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A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
Abstract

Cited by 28 (11 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
A Practical Approach to Partial Functions in CVC Lite
, 2004
"... Most verification approaches assume a mathematical formalism in which functions are total, even though partial functions occur naturally in many applications. Furthermore, although there have been various proposals for logics of partial functions, there is no consensus on which is "the right" logic ..."
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Cited by 14 (7 self)
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Most verification approaches assume a mathematical formalism in which functions are total, even though partial functions occur naturally in many applications. Furthermore, although there have been various proposals for logics of partial functions, there is no consensus on which is "the right" logic to use for verification applications. In this paper, we propose using a threevalued Kleene logic, where partial functions return the "undefined" value when applied outside of their domains. The particular semantics are chosen according to the principle of least surprise to the user; if there is disagreement among the various approaches on what the value of the formula should be, its evaluation is undefined. We show that the problem of checking validity in the threevalued logic can be reduced to checking validity in a standard twovalued logic, and describe how this approach has been successfully implemented in our tool, CVC Lite.
Mechanising Partiality without ReImplementation
 IN 21ST ANNUAL GERMAN CONFERENCE ON ARTIFICIAL INTELLIGENCE, VOLUME 1303 OF LNAI
, 1997
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago. This approach allows rejecting certain unwanted formul ..."
Abstract

Cited by 9 (4 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago. This approach allows rejecting certain unwanted formulae as faulty, which the simpler twovalued ones accept. We have developed resolution and tableau calculi for automated theorem proving that take the restrictions of the threevalued logic into account, which however have the severe drawback that existing theorem provers cannot directly be adapted to the technique. Even recently implemented calculi for manyvalued logics are not wellsuited, since in those the quantification does not exclude the undefined element. In this work we show, that it is possible to enhance a twovalued theorem prover by a simple strategy so that it can be used to generate proofs for the theorems of the threevalued setting. By this we are able to use an existing t...
A Tableau Calculus for Partial Functions
, 1996
"... . Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
Abstract

Cited by 6 (5 self)
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. Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. Keywords: Partial functions, manyvalued logic, sorted logic, tableau. 1 Introduction Many practical applications of deduction systems in mathematics and computer science rely on the correct and efficient treatment of partial functions. For this purpose...
Typed lambda calculi and possible worlds models
"... Course overview (current plan; adjustable) i. Typed lambda calculi and possible worlds models ii. Alternative approaches to meaning iii. Grounded language understanding iv. Question answering: Grounding in databases v. Stochastic lambda calculus vi. Distributional approaches to word meanings vii. Co ..."
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Course overview (current plan; adjustable) i. Typed lambda calculi and possible worlds models ii. Alternative approaches to meaning iii. Grounded language understanding iv. Question answering: Grounding in databases v. Stochastic lambda calculus vi. Distributional approaches to word meanings vii. Composition in vectorspace models 1: tensors viii. Composition in vectorspace models 2: recursive neural networks Plan for today 1 Foundational issues and controversies 2