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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally e ..."
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Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Iterative Belief Revision in Extended Logic Programming
 Theoretical Computer Science
, 1996
"... Extended logic programming augments conventional logic programming with both default and explicit negation. Several semantics for extended logic programs have been proposed that extend the wellfounded semantics for logic programs with default negation (called normal programs). We show that two of t ..."
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Cited by 7 (5 self)
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Extended logic programming augments conventional logic programming with both default and explicit negation. Several semantics for extended logic programs have been proposed that extend the wellfounded semantics for logic programs with default negation (called normal programs). We show that two of these extended semantics are intractable; both Dung's grounded argumentation semantics and the wellfounded semantics of Alferes et al. are NPhard. Nevertheless, we also show that these two semantics have a common core, a more restricted form of the grounded semantics, which is tractable and can be computed iteratively in quadratic time. Moreover, this semantics is a representative of a rich class of tractable semantics based on a notion of iterative belief revision. 1 Introduction The semantics of logic programs with two kinds of negation, default negation and explicit negation, has been studied extensively in the recent literature (e.g. [1, 4, 9, 12, 16, 18, 19, 24, 25, 26, 27]). Logic ...
Inductive invariants for nested recursion
 Theorem Proving in Higher Order Logics (TPHOLS'03), volume 2758 of LNCS
, 2003
"... Abstract. We show that certain inputoutput relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higherorder logic mechanizations. We d ..."
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Cited by 4 (2 self)
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Abstract. We show that certain inputoutput relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higherorder logic mechanizations. We demonstrate the usefulness of inductive invariants on a large example of the BDD algorithm Apply. Finally, we introduce a related concept of inductive fixpoints with the property that for every functional in higherorder logic there exists a largest partial function that is such a fixpoint. 1
A Theory of Truth that prefers falsehood
 Journal of Philosophical Logic
, 1994
"... We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to h ..."
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We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to have many nice features. It is noted that a similar class of fixed points, preferring truth, can also be studied. The notion of intrinsic is shown to relativize to these two subclasses. The mathematical ideas presented here originated in investigations of socalled stable models in the semantics of logic programming. 1 Introduction Briefly stated, the job of a theory of truth is to assign truth values to sentences in a language allowing selfreference, in a way that respects intuition while avoiding paradox. Of course this can not be done in the framework of classical, twovalued logic because of liar sentences. Some generalization allowing partial truth assignments, or perhaps contradic...
Some issues and trend in the semantics of logic programming
 In Research Report of IBM TJ Watson Research Center (Armonk: IBM
, 1986
"... A major problem facing the designers of programming languages is the conflict between expressive power (or “highlevelness”) and efficiency of execution. In the conventional approach expressive power has been consistently sacrificed in an ad hoc manner to efficiency and often has been confused with ..."
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A major problem facing the designers of programming languages is the conflict between expressive power (or “highlevelness”) and efficiency of execution. In the conventional approach expressive power has been consistently sacrificed in an ad hoc manner to efficiency and often has been confused with the accumulation of “useful ” features. This has resulted in the design of intricate languages such as ADA. Both Backus and Hoare in their Turing award addresses have been highly critical of this approach. The hope that the complexity of such languages could be mastered by using the sophisticated theoretical tools of denotational semantics has not been fulfilled. Logicbased programming languages, on the other hand, have the great advantage that formal tools such as models and resolution used to capture their semantic properties do so in a simple and natural manner. In the first part we will review the main semantic properties of definite clauses, which form a theoretical basis for the study of PROLOG. These are model theoretic semantics, least fixedpoint semantics, finite failure and
On Prudent Bravery and Other Abstractions
 In preparation
, 1994
"... ions Melvin Fitting fitting@alpha.lehman.cuny.edu Dept. Mathematics and Computer Science Lehman College (CUNY), Bronx, NY 10468 Depts. Computer Science, Philosophy, Mathematics Graduate Center (CUNY), 33 West 42nd Street, NYC, NY 10036 # October 13, 1994 Abstract A special class of partial ..."
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ions Melvin Fitting fitting@alpha.lehman.cuny.edu Dept. Mathematics and Computer Science Lehman College (CUNY), Bronx, NY 10468 Depts. Computer Science, Philosophy, Mathematics Graduate Center (CUNY), 33 West 42nd Street, NYC, NY 10036 # October 13, 1994 Abstract A special class of partial stable models, the intrinsic ones, is singled out for consideration, and attention is drawn to the largest one, which we designate as the prudently brave one. It is the largest partial stable model that is compatible with every partial stable model. As such, it is an object of natural interest. Its existence follows from general properties of monotonic functions, so it is a robust notion. The proofs given concerning intrinsic stable models are not new, but they appeared earlier in quite di#erent contexts. What we do, essentially, is call the attention of the logic programming and nonmonotonic reasoning community to them. We further show the entire development fits into the bilattice ...
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Implicit Programming and Computable Optimal Fixed Points
, 1997
"... If a program has a unique solution then programming semantics should return that solution. Optimal fixed points represent this unique solution; that is, if the only choice is between undefined and a specific defined value then we choose the defined value. However, optimal fixed points can be unco ..."
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If a program has a unique solution then programming semantics should return that solution. Optimal fixed points represent this unique solution; that is, if the only choice is between undefined and a specific defined value then we choose the defined value. However, optimal fixed points can be uncomputable. For this reason they have been relatively underutilized. Here we provide a method which computes the optimal fixed point when it is computable. To do this we use proofs of uniqueness in an extension of intuitionistic logic together with its associated CurryHowardDe Bruijn isomorphism. Using this correspondence, we extract programs in a confluent strongly normalizable lambda calculus that computes the optimal fixed point of a recursively defined function. When this uniqueness proof exists, the optimal fixed point is computable. We give several examples of where this fixed point gives intuitive answers. This new semantics is especially natural for functions on corecursivel...
Implicit Programming and the Logic of Constructible Duality
"... ABSTRACT We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a selfdual logic constructible duality. We develop a selfdual model by considering an interval of worlds in an intuitionistic Kr ..."
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ABSTRACT We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a selfdual logic constructible duality. We develop a selfdual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards. We develop a selfdual algebra such that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a Heyting algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a Heyting algebra, the second Dialectica construction applied to a Heyting algebra, and as an algebra for the study of recursion and corecursion. Thus the algebra provides a common base for these constructions, and suggests itself as an important part of any constructive logical treatment of duality. Implicit programming is suggested as a new paradigm for computing with constructible duality as its formal system. We show that all the operators that have computable least fixed points are definable explicitly and all operators with computable optimal fixed points are definable implicitly within constructible duality. Implicit programming adds a novel definitional mechanism that allows functions to be defined implicitly. This new programming feature is especially useful for programming with corecursively defined datatypes such as circular lists. iii DEDICATION To my cousin Jordan Lackey (19631995) whose courage with AIDS was an inspiration. iv