Results 1  10
of
14
PairDense Relation Algebras
 Transactions of the American Mathematical Society
, 1991
"... The central result of this paper is that every pairdense relation algebra is completely representable. A relation algebra is said to be pairdense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big ..."
Abstract

Cited by 62 (8 self)
 Add to MetaCart
The central result of this paper is that every pairdense relation algebra is completely representable. A relation algebra is said to be pairdense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big (with a = b allowed). In a simple pairdense relation algebra, every pair is either a "point" (an algebraic analogue of fha; aig) or a "twin" (a pair which contains no point). In fact, every simple pairdense relation algebra A is completely representable over a set U iff jU j = + 2, where is the number of points of A and is the number of twins of A.
A Compositional Logic for Polymorphic HigherOrder Functions
 PPDP'04
, 2004
"... This paper introduces a compositional program logic for higherorder polymorphic functions and standard data types. The logic enables us to reason about observable properties of polymorphic programs starting from those of their constituents. Just as types attached to programs offer information on the ..."
Abstract

Cited by 26 (11 self)
 Add to MetaCart
This paper introduces a compositional program logic for higherorder polymorphic functions and standard data types. The logic enables us to reason about observable properties of polymorphic programs starting from those of their constituents. Just as types attached to programs offer information on their composability so as to guarantee basic safety of composite programs, formulae of the proposed logic attached to programs offer information on their composability so as to guarantee finegrained behavioural properties of polymorphic programs. The central feature of the logic is a systematic usage of names and operations on them, whose origin is in the logics for typed πcalculi. The paper introduces the program logic and its proof rules and illustrates their usage by nontrivial reasoning examples, taking a prototypical callbyvalue functional language with impredicative polymorphism and recursive types as a target language.
Some Theorems We Should Prove
 In Proc., 1993 Intern. Conf. on HOL Theorem Proving and Its Applications
, 1993
"... ..."
Supervaluationism and Its Logics
"... If we adopt a supervaluational semantics for vagueness, what sort of logic results? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recast. There are several ways of doing this within a supervaluational framework, the main alternative bei ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
If we adopt a supervaluational semantics for vagueness, what sort of logic results? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recast. There are several ways of doing this within a supervaluational framework, the main alternative being between ‘global ’ construals (e.g. an argument is valid if and only if it preserves truthunderallprecisifications) and ‘local’ construals (an argument is valid if and only if, under all precisifications, it preserves truth). The former alternative is by far more popular, but I argue in favour of the latter, for (i) it does not suffer from a number of serious objections, and (ii) it makes it possible to restore global validity as a defined notion. Supervaluationism is a mixed bag. It is sometimes described as the ‘standard ’ theory of vagueness, at least in so far as vagueness is construed as a semantic phenomenon, but exactly what that standard theory amounts to is far from clear. In fact, it is pretty clear that there is not just one supervaluational semantics out there—there are lots of such semantics; and although it is true that they all exploit the same insight, their relative differences are by no means immaterial. For one thing, a lot depends on how exactly supervaluations are constructed, that is, on how exactly we come to establish the truthvalue of a given statement. (And when I say that a lot depends on this I mean to say that different explanations may give rise to different philosophical worries, or justify different reactions.) Secondly, and equally importantly, a lot depends on how a given supervaluationary machinery is brought into play when it comes to explaining the logic of the language, that is, not the notion of truth, or ‘supertruth’, as it applies to individual statements, but the notion of validity, or ‘supervalidity’, as it applies to whole arguments. (I am thinking for instance of how different explanations may bear on the question of whether, or to what extent, vagueness involves a departure from classical logic.) Here I want to focus on this second part of the story. However, since the notion of validity depends on the notion of truth—or so one may argue—I also want to comment briefly on the first.
