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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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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 10 (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. 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. R ..."
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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.
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 ..."
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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...
Partiality without the Cost
 CADE13 WORKSHOP ON MECHANIZATION OF 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 mechanisatio ..."
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Cited by 2 (1 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 mechanisation. Based on this, we have developed resolution and tableau calculi for automated theorem proving. The threevalued approach is more restrictive and allows rejecting certain unwanted formulae as faulty, which the simpler twovalued accept. It is commonly assumed that this finer analysis has to be payed for by greater computational complexity of proof search. However, for a large class of theorems that hold with respect to Kleene logic, the proofs can be transformed into classical ones and vice versa conserving the structure and size of the proof. Another main objective against a threevalued approach are the costs to implement a corresponding theorem prover. We show, that it is po...
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
RasiowaSikorski Deduction Systems  Foundations and Applications
, 2000
"... Introduction The aim of this tutorial is to present a powerful and exible, yet simple methodology of developing deduction systems for various logics based on the analysis of their semantics. The paper will present a general outline of this methodology, and show examples of its applications to vario ..."
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Introduction The aim of this tutorial is to present a powerful and exible, yet simple methodology of developing deduction systems for various logics based on the analysis of their semantics. The paper will present a general outline of this methodology, and show examples of its applications to various brands of computer science logics. The said methodology is inherently connected with the semantics of the given logic, and its key concept is to obtain an adequate and complete proof mechanism for the logic in a systematic way from the said semantics. This is achieved by "mirroring" the semantics of all the logical constructs (connectives, quantiers, modalities, . . . ) through invertible rules operating on sequences of formulae of the logic. The methodology is based on the use of a simple and universal deduction mechanism developed by Polish logicians about 40 years ago: the RasiowaSikorski (RS) deduction system [RS63]  and hence from now on it will be called RS methodology
several conferences. Acknowledgements
, 1991
"... This report is a more detailed and elaborated version of a number of papers submitted to ..."
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This report is a more detailed and elaborated version of a number of papers submitted to
Acknowledgements
, 1994
"... I would like to thank my colleages at the MaxPlanckInstitut for many helpful comments on this paper. In this article I investigate the properties of unification in sort theories. The usual notion of a sort consisting of a sort symbol is extended to a set of sort symbols. In this language sorted u ..."
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I would like to thank my colleages at the MaxPlanckInstitut for many helpful comments on this paper. In this article I investigate the properties of unification in sort theories. The usual notion of a sort consisting of a sort symbol is extended to a set of sort symbols. In this language sorted unification in elementary sort theories is of unification type finitary. The rules of standard unification with the addition of four sorted rules form the new sorted unification algorithm. The algorithm is proved sound and complete. The rule based form of the algorithm is not suitable for an implementation because there is no control and the used data structures are weak. Therefore we transform the algorithm into a deterministic sorted unification procedure. For the procedure sorted unification in pseudolinear sort theories is proved decidable. The notions of a sort and a sort theory are developed in a way such that a standard calculus can be turned into a sorted calculus by replacing standard unification with sorted unification. To this end sorts may denote the empty set. Sort theories may contain clauses with more than one declaration and may change dynamically during the deduction process. The applicability of the approach is exemplified for the resolution and the tableau calculus.