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A TemporalInteractivist Perspective on the Dynamics of Mental States
 COGNITIVE SYSTEMS RESEARCH JOURNAL
, 2003
"... This paper addresses the dynamics of mental states in relation to the dynamics of the interaction with the external world. It contributes a formalised temporalinteractivist approach to these dynamics based on temporal traces for semantics and a temporal trace language to provide an expressive means ..."
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Cited by 39 (27 self)
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This paper addresses the dynamics of mental states in relation to the dynamics of the interaction with the external world. It contributes a formalised temporalinteractivist approach to these dynamics based on temporal traces for semantics and a temporal trace language to provide an expressive means to formulate dynamic properties. The approach provides both a foundation for the dynamical and interactivist perspective on cognitive phenomena, and a supporting software environment for practical application.
Continuity and differentiability in ACL2
 ComputerAided Reasoning: ACL2 Case Studies, chapter 18
, 2000
"... This case study shows how ACL2 can be used to reason about the real and complex numbers, using nonstandard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems ..."
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Cited by 14 (9 self)
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This case study shows how ACL2 can be used to reason about the real and complex numbers, using nonstandard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems of analysis, such as the intermediatevalue and meanvalue theorems.
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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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
Mechanical Verification of a Square Root Algorithm Using Taylor’s Theorem
 In Formal Methods in Computer Aided Design (FMCAD'02
, 2002
"... Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theore ..."
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Cited by 10 (4 self)
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Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of nonstandard analysis, as implemented in ACL2(r). Since Taylor series of a given order have less accuracy than Chebyshev series in general, we used hundreds of Taylor series generated by ACL2(r) to evaluate the error of a Chebyshev series. 1
Improving Real Analysis in Coq: a UserFriendly Approach to Integrals and Derivatives ⋆
"... Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, wh ..."
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Cited by 5 (2 self)
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Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, which may be used for this purpose. Unfortunately, its definitions of derivative and integral are unpractical as they are partial functions that demand a proof term. This proof term makes the handling of mathematical formulas cumbersome and does not conform to traditional analysis. Other proof assistants usually do not suffer from this issue; for instance, they may rely on Hilbert’s epsilon to get total operators. In this paper, we propose a way to define total operators for derivative and integral without having to extend Coq’s standard axiomatization of real numbers. We proved the compatibility of our definitions with the standard library’s in order to leverage existing results. We also greatly improved automation for real analysis proofs that use Coq standard definitions. We exercised our approach on lemmas involving iterated partial derivatives and differentiation under the integral sign, that were missing from the formal proof of a numerical program solving the wave equation. 1
Taylor's Formula with Remainder
 In Proceedings of the Third International Workshop of the ACL2 Theorem Prover and its Applications
, 2002
"... In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder. ..."
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In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder.
Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals
"... Abstract. Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this re ..."
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Abstract. Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thom’s Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples. 1 Overview and Related Work Decision methods for nonlinear real arithmetic are essential to the formal verification of cyberphysical systems and formalized mathematics. Classically, these
An ACL2 Tutorial
"... Abstract. We describe a tutorial that demonstrates the use of the ACL2 theorem prover. We have three goals: to enable a motivated reader to start on a path towards effective use of ACL2; to provide ideas for other interactive theorem prover projects; and to elicit feedback on how we might incorporat ..."
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Abstract. We describe a tutorial that demonstrates the use of the ACL2 theorem prover. We have three goals: to enable a motivated reader to start on a path towards effective use of ACL2; to provide ideas for other interactive theorem prover projects; and to elicit feedback on how we might incorporate features of other proof tools into ACL2. 1
Theory Extension in ACL2(r)
"... Abstract. ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper pres ..."
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Abstract. ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper presents the logical foundations of ACL2(r). These foundations are also used to justify significant enhancements to ACL2(r). 1.
Formally Verifying an Algorithm Based on Interval Arithmetic for Checking Transversality
 IN: FIFTH INTERNATIONAL WORKSHOP ON THE ACL2 THEOREM PROVER AND ITS APPLICATIONS
, 2004
"... In this paper we use ACL2 to formally verify the correctness of an algorithm used in the analysis of dynamical systems. The algorithm uses interval arithmetic to check that a given vector field is transverse (nontangential) to an edge (line segment). Instead of operating on numbers, interval opera ..."
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Cited by 3 (0 self)
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In this paper we use ACL2 to formally verify the correctness of an algorithm used in the analysis of dynamical systems. The algorithm uses interval arithmetic to check that a given vector field is transverse (nontangential) to an edge (line segment). Instead of operating on numbers, interval operations operate on intervals, and they are guaranteed to return an overapproximation of the actual answer, thereby allowing us to use floating point arithmetic in a safe way. In this paper we prove that if the algorithm identifies an edge as transverse, then it is in fact transverse, as long as the underlying interval arithmetic operations are correctly implemented.