The Satisfiability problem for the SchöenfinkelBernays fragment: Partial Instantiation and Hypergraph Algorithms
 Proceedings of 8th subcommission "Magnetic
, 1994
"... A partial instantiation approach to the solution of the satisfiability problem in the SchoenfinkelBernays fragment of 1 st order logic is presented. It is based on a reduction of the problem to a finite sequence of satisfiability problems in the propositional logic and it improves upon the ori ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
A partial instantiation approach to the solution of the satisfiability problem in the SchoenfinkelBernays fragment of 1 st order logic is presented. It is based on a reduction of the problem to a finite sequence of satisfiability problems in the propositional logic and it improves upon the original idea of partial instantiation, as proposed by Jeroslow. In the second part of the paper a new interpretation of the partial instantiation approach in terms of Directed Hypergraphs is proposed and a particular implementation for the Datalog case is described in detail. 1 Introduction. The problem of Logical Inference plays a fundamental role in Decision Sciences and has several applications in fields such as decision support systems, logic circuit design, data bases, and programming languages. Although classical approaches to formalize and solve inference problems have been of symbolic nature, in the last few years many scientists in the Operations Research community have studied ...
Unified Semantics for Modality and lambdaterms via Proof Polynomials
"... It is shown that the modal logic S4, simple calculus and modal calculus admit a realization in a very simple propositional logical system LP , which has an exact provability semantics. In LP both modality and terms become objects of the same nature, namely, proof polynomials. The provability inte ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
It is shown that the modal logic S4, simple calculus and modal calculus admit a realization in a very simple propositional logical system LP , which has an exact provability semantics. In LP both modality and terms become objects of the same nature, namely, proof polynomials. The provability interpretation of modal terms presented here may be regarded as a systemindependent generalization of the CurryHoward isomorphism of proofs and terms. 1 Introduction The Logic of Proofs (LP , see Section 2) is a system in the propositional language with an extra basic proposition t : F for "t is a proof of F ". LP is supplied with a formal provability semantics, completeness theorems and decidability algorithms ([3], [4], [5]). In this paper it is shown that LP naturally encompasses calculi corresponding to intuitionistic and modal logics, and combinatory logic. In addition, LP is strictly more expressive because it admits arbitrary combinations of ":" and propositional connectives. The id...
Truth Definitions, Skolem Functions And Axiomatic Set Theory
 Bulletin of Symbolic Logic
, 1998
"... this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary firstorder logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the fi ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary firstorder logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the firstorder level. This eliminates once and for all the need of set theory for the purposes of a metatheory of logic.
Proving FirstOrder Equality Theorems with HyperLinking
, 1995
"... Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving p ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving problems. However, Lee's roundbyround implementation of hyperlinking is not particularly well suited for the addition of special methods in support of equality. In this dissertation, we describe, as alternative to the roundbyround hyperlinking implementation of Lee, a smallest instance first implementation of hyperlinking which addresses many of the inefficiencies of roundbyround hyperlinking encountered when adding special methods in support of equality. Smallest instance first hyperlinking is based on the formalization of generating smallest clauses first, a heuristic widely used in other automated theorem provers. We prove both the soundness and logical completeness of smallest instance first hyperlinking and show that it always generates smallest clauses first under
BEYOND UNDECIDABLE
, 2006
"... Abstract. The predicate complementary to the wellknown Gödel’s provability predicate is defined. From its recursiveness new consequences concerning the incompleteness argumentation are drawn and extended to new results of consistency, completeness and decidability with regard to Peano Arithmetic an ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. The predicate complementary to the wellknown Gödel’s provability predicate is defined. From its recursiveness new consequences concerning the incompleteness argumentation are drawn and extended to new results of consistency, completeness and decidability with regard to Peano Arithmetic and the first order predicate calculus.
Operations on Proofs That Can Be Specified By Means of Modal Logic
"... Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which constitute the system of proof terms in LP. Proof polynomials are built from variables and constants by three operations on proofs: "\Delta" (application), "!" (proof checker), and "+" (choice). Here constants stand for canonical proofs of "simple facts", namely instances of propositional axioms and axioms of LP in a given proof system. We show that every operation on proofs that (i) can be specified in a propositional modal language and (ii) is invariant with respect to the choice of a proof system is realized by a proof polynomial.