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12
A thread of HOL development
 Computer Journal
"... The HOL system is a mechanized proof assistant for higher order logic that has been under continuous development since the mid1980s, by an everchanging group of developers and external contributors. We give a brief overview of various implementations of the HOL logic before focusing on the evoluti ..."
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Cited by 13 (7 self)
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The HOL system is a mechanized proof assistant for higher order logic that has been under continuous development since the mid1980s, by an everchanging group of developers and external contributors. We give a brief overview of various implementations of the HOL logic before focusing on the evolution of certain important features available in a recent implementation. We also illustrate how the module system of Standard ML provided security and modularity in the construction of the HOL kernel, as well as serving in a separate capacity as a useful representation medium for persistent, hierarchical logical theories.
An Even Closer Integration of Linear Arithmetic into Inductive Theorem Proving
 Proc. Calculemus 2005: 12 th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning
, 2006
"... To broaden the scope of decision procedures for linear arithmetic, they have to be integrated into theorem provers. Successful approaches e.g. in NQTHM or ACL2 suggest a close integration scheme which augments the decision procedures with lemmas about userdefined operators. We propose an even close ..."
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Cited by 6 (1 self)
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To broaden the scope of decision procedures for linear arithmetic, they have to be integrated into theorem provers. Successful approaches e.g. in NQTHM or ACL2 suggest a close integration scheme which augments the decision procedures with lemmas about userdefined operators. We propose an even closer integration providing feedback about the state of the decision procedure in terms of entailed formulas for three reasons: First, to provide detailed proof objects for proof checking and archiving. Second, to analyze and improve the interaction between the decision procedure and the theorem prover. Third, to investigate whether the communication of the state of a failed proof attempt to the human user with the comprehensible standard GUI mechanisms of the theorem prover can enhance the speculation of auxiliary lemmas.
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...
J.S.: Integrating nonlinear arithmetic into into ACL2
 In: Fifth International Workshop on the ACL2 Theorem Prover and Its Applications
, 2004
"... Abstract. In this paper we present an overview of the integration of a nonlinear arithmetic reasoning package into ACL2. We provide a brief operational description of the entire arithmetic package and describe how it fits into ACL2’s operation, including what was needed for the successful introducti ..."
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Cited by 3 (0 self)
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Abstract. In this paper we present an overview of the integration of a nonlinear arithmetic reasoning package into ACL2. We provide a brief operational description of the entire arithmetic package and describe how it fits into ACL2’s operation, including what was needed for the successful introduction of such a facility into an existing automated theorem prover. We describe most of the changes we made to the previous version of ACL2 as well as a couple of recent improvements to the nonlinear package we made based upon our experiences using the same. The resulting system lessens the human effort required to construct a large arithmetic proof by reducing the number of intermediate lemmas that must be proven. 1
A comparison of decision procedures
 in Presburger arithmetic. LIRA '97, Univ. of Novi Sad
, 1997
"... The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In ..."
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Cited by 2 (0 self)
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The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In this paper we introduce a general setting for describing different schemes for both combining and augmenting decision procedures. This setting is based on the macro inference rules used in different approaches. Some of these rules are abstraction, entailment, congruence closure and lemma invoking. The general setting gives a simple description and the key ideas of one scheme and makes different schemes comparable. Also, it makes easier combining ideas from different schemes. In this paper we describe several schemes via introduced macro inference rules and report on our prototype implementation.
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.
Reasoning About Linear Systems
 In SEFM’07
, 2007
"... We consider reasoning about linear systems expressed as block diagrams in a general relational setting. Using the notion of additive relation borrowed from homological algebra, the theory of weakest preconditions for these systems turns out to be very tractable and gives simple Hoarestyle rules for ..."
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Cited by 1 (1 self)
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We consider reasoning about linear systems expressed as block diagrams in a general relational setting. Using the notion of additive relation borrowed from homological algebra, the theory of weakest preconditions for these systems turns out to be very tractable and gives simple Hoarestyle rules for the block diagram constructors. Many natural choices for the logical language used to express properties of linear systems admit a high degree of automation. We show by example how the rules can be used to inform development of a proof while a decision procedure automates the routine work. 1
Mechanizing Proof for the Z Toolkit
"... This paper reports on theorems and proof procedures for working with the Z mathematical toolkit developed using the ProofPower system. This development has taken place in parallel with work on ProofPower itself over the last ve or six years. ..."
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This paper reports on theorems and proof procedures for working with the Z mathematical toolkit developed using the ProofPower system. This development has taken place in parallel with work on ProofPower itself over the last ve or six years.
Mechanized Reasoning for Continuous Problem Domains (Extended Abstract)
"... Abstract. Specification and verification in continuous problem domains are key topics for the practical application of formal methods and mechanized reasoning. I discuss one approach to linear continuous control systems and consider the challenges and opportunities raised for mechanized reasoning. T ..."
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Abstract. Specification and verification in continuous problem domains are key topics for the practical application of formal methods and mechanized reasoning. I discuss one approach to linear continuous control systems and consider the challenges and opportunities raised for mechanized reasoning. These include practical implementation and integration issues, algorithms in computational real algebraic geometry and hard open questions such as the Schanuel conjecture. I conclude with an overview of some recent new results on decidability and undecidability for vector spaces and related theories. 1