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14
Computer algebra meets automated theorem proving: Integrating Maple and pvs
 Theorem Proving in Higher Order Logics (TPHOLs 2001), volume 2152 of LNCS
, 2001
"... ..."
NonStandard Analysis in ACL2
, 2001
"... ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complexrational numbers and excluding the real and complex irrationals. In conjunction ..."
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Cited by 18 (7 self)
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ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complexrational numbers and excluding the real and complex irrationals. In conjunction with the arithmetic completion axioms, this numeric type system makes it possible to prove the nonexistence of specific irrational numbers, such as √2. This paper describes ACL2(r), a version of ACL2 with support for the real and complex numbers. The modifications are based on nonstandard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.
Evaluating, testing, and animating PVS specifications
, 2001
"... We explore ways to enhance the utility of PVS for evaluating, testing, and animating PVS specifications. The PVS ground evaluator is the focus of the work. We describe a mechanism to provide semantic attachments for PVS symbols while preserving soundness, and discuss strategies to provide a generic ..."
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Cited by 13 (5 self)
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We explore ways to enhance the utility of PVS for evaluating, testing, and animating PVS specifications. The PVS ground evaluator is the focus of the work. We describe a mechanism to provide semantic attachments for PVS symbols while preserving soundness, and discuss strategies to provide a generic framework for integrating independently developed applications with PVS. We explore these capabilities in the current system, but conclude that more effective functionality requires extensions to PVS. Recommendations for these extensions are outlined.
Defining Differentiation and Integration in Z
 IEEE International Conference on Formal Engineering Methods (ICFEM '98
, 1998
"... s and compressed postscript files are available via http://svrc.it.uq.edu.au Defining Differentiation and Integration in Z C. J. Fidge I. J. Hayes y B. P. Mahony z Abstract We show how familiar mathematical concepts from differential and integral calculus can be represented in the Z specifi ..."
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Cited by 11 (3 self)
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s and compressed postscript files are available via http://svrc.it.uq.edu.au Defining Differentiation and Integration in Z C. J. Fidge I. J. Hayes y B. P. Mahony z Abstract We show how familiar mathematical concepts from differential and integral calculus can be represented in the Z specification language. Digital computer systems involve hardware devices and software variables that can adopt a limited range of values only, and may be temporarily inaccessible or illdefined. Emphasis is therefore given to supporting discrete range types and partial functions. 1 Introduction Calculus is the study of change and motion, and a cornerstone of modern mathematics. It provides methods for solving two large classes of problems. Differential calculus allows us to find the rate at which a variable quantity changes, given its characteristic function. Integral calculus allows us to find the function defining a variable quantity, given its rate of change. Considering the importance of th...
Formal Verification of Conflict Detection Algorithms
 in Proceedings of the 11th Working Conference on Correct Hardware Design and Verification Methods CHARME 2001
, 2001
"... Safety assessment of new air traffic management systems is a main issue for civil aviation authorities. Standard techniques such as testing and simulation have serious limitations in new systems which are significantly more autonomous than the older ones. In this paper, we present an innovative appr ..."
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Cited by 11 (4 self)
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Safety assessment of new air traffic management systems is a main issue for civil aviation authorities. Standard techniques such as testing and simulation have serious limitations in new systems which are significantly more autonomous than the older ones. In this paper, we present an innovative approach for establishing the correctness of conflict detection systems. Fundamental to our approach is the concept of trajectory, which is described by a continuous path in the xy plane constrained by physical laws and operational requirements. From the model of trajectories, we extract, and formally prove, higher level properties that can serve as a framework to analyze conflict scenarios. We use the AILS (Airborne Information for Lateral Spacing) alerting algorithm as a case study of our approach.
Real number calculations and theorem proving
 Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. ..."
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Cited by 11 (4 self)
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Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 1
Automated Theorem Proving in Support of Computer Algebra: Symbolic Definite Integration as a Case Study
"... We assess the current state of research in the application of computer aided formal reasoning to computer algebra, and argue that embedded verification support allows users to enjoy its benefits without wrestling with technicalities. We illustrate this claim by considering symbolic definite integrat ..."
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Cited by 10 (5 self)
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We assess the current state of research in the application of computer aided formal reasoning to computer algebra, and argue that embedded verification support allows users to enjoy its benefits without wrestling with technicalities. We illustrate this claim by considering symbolic definite integration, and present a verifiable symbolic definite integral table look up: a system which matches a query comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of lemmas about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. We present the full model of such a system as well as a description of our prototype implementation showing the efficacy of such a system: for example, the prototype is able to obtain correct answers in cases where computer algebra systems [CAS] do not. We extend upon Fateman's webbased table by including parametric limits of integration and queries w...
Formal Proof of a Wave Equation Resolution Scheme: the Method Error ⋆
"... Abstract. Popular finite difference numerical schemes for the resolution of the onedimensional acoustic wave equation are wellknown to be convergent. We present a comprehensive formalization of the simplest scheme and formally prove its convergence in Coq. The main difficulties lie in the proper d ..."
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Cited by 4 (2 self)
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Abstract. Popular finite difference numerical schemes for the resolution of the onedimensional acoustic wave equation are wellknown to be convergent. We present a comprehensive formalization of the simplest scheme and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical penandpaper proofs. To our knowledge, this is the first time this kind of mathematical proof is machinechecked. Key words: partial differential equation, acoustic wave equation, numerical scheme, Coq formal proofs 1
VSDITLU: a verifiable symbolic definite integral table lookup
, 1999
"... We present a verifiable symbolic de nite integral table lookup: a system which matches a query, comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of facts about the reals in the theorem prover PVS to aid in the tra ..."
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Cited by 2 (1 self)
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We present a verifiable symbolic de nite integral table lookup: a system which matches a query, comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of facts about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. Our system is able to obtain correct answers in cases where standard techniques implemented in computer algebra systems fail. We present the full model of such a system as well as a description of our prototype implementation showing the efficacy of such a system: for example, the prototype is able to obtain correct answers in cases where computer algebra systems [CAS] do not. We extend upon Fateman's webbased table by including parametric limits of integration and queries with side conditions.