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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.
Partial Logics Reconsidered: A Conservative Approach
, 1997
"... Partial functions play an important role in computer science. In order to reason about partial functions one may extend classical logic to a logic supporting partial functions, a socalled partial logic. Usually such an extension necessitates sideconditions on classical proof rules in order to ensu ..."
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Cited by 27 (2 self)
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Partial functions play an important role in computer science. In order to reason about partial functions one may extend classical logic to a logic supporting partial functions, a socalled partial logic. Usually such an extension necessitates sideconditions on classical proof rules in order to ensure consistency, and introduces nonclassical proof rules in order to maintain completeness. These complications depend on the choice of consequence relation and nonmonotonic operators. In computer science applications such complications are undesirable, because they affect (semi) mechanical reasoning methods, and make manual reasoning difficult for computer scientists who are not logicians. By carefully choosing the consequence relation and nonmonotonic operators, a simple calculus for partial functions arises. The resulting logic is "healthy" in the sense that "meaningless" formulas (such as top(emptystack) > 1) cannot be concluded, except from contradictory or false assumptions, and a...
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 ..."
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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...
Free Variable Tableaux for a Many Sorted Logic with Preorders
, 1996
"... The proof of properties of formal systems including inequalities is currently evolving into an increasingly appealing workline in different areas of computer science. We propose a sound and complete semantic tableau method for handling manysorted preorders. As logical framework a manysorted first ..."
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Cited by 2 (2 self)
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The proof of properties of formal systems including inequalities is currently evolving into an increasingly appealing workline in different areas of computer science. We propose a sound and complete semantic tableau method for handling manysorted preorders. As logical framework a manysorted first order logic is supplied, where functions and predicates behave monotonically or antimonotonically on their arguments. We formulate additional expansion tableau rules as a more efficient alternative to adding the axioms characterizing a preordered structure. Efficiency of the method is improved using a free variable tableau version. Completeness of the system is proved in detail; examples and applications are introduced. 1 Introduction Almost every formal theory of interest uses the equality relation in its definition, for this reason one of the main goals of automated deduction concerns the efficient proof of properties in the context of some formal framework including equality. On the othe...
C.A.: Inversive meadows and divisive meadows
, 2009
"... Abstract. An inversive meadow is a commutative ring with identity and a total multiplicative inverse operation satisfying 0 −1 = 0. Previously, inversive meadows were shortly called meadows. In this paper, we introduce divisive meadows, which are inversive meadows with the multiplicative inverse ope ..."
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Cited by 2 (2 self)
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Abstract. An inversive meadow is a commutative ring with identity and a total multiplicative inverse operation satisfying 0 −1 = 0. Previously, inversive meadows were shortly called meadows. In this paper, we introduce divisive meadows, which are inversive meadows with the multiplicative inverse operation replaced by a division operation. We introduce a translation from the terms over the signature of divisive meadows into the terms over the signature of inversive meadows and a translation the other way round to show that it depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. Divisive meadows are more basic if variants with a partial multiplicative inverse or division operation are considered as well. We also take a survey of firstorder logics that are appropriate to handle those partial variants of inversive and divisive meadows.
A Treatment of Partiality: its Application to the B Method
 In CADE15 Workshop on Mechanization of Partial Functions. http://www.cs.bham.ac.uk/~mmk/cade98partiality/index.html
, 1998
"... This paper presents the solution that has been developed at Matra Transport International to treat the illdefinedness problem occurring in the B method. The B method is a formal method that permits to develop software from the specification to the code generation by successive refinement steps. The ..."
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Cited by 1 (0 self)
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This paper presents the solution that has been developed at Matra Transport International to treat the illdefinedness problem occurring in the B method. The B method is a formal method that permits to develop software from the specification to the code generation by successive refinement steps. The validity of each refinement step is ensured by the proof of automatically produced lemmas. We have constructed a logic that guarantees that the proof of the refinement lemmas is sufficient to ensure the validity of all the sources (from the specification to the code) of a B development. The main problem comes from the fact that the B language contains partial functions. This implies that one could construct a model with an undefined value (the term 1=0 could be interpreted as undefined), thus our interpretation is threevalued (a formula is either true, false or illdefined). Different threev alued interpretations and different consequence relations exist for threevalued logics. Thus, we have to make some choices.
Reasoning with Higher Order Partial Functions
, 1992
"... . In this paper we introduce the logic PHOL, which embodies higherorder functions through a simplytyped calculus and deals with partial objects by using partially ordered domains and three truth values. We define a refutationally complete tableaux method for PHOL and we show how to derive a sound ..."
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Cited by 1 (0 self)
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. In this paper we introduce the logic PHOL, which embodies higherorder functions through a simplytyped calculus and deals with partial objects by using partially ordered domains and three truth values. We define a refutationally complete tableaux method for PHOL and we show how to derive a sound and complete cut free sequent calculus through a systematic analysis of the rules for tableaux construction. 1 Introduction The formal methodologies used for the specification and verification of software (and also hardware) systems have motivated during the last years the development of so called partial logics, where partial functions can be used to argue about errors, diverging computations, and similar phenomena. Following some earlier precedents, such as [16], [21] and [5], quite many recent papers (as e.g. [2], [24], [17], [14], [19], [13]) have proposed partial versions of first order predicate logic aiming at this field of application. Partiality has been also investigated within t...
Automatic Deduction for Theories of Algebraic Data Types
, 2011
"... In this thesis we present formal logical systems, concerned with reasoning about algebraic data types. The first formal system is based on the quantifierfree calculus (outermost universally quantified). This calculus is comprised of state change rules, and computations are performed by successive a ..."
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In this thesis we present formal logical systems, concerned with reasoning about algebraic data types. The first formal system is based on the quantifierfree calculus (outermost universally quantified). This calculus is comprised of state change rules, and computations are performed by successive applications of these rules. Thereby, our calculus gives rise to an abstract decision procedure. This decision procedure determines if a given formula involving algebraic type members is valid. It is shown that this calculus is sound and complete. We also examine how this system performs practically and give experimental results. Our main contribution, as compared to previous work on this subject, is a new and more efficient decision procedure for checking satisfiability of the universal fragment within the theory of algebraic data types. The second formal system, called Term Builder, is the deductive system based on higher order type theory, which subsumes second order and higher order logics. The main purpose of this calculus is to formulate and prove theorems about algebraic or other arbitrary userdefined types. Term Builder supports proof objects and is