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19
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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Cited by 57 (2 self)
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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 ...
ON PROVABILITY LOGIC
, 2000
"... This is an introductory paper about provability logic, a modal propositional logic in which necessity is interpreted as formal provability. I discuss the ideas that led to establishing this logic, I survey its history and the most important results, and I emphasize its applications in metamathematic ..."
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Cited by 1 (0 self)
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This is an introductory paper about provability logic, a modal propositional logic in which necessity is interpreted as formal provability. I discuss the ideas that led to establishing this logic, I survey its history and the most important results, and I emphasize its applications in metamathematics. Stress is put on the use of Gentzen calculus for provability logic. I sketch my version of a decision procedure for provability logic and mention some connections to computational complexity.
Iterated Local Reflection vs Iterated Consistency
, 1995
"... For "natural enough" systems of ordinal notation we show that times iterated local reflection schema over a sufficiently strong arithmetic T 0 proves the same 1sentences as! times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exact ..."
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Cited by 1 (0 self)
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For "natural enough" systems of ordinal notation we show that times iterated local reflection schema over a sufficiently strong arithmetic T 0 proves the same 1sentences as! times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly atnumbers. We also derive the following more general "mixed" formulas estimating the consistency strength of iterated local reflection: for all ordinals 1 and all,
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
Incompleteness in a General Setting
"... Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details ..."
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Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel‘s theorems without getting mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the HilbertBernaysLöb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel‘s second incompleteness theorem for it. A key role in Löb‘s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a
On Modal Systems with Rosser Modalities
"... Sufficiently strong axiomatic theories allow for the construction of selfreferential sentences, i.e. sentences saying something about themselves. After the Gödel’s paper on incompleteness (Gödel, 1931) the selfreference method found further applications—some are listed below—and became even more ..."
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Sufficiently strong axiomatic theories allow for the construction of selfreferential sentences, i.e. sentences saying something about themselves. After the Gödel’s paper on incompleteness (Gödel, 1931) the selfreference method found further applications—some are listed below—and became even more