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35
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 66 (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 ...
Proof Planning with Multiple Strategies
 In Proc. of the First International Conference on Computational Logic
, 2000
"... . Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introd ..."
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Cited by 59 (37 self)
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. Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introduce multistrategy proof planning that allows for combining a number of strategies and for switching flexibly between strategies in a proof planning process. Thereby proof planning becomes more robust since it does not necessarily fail if one problem solving mechanism fails. Rather it can reason about preference of strategies and about failures. Moreover, our strategies provide a means for structuring the vast amount of knowledge such that the planner can cope with the otherwise overwhelming knowledge in mathematics. 1 Introduction The choice of an appropriate problem solving strategy is a crucial human skill and is typically guided by some metalevel reasoning. Trained mathematicia...
Beyond The Universal Turing Machine
, 1998
"... We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a phi ..."
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Cited by 31 (1 self)
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We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 21 (8 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
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 20 (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
Deciding Presburger Arithmetic by Model Checking and Comparisons with Other Methods
 In Proceedings of FMCAD 02
, 2002
"... We present a new way of using Binary Decision Diagrams in automata based algorithms for solving the satisfiability problem of quantifierfree Presburger arithmetic. Unlike in previous approaches [5, 2, 19], we translate the satisfiability problem into a model checking problem and use the existing ..."
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Cited by 18 (1 self)
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We present a new way of using Binary Decision Diagrams in automata based algorithms for solving the satisfiability problem of quantifierfree Presburger arithmetic. Unlike in previous approaches [5, 2, 19], we translate the satisfiability problem into a model checking problem and use the existing BDDbased model checker SMV [13] as our primary engine.
A Proof Planning Framework for Isabelle
, 2005
"... Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully ..."
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Cited by 13 (9 self)
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Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully formal proofs. This thesis concerns the development and analysis of a novel approach to proof planning that focuses on an explicit representation of choices during search. We embody our approach as a proof planner for the generic proof assistant Isabelle and use the Isar language, which is humanreadable and machinecheckable, to represent proof plans. Within this framework we develop an inductive theorem prover as a case study of our approach to proof planning. Our prover uses the difference reduction heuristic known as rippling to automate the step cases of the inductive proofs. The development of a flexible approach to rippling that supports its various modifications and extensions is the second major focus of this thesis. Here, our inductive theorem prover provides a context in which to evaluate rippling experimentally. This work results in an efficient and powerful inductive theorem prover for Isabelle as well as proposals for further improving the efficiency of rippling. We also draw observations in order
Extensions to a Generalization Critic for Inductive Proof
 13th Conference on Automated Deduction
, 1996
"... In earlier papers a critic for automatically generalizing conjectures in the context of failed inductive proofs was presented. The critic exploits the partial success of the search control heuristic known as rippling. Through empirical testing a natural generalization and extension of the basic crit ..."
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Cited by 12 (7 self)
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In earlier papers a critic for automatically generalizing conjectures in the context of failed inductive proofs was presented. The critic exploits the partial success of the search control heuristic known as rippling. Through empirical testing a natural generalization and extension of the basic critic emerged. Here we describe our extended generalization critic together with some promising experimental results. 1 Introduction A major obstacle to the automation of proof by mathematical induction is the need for generalization. A generalization is underpinned by the cutrule of inference. In a goaldirected framework, therefore, a generalization introduces an infinite branching point into the search space. It is known [13] that the cutelimination theorem does not hold for inductive theories. Consequently heuristics for controlling generalization play an important role in the automation of inductive proof. There are a number of different kinds of generalization. In this paper we present...
Hierarchical Overlapped Tiling
"... This paper introduces hierarchical overlapped tiling, a transformation that applies loop tiling and fusion to conventional loops. Overlapped tiling is a useful transformation to reduce communication overhead, but it may also generate a significant amount of redundant computation. Hierarchical overla ..."
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Cited by 11 (1 self)
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This paper introduces hierarchical overlapped tiling, a transformation that applies loop tiling and fusion to conventional loops. Overlapped tiling is a useful transformation to reduce communication overhead, but it may also generate a significant amount of redundant computation. Hierarchical overlapped tiling performs overlapped tiling hierarchically to balance communication overhead and redundant computation, and thus has the potential to provide better performance. In this paper, we describe the hierarchical overlapped tiling optimization and its implementation in an OpenCL compiler. We also evaluate the effectiveness of this optimization using 8 programs that implement different forms of stencil computation. Our results show that hierarchical overlapped tiling achieves an average 37 % speedup over traditional tiling on a 32core workstation. Categories and Subject Descriptors