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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. ..."
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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.
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 ..."
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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. ..."
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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...
RasiowaSikorski Deduction Systems: a Handy Tool for Computer Science Logics
 Recent Trends in Algebraic Specification Techniques, volume 1589 of LNCS
, 1998
"... . A RasiowaSikorski system is a sequencetype formalization of logics based on building decomposition trees of formulae labelled with sequences of formulae. Proofs are nite decomposition trees with leaves having \fundamental", valid labels. The system is dual to the tableau system. The author gives ..."
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Cited by 3 (1 self)
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. A RasiowaSikorski system is a sequencetype formalization of logics based on building decomposition trees of formulae labelled with sequences of formulae. Proofs are nite decomposition trees with leaves having \fundamental", valid labels. The system is dual to the tableau system. The author gives examples of applying the RS formalism to various C.S and A.I. logic, including a logic for reasoning about relative similarity, a threevalued software specication logic with McCarthy's connectives, and a logic for nondeterministic specications. As a new result, an RS system for manysorted rst order logic with possibly empty carriers of some sorts is developed. 1 Introduction An issue in computer science logics that has gained much popularity lately are the socalled labelled deductive systems [5]. The predecessors of this type of deductive systems were Beth's tableau systems [1] and RasiowaSikorski (RS) deduction systems [12], both developed over thirty years ago. Their important...
A Gentzen calculus system for a contextual consequence relation in manysorted first order logic
"... Introduction In [2], we presented two deductions systems for manysorted firstorder logic allowing empty carriers of some sorts based on the use of context formulae. Here we shall present a somewhat different approach: instead of equipping each individual formula with a context, we shall consider ..."
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Introduction In [2], we presented two deductions systems for manysorted firstorder logic allowing empty carriers of some sorts based on the use of context formulae. Here we shall present a somewhat different approach: instead of equipping each individual formula with a context, we shall consider normal contextless formulae (so preformulae in the sense of [2]) with a natural partial semantics generalizing in a way that defined in [2], but the deduction system will use context sequents; so all the formulae in a sequent will be considered in a single common context assuring definedness of the formulae on the lefthand side of the turnstile symbol. Thus, in a way, we shall axiomatize a generalized contextual consequence relation of the discussed logic, labelled by a context for formulae. Like in [2], we assume our models to be total in the sense that all functions are to be interpreted as the total ones, which in view of possible emptiness of some carriers puts bounds on the cl