he obtains is not satisfactory from the logical point of view: the reduction is in fact a reduction strategy and the type assigned to C doesn't fit in general the reduction rule for C. C. Murthy further analysed the connections t91 between control operators, classical constructs and translations from classical logic to intuitionistic logic (see [4]). The difficulties met in trying to use-~-A ~ A (or the classical absurdity rule) as a type for control operators is not really due to classical logic, but much nore to …

A logical analysis of modules in logic programming

Abstract: We present a logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses. The operational meaning of a goal which is an implication is given by the deduction theorem: a goal D ⊃ G is provable from a program P if the goal G is provable from the larger program P ∪ {D}. This paper explores the qualitative nature of this extension to logic programming. For example, if the formula D is the conjunction of universally quantified clauses, we interpret the goal D ⊃ G as a request to load the code in D prior to attempting G and then unload that code after G succeeds or fails. This extended use of implication provides a logical explanation of parametric modules, some uses of PROLOG's assert predicate, and aspects of abstract datatypes. Both a model theory and proof theory are presented for this logical language. In particular, we show how to build a Kripke-like model for programs by a fixed-point construction and show that the operational meaning of implication mentioned above is sound and complete for intuitionistic logic. We also examine a weak notion of negation which is easily implemented in this language and show how database constraints can be represented by it.

Quantitative deduction and its fixpoint theory

Abstract: Logic programming provides a model for rule-based reasoning in expert systems. The advantage of this formal model is that it makes available many results from the semantics and proof theory of first-ordet predicate logic. A disadvantage is that in expert systems one often wants to use, instead of the usual two truth values, an entire continuum of “uncertainties” in between. That is, instead of the usual “qualitative” deduction, a form of “quantitative” deduction is required. We present an approach to generalizing the Tarskian semantics of Horn clause rules to justify a form of quantitative deduction. Each clause receives a numerical attenuation factor. Herbrand interpretations, which are subsets of the Herbrand base, are generalized to subsets which are fuzzy in the sense of Zadeh. We show that as result the fixpoint method in the semantics of Horn clause rules can be developed in much the same way for the quantitative case. As for proof theory, the interesting phenomenon is that a proof should be viewed as a two-person game. The value of the game turns out to be the truth value of the atomic formula to be proved, evaluated in the minimal fixpoint of the rule set. The analog of the PROLOG interpreter for quantitative deduction becomes a search of the game tree ( = proof tree) using the alpha-beta heuristic well known in game theory.

Lattice-Valued Logic: An Alternative Approach to Treat Fuzziness and Incomparability

