A Complete Axiomatization of the Three valued Completion of Logic Programs

We prove the completeness of extended SLDNF-resolution for the new class of e-programs with respect to the three-valued completion of a logic program. Not only the class of allowed programs but also the class of definite programs are contained in the class of ε-programs. To understand better the three-valued completion of a logic program we introduce a formal system for three-valued logic in which one can derive exactly the three-valued consequences of the completion of a logic program. The system is proof theoretically interesting, since it is a fragment of Gentzen's sequent calculus L

A linear axiomatization of negation as failure

AbstractThis paper is concerned with the axiomatization of success and failure in propositional logic programming. It deals with the actual implementation of SLDNF in PROLOG, as opposed to the general nondeterministic SLDNF evaluation method. Given any propositional program P, a linear theory LTP is defined (the linear translation of P) and the following results are proved for any literal A: soundness of PROLOG evaluation (if the goal A PROLOG-succeeds on P, then LTP⊢lin A, and if A PROLOG-fails on P, then LTP⊢lin A⊥), and completeness of PROLOG evaluation (if LTP⊢lin A, then the goal A PROLOG-succeeds on P, and if LTP⊢lin A⊥, then A PROLOG-fails on P). Here ⊢lin means provability in linear logic, and A⊥ is the linear negation of A

Free Higher-Order Logic - Notion, Definition and Embedding in HOL

Free logics are a family of logics that are free of any existential assumptions. This family can roughly be divided into positive, negative, neutral and supervaluational free logic whose semantics differ in the way how nondenoting terms are treated. While there has been remarkable work done concerning the definition of free first-order logic, free higher-order logic has not been addressed thoroughly so far. The purpose of this thesis is, firstly, to give a notion and definition of free higher-order logic based on simple type theory and, secondly, to propose faithful shallow semantical embeddings of free higher-order logic into classical higher order logic found on this definition. Such embeddings can then effectively be utilized to enable the application of powerful state-of-the-art higher-order interactive and automated theorem provers for the formalization and verification and also the further development of increasingly important free logical theories

First-order Goedel logics

First-order Goedel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Goedel logics G_V (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that G_V is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Goedel logics are also characterized.Comment: 37 page

Towards a complete transformational toolkit for compilers

PIM is an equational logic designed to function as a ``transformational toolkit'' for compilers and other programming tools that analyze and manipulate imperative languages.It has been applied to such problems as program slicing, symbolic evaluation, conditional constant propagation, and dependence analysis.PIM consists of the untyped lambda calculus extended with an algebraic data type that characterizes the behavior of lazy stores and generalized conditionals.A graph form of PIM terms is by design closely related to several intermediate representations commonly used in optimizing compilers. In this paper, we show that PIM's core algebraic component, PIM$_t$, possesses a complete equational axiomatization (under the assumption of certain reasonable restrictions on term formation). This has the practical consequence of guaranteeing that every semantics-preserving transformation on a program representable in PIM$_t$ can be derived by application of PIM$_t$ rules. We systematically derive the complete PIM$_t$ logic as the culmination of a sequence of increasingly powerful equational systems starting from a straightforward ``interpreter'' for closed PIM$_t$ terms. This work is an intermediate step in a larger program to develop a set of well-founded tools for manipulation of imperative programs by compilers and other systems that perform program analysis

Metric 1-spaces

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and equivalent, axiomatization of metric space is given which is then generalized from a fresh point of view. Naturally arising examples from metric geometry are presented