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13
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 ...
A proofproducing decision procedure for real arithmetic
 Automated deduction – CADE20. 20th international conference on automated deduction
, 2005
"... Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate conv ..."
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Cited by 24 (3 self)
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Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving. 1 Overview and related work Arguably the first automated theorem prover ever written was for a theory of lineararithmetic [8]. Nowadays many theorem proving systems, even those normally classified as `interactive ' rather than `automatic', contain procedures to automate routinearithmetical reasoning over some of the supported number systems like N, Z, Q, R and C. Experience shows that such automated support is invaluable in relieving users ofwhat would otherwise be tedious lowlevel proofs. We can identify several very common limitations of such procedures: Often they are restricted to proving purely universal formulas rather than dealingwith arbitrary quantifier structure and performing general quantifier elimination. Often they are not complete even for the supported class of formulas; in particular procedures for the integers often fail on problems that depend inherently on divisibility properties (e.g. 8x y 2 Z. 2x + 1 6 = 2y) They seldom handle nontrivial nonlinear reasoning, even in such simple cases as 8x y 2 R. x> 0 ^ y> 0) xy> 0, and those that do [18] tend to use heuristicsrather than systematic complete methods. Many of the procedures are standalone decision algorithms that produce no certificate of correctness and do not produce a `proof ' in the usual sense. The earliest serious exception is described in [4]. Many of these restrictions are not so important in practice, since subproblems arising in interactive proof can still often be handled effectively. Indeed, sometimes the restrictions are unavoidable: Tarski's theorem on the undefinability of truth implies thatthere cannot even be a complete semidecision procedure for nonlinear reasoning over
A framework for the flexible integration of a class of decision procedures into theorem provers
 FEDRA, K., GIS AND ENVIRONMENTAL MODELING
, 1999
"... The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision proc ..."
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Cited by 5 (2 self)
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The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision procedures. Some of these conjectures could in an informal sense fall ‘just outside’ that scope. In these situations a problem arises because lemmas have to be invoked or the decision procedure has to communicate with the heuristic component of a theorem prover. This problem is also related to the general problem of how to exibly integrate decision procedures into heuristic theorem provers. In this paper we address such problems and describe a framework for the exible integration of decision procedures into other proof methods. The proposed framework can be used in different theorem provers, for different theories and for different decision procedures. New decision procedures can be simply ‘pluggedin’ to the system. As an illustration, we describe an instantiation of this framework within the Clam proofplanning system, to which it is well suited. We report on some results using this implementation.
A comparison of decision procedures in Presburger arithmetic
 University of Novi Sad
, 1997
"... It is part of the tradition and folklore of automated reasoning that the intractability of Cooper's decision procedure for Presburger integer arithmetic makes is too expensive for practical use. More than 25 years of work has resulted in numerous approximate procedures via rational arithmetic, ..."
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Cited by 4 (1 self)
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It is part of the tradition and folklore of automated reasoning that the intractability of Cooper's decision procedure for Presburger integer arithmetic makes is too expensive for practical use. More than 25 years of work has resulted in numerous approximate procedures via rational arithmetic, all of which are incomplete and restricted to the quantifierfree fragment. In this paper we report on an experiment which strongly questions this tradition. We measured the performance of procedures due to Hodes, Cooper (and heuristic variants thereof which detect counterexamples), across a corpus of 10 000 randomly generated quantifierfree Presburger formulae. The results are startling: a variant of Cooper's procedure outperforms Hodes' procedure on both valid and invalid formulae, and is fast enough for practical use. These results contradict much perceived wisdom that decision procedures for integer arithmetic are too expensive to use in practice. 1 Introduction A decis...
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Cited by 4 (0 self)
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1
Strict General Setting for Building Decision Procedures into Theorem Provers
 THE 1ST INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONING (IJCAR2001) — SHORT PAPERS
, 2001
"... The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking ..."
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Cited by 2 (1 self)
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The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of wellknown or new schemes within this framework a very easy task even for a nonexpert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas.
Compositionality  With an appendix by B. Partee
 IN: VAN BENTHEM & TER MEULEN
, 1997
"... The first topic of the paper is to provide a formalization of the principle of compositionality of meaning. A mathematical model (based upon universal algebra) is presented, and its properties are investigated. The second topic is to discuss arguments from the literature against compositionality (of ..."
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The first topic of the paper is to provide a formalization of the principle of compositionality of meaning. A mathematical model (based upon universal algebra) is presented, and its properties are investigated. The second topic is to discuss arguments from the literature against compositionality (of Hintikka, Higginbotham, Pelletier, Partee, Schiffer and others). Methods are presented that help to obtain compositionality. It is argued that the principle is should not be considered an empirical verifyable restriction, but a methodological principle that describes how a system for syntax and semantics should be designed. The paper has an appendix by B. Partee on the compositional treatment of genitives.
Testing Deadlockfreeness in Realtime Systems; A Formal Approach
"... Abstract. A Time Action Lock is a state of a Realtime system at which neither time can progress nor an action can occur. Time Action Locks are often seen as signs of errors in the model or inconsistencies in the specification. As a result, finding out and resolving Time Action Locks is a major task ..."
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Abstract. A Time Action Lock is a state of a Realtime system at which neither time can progress nor an action can occur. Time Action Locks are often seen as signs of errors in the model or inconsistencies in the specification. As a result, finding out and resolving Time Action Locks is a major task for the designers of Realtime systems. Verification is one of the methods of discovering deadlocks. However, due to state explosion, the verification of deadlock freeness is computationally expensive. The aim of this paper is to present a computationally cheap testing method for Timed Automata models and pointing out any source of possible Time Action Locks to the designer. We have implemented the approach presented in the paper, which is based on the geometry of Timed Automata, via a Testing Tool called TALC (Time Action Lock Checker). TALC, which is used in the conjunction with the model checker UPPAAL, tests the UPPAAL model and provides feedback to the designer. We have illustrated our method by applying TALC to a model of a simple communication protocol.
The History of Mathematical Logic (vastly abbreviated and horribly simplified)
, 1997
"... F11.95> manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate "building blocks". In ..."
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F11.95> manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate "building blocks". In particular, one has to ask the questions about the meaning of such building blocks, of various terms and categories of terms and, furthermore, of their combinations. 3. Finally, there is the question of how to represent these patterns. Although apparently of secondary importance, it is the answer to this question which can be, to a high degree, considered the beginning of modern mathematical logic. The first three sections sketch the development along the respective lines until Renessance. In section 4, we indicate the development in modern era, with particular emphasis on the last two centuries. Section 5 indicates some basic aspect