Abstract: I Introduction.- 1 Introduction.- 1.1 Major Methodologies in Artificial Intelligence.- 1.2 Basic Academic Ideas.- 1.3 Some Related Concepts.- 1.4 Many-Valued Logic and Lattice-Valued Logic.- 1.5 Uncertainty Inference.- 1.5.1 Probability-Based Uncertainty Reasoning.- 1.5.2 Fuzzy Set Based Uncertainty Reasoning.- 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning.- 1.6 Automated Reasoning in Many-Valued Logic.- II Lattice Implication Algebras.- 2 Concepts and Properties.- 2.1 Lattice Implication Algebras.- 2.1.1 Concepts and Examples.- 2.1.2 Basic Properties.- 2.2 Lattice H Implication Algebras.- 2.3 Lattice Properties.- 2.4 Homomorphisms.- 3 Filters.- 3.1 Filters and Implicative Filters.- 3.2 Generated Filters.- 3.3 Positive Implicative Filters and Associative Filters.- 3.4 Prime Filters and Ultra-Filters.- 3.5 I-Filters, Involution Filters and Obstinate Filters.- 3.6 Fuzzy Filters.- 4 LI-Ideals.- 4.1 LI-Ideals.- 4.2 Fuzzy LI-Ideals.- 4.3 Normal Fuzzy LI-Ideals.- 4.4 Intuitionistic Fuzzy LI-Ideals.- 5 Homomorphisms and Representations.- 5.1 Congruence Relations.- 5.1.1 Congruence Relations Induced by Filters.- 5.1.2 Congruences Relations Induced by LI-ideals.- 5.1.3 Congruence Relations Induced by Fuzzy Filters.- 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals.- 5.2 Proper Lattice Implication Algebras.- 5.3 Representations.- 6 Topological Structure of Filter Spaces.- 6.1 Filter Spaces.- 6.1.1 Basic Concepts.- 6.1.2 Topological Properties.- 6.2 Product Topology and Quotient Topology.- 6.3 Lattice Topology.- 6.4 Prime Spaces.- 7 Connections with Related Algebras.- 7.1 Lattice Implication Algebras and BCK-Algebras.- 7.2 Lattice Implication Algebras and MV-Algebras.- 7.3 Lattice Implication Algebras and Related Algebras.- 8 Related Issues.- 8.1 Category of Lattice Implication Algebras.- 8.2 Category of Fuzzy Lattice Implication Algebras.- 8.3 Fuzzy Power Sets.- 8.4 Adjoint Semigroups.- 8.5 Logical Properties.- III Lattice-Valued Logic Systems.- 9 Lattice-Valued Propositional Logics.- 9.1 Lattice-Valued Propositional Logic LP(X).- 9.1.1 Language.- 9.1.2 Semantics.- 9.1.3 Syntax.- 9.1.4 Examples.- 9.2 Gradational Lattice-Valued Propositional Logic Lvpl.- 9.2.1 Language.- 9.2.2 Rules of Inference.- 9.2.3 Semantics.- 9.2.4 Syntax.- 9.2.5 Satisfiability and Consistency.- 9.2.6 Deduction Theorem.- 9.2.7 Compactness.- 9.2.8 Examples.- 10 Lattice-Valued First-Order Logics.- 10.1 Lattice-Valued First-Order Logic LF(X).- 10.1.1 Language.- 10.1.2 Interpretation.- 10.1.3 Semantics.- 10.1.4 Syntax.- 10.1.5 Properties of Model Theory.- 10.2 Gradational Lattice-Valued First-Order Logic Lvfl.- 10.2.1 Language.- 10.2.2 Interpretation.- 10.2.3 Semantics.- 10.2.4 Standardization of Formulae.- 10.2.5 Syntax.- 10.2.6 Soundness and Completeness.- 10.2.7 Satisfiability and Consistency.- 10.2.8 Deduction Theorem.- 10.2.9 Compactness.- 10.2.10Examples.- 11 Uncertainty and Automated Reasoning.- 11.1 Uncertainty Reasoning Based on LP(X).- 11.2 Uncertainty Reasoning Based on Lvpl.- 11.2.1 Another Kind of Interpretation of X ? Y.- 11.2.2 Basic Theory.- 11.2.3 Examples.- 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning.- Models and Methods.- Semantical Interpretation and Syntactical Proof.- 11.3 ?-Resolution Principle Based on LP(X).- 11.3.1 ?-Resolution Principle.- 11.3.2 Soundness and Completeness.- 11.4 ?-Resolution Principle Based on LF(X).- 11.4.1 Interpretation of Formulae.- 11.4.2 ?-Resolution Principle.- References.

Probabilistic Logic Programming

Abstract: We present a new approach to probabilistic logic pro- grams with a possible worlds semantics. Classical program clauses are extended by a subinterval of that describes the range for the conditional probability of the head of a clause given its body. We show that deduction in the defined probabilistic logic pro grams is computationally more complex than deduction in classical logic programs. More precisely, restricted deduction problems t hat are P- complete for classical logic programs are already NP-hard f or proba- bilistic logic programs. We then elaborate a linear program ming ap- proach to probabilistic deduction that is efficient in inter esting spe- cial cases. In the best case, the generated linear programs h ave a num- ber of variables that is linear in the number of ground instan ces of purely probabilistic clauses in a probabilistic logic prog ram. As a second contribution, we show that deduction in probabilistic logic programs is computationally more complex than deduction in classical logic programs: restricted deduction problems t hat are P- complete for classical logic programs are already NP-hard f or prob- abilistic logic programs. Hence, any attempt towards effici ent deduc- tion in probabilistic logic programs should be guided by looking for efficient special-case, average-case, or approximation te chniques. As a third contribution, by generalizing own work from (18), we elaborate a linear programming approach to deduction in probabilis- tic logic programs, which is efficient in interesting specia l cases. In our framework, probabilistic deduction problems can easil y be rep- resented by linear programs. However, these initial linear programs have a number of variables that is exponential in the cardina lity of the Herbrand base. Moreover, also the Herbrand base of a probabilistic logic program is generally quite large. Motivated by this ob servation, we elaborate a technique that, in the best case, yields linea r programs with a number of variables that is linear in the number of ground in- stances of purely probabilistic clauses. This result is ver y promising. The work in (22), in contrast, is based on solving an exponential number of linear programs over an exponential number of variables (both in the cardinality of the Herbrand base) in each fixpoin t itera- tion step and in each compilation step for SLDp-refutation. The rest of this paper is organized as follows. In Section 2, w e present the syntax and the semantics of probabilistic logic programs and of queries addressed to them. Section 3 gives an illustra tive ex- ample. Section 4 concentrates on the computational complexity of probabilistic deduction in our framework. In Sections 5 and 6, we present and discuss an optimized linear programming approach to probabilistic deduction. Section 7 summarizes the main results.