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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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Cited by 53 (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 ...
Reflection in logic, functional and objectoriented programming: a short comparative study
 Proc. of the IJCAI’95 Workshop on Reflection and Metalevel Architectures andtheir Applications in AI,1995
"... Département d’informatique et de recherche opérationnelle ..."
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Cited by 38 (1 self)
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Département d’informatique et de recherche opérationnelle
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
Algorithmic Information Theory
, 1989
"... We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on G6del's first incompleteness theorem. ..."
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Cited by 13 (0 self)
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We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on G6del's first incompleteness theorem.
ON INTERPRETING CHAITIN’S INCOMPLETENESS THEOREM
, 1998
"... The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number ..."
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Cited by 9 (0 self)
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The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.
A Finitary Treatment of the Closed Fragment of
, 2005
"... We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev (see [11, 7]), heavily relied on some nonfinitary principles such as transfinite induction up to #0 or reflection principles. In fact, the cl ..."
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Cited by 6 (6 self)
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We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev (see [11, 7]), heavily relied on some nonfinitary principles such as transfinite induction up to #0 or reflection principles. In fact, the closed fragment of GLP gives rise to a natural system of ordinal notation for #0 that was used in [1] for a prooftheoretic analysis of Peano arithmetic and for constructing simple combinatorial independent statements.
Case Studies in MetaLevel Theorem Proving
 PROC. INTL. CONF. ON THEOREM PROVING IN HIGHER ORDER LOGICS (TPHOLS), LECTURE
, 1998
"... We describe an extension of the Pvs system that provides a reasonably efficient and practical notion of reflection and thus allows for soundly adding formalized and verified new proof procedures. These proof procedures work on representations of a part of the underlying logic and their correct ..."
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Cited by 4 (1 self)
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We describe an extension of the Pvs system that provides a reasonably efficient and practical notion of reflection and thus allows for soundly adding formalized and verified new proof procedures. These proof procedures work on representations of a part of the underlying logic and their correctness is expressed at the object level using a computational reflection function. The implementation of the Pvs system has been extended with an efficient evaluation mechanism, since the practicality of the approach heavily depends on careful engineering of the core system, including efficient normalization of functional expressions. We exemplify the process of applying metalevel proof procedures with a detailed description of the encoding of cancellation in commutative monoids and of the kernel of a BDD package.