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22
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 ...
SelfReference and Validity revisited
"... This paper is a revision and expansion of my SelfReference and Validity, Synthese, 42, 1979, pp. 26574. The major changes consist in the addition of new //2, 4 and 7. The reader may be interested to see subsequent discussion of the issues in Priest and Routley 1982, Sorensen 1988, pp. 299310, Dra ..."
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This paper is a revision and expansion of my SelfReference and Validity, Synthese, 42, 1979, pp. 26574. The major changes consist in the addition of new //2, 4 and 7. The reader may be interested to see subsequent discussion of the issues in Priest and Routley 1982, Sorensen 1988, pp. 299310, Drange 1990 and Jacquette 19??
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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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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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,
Computation, consciousness and the quantum
 Teorie e Modelli
, 2001
"... Abstract: It is sometimes said that Everett’s formulation of Quantum Mechanics dispenses us with the need of a theory of consciousness in the foundation of physics. This is false as Everett himself clearly recognized in his paper. Indeed he has build its quantum mechanics formulation by using expli ..."
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Abstract: It is sometimes said that Everett’s formulation of Quantum Mechanics dispenses us with the need of a theory of consciousness in the foundation of physics. This is false as Everett himself clearly recognized in his paper. Indeed he has build its quantum mechanics formulation by using explicitly the mechanist or computationalist hypothesis in psychology. Everett and his followers have then derived the subjective appearance, in the mind of machineobservers, of indeterminacy and nonlocality from the Schrödinger Equation. I argue in this paper that if we take the computationalist hypothesis seriously enough then the Schrödinger equation itself should be derivable from the computationalist theory of consciousness, making ultimately physics a branch of machine’s psychology. I sketch the basic argument and illustrate it with two embryonic derivations. In some sense I criticize Everett for his lack of radicality. 1 Quantum Realism Let me put it in this way: all sufficiently realist1 interpretations of quantum
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