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Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
Iterated Local Reflection vs Iterated Consistency
, 1995
"... For "natural enough" systems of ordinal notation we show that ff times iterated local reflection schema over a sufficiently strong arithmetic T proves the same \Pi 0 1 sentences as ! ff times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability ..."
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For "natural enough" systems of ordinal notation we show that ff times iterated local reflection schema over a sufficiently strong arithmetic T proves the same \Pi 0 1 sentences as ! ff times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at fflnumbers. We also derive the following more general "mixed" formulas estimating the consistency strength of iterated local reflection: for all ordinals ff 1 and all fi, (T ff ) fi j \Pi 0 1 T ! ff \Delta(1+fi) ; (T fi ) ff j \Pi 0 1 T fi+! ff : Here T ff stands for ff times iterated local reflection over T , T fi stands for fi times iterated consistency, and j \Pi 0 1 denotes (provable in T ) mutual \Pi 0 1 conservativity. In an appendix to this paper we develop our notion of "natural enough" system of ordinal notation and show that such systems do exist for every recursive ordinal. 1 Introduction Since the fundamental works of A.Turing [17] and S.Feferman [6...
Reflection Using the Derivability Conditions
"... We extend arithmetic with a new predicate, Pr, giving axioms for Pr based on firstorder versions of Lob's derivability conditions. We hoped that the addition of a reflection schema mentioning Pr would then give a nonconservative extension of the original arithmetic theory. The paper investigates t ..."
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We extend arithmetic with a new predicate, Pr, giving axioms for Pr based on firstorder versions of Lob's derivability conditions. We hoped that the addition of a reflection schema mentioning Pr would then give a nonconservative extension of the original arithmetic theory. The paper investigates this possibility. It is shown that, under special conditions, the extension is indeed nonconservative. However, in general such extensions turn out to be conservative. 1 Introduction In any recursively axiomatized theory of arithmetic, T , one can follow Godel's construction to obtain a `provability predicate', a \Sigma 1 formula Bew T (x) such that Bew T (pAq) is true if and only if T ` A, where pAq is the Godel number of the formula A. Moreover, if T is sufficiently strong then Bew T satisfies the following predicate (or `uniform') versions of Lob's derivability conditions [7]: (D1) if T ` 8xA then T ` 8xBew T (pAhxiq); (D2) T ` 8x(Bew T (p(A ! B)hxiq) ! (Bew T (pAhxiq) ! Bew T (pBhxiq)...
Dialetheic Truth Theory: Inconsistency, NonTriviality, Soundness, Incompleteness
"... Abstract. The bestknown application of dialetheism is to semantic paradoxes such as the Liar. In particular, Graham Priest has advocated the adoption of an axiomatic truth theory in which contradictions arising from the Liar paradox can be accepted as theorems, while the Liar sentence itself is eva ..."
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Abstract. The bestknown application of dialetheism is to semantic paradoxes such as the Liar. In particular, Graham Priest has advocated the adoption of an axiomatic truth theory in which contradictions arising from the Liar paradox can be accepted as theorems, while the Liar sentence itself is evaluated as being both true and false. Such eccentricities might be tolerated, in exchange for great rewards. But in this note I show that it is not possible in Priest’s truth theory to express certain semantic facts about that very theory, and thus that it enjoys no definite advantage over more orthodox approaches to semantic paradox.
Robust Cooperation in the Prisoner’s Dilemma: Program Equilibrium via Provability Logic
, 2013
"... We consider the oneshot Prisoner’s Dilemma between algorithms with access to one anothers ’ source codes, and apply the modal logic of provability to achieve a flexible and robust form of mutual cooperation. We discuss some variants, and point out obstacles to definitions of optimality. 1 ..."
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We consider the oneshot Prisoner’s Dilemma between algorithms with access to one anothers ’ source codes, and apply the modal logic of provability to achieve a flexible and robust form of mutual cooperation. We discuss some variants, and point out obstacles to definitions of optimality. 1
Tiling Agents for SelfModifying AI, and the Löbian Obstacle *
, 2013
"... (Early Draft) We model selfmodification in AI by introducing “tiling ” agents whose decision systems will approve the construction of highly similar agents, creating a repeating pattern (including similarity of the offspring’s goals). Constructing a formalism in the most straightforward way produce ..."
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(Early Draft) We model selfmodification in AI by introducing “tiling ” agents whose decision systems will approve the construction of highly similar agents, creating a repeating pattern (including similarity of the offspring’s goals). Constructing a formalism in the most straightforward way produces a Gödelian difficulty, the “Löbian obstacle. ” By technical methods we demonstrate the possibility of avoiding this obstacle, but the underlying puzzles of rational coherence are thus only partially addressed. We extend the formalism to partially unknown deterministic environments, and show a very crude extension to probabilistic environments and expected utility; but the problem of finding a fundamental decision criterion for selfmodifying probabilistic agents remains open. 1