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93
Automatically Generating Loop Invariants Using Quantifier Elimination
 APPLICATIONS OF COMPUTER ALGEBRA (ACA2004)
, 2004
"... An approach for automatically generating loop invariants using quantifierelimination is proposed. An invariant of a loop is hypothesized as a parameterized formula. Parameters in the invariant are discovered by generating constraints on the parameters by ensuring that the formula is indeed preser ..."
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An approach for automatically generating loop invariants using quantifierelimination is proposed. An invariant of a loop is hypothesized as a parameterized formula. Parameters in the invariant are discovered by generating constraints on the parameters by ensuring that the formula is indeed preserved by the execution path corresponding to every basic cycle of the loop. The parameterized formula can be successively refined by considering execution paths one by one; heuristics can be developed for determining the order in which the paths are considered. Initialization of program variables as well as the precondition and postcondition of the loop, if available, can also be used to further refine the hypothesized invariant. Constraints on parameters generated in this way are solved for possible values of parameters. If no solution is possible, this means that an invariant of the hypothesized form does not exist for the loop. Otherwise, if the parametric constraints are solvable, then under certain conditions on methods for generating these constraints, the strongest possible invariant of the hypothesized form can be generated from most general solutions of the parametric constraints. The approach is illustrated using the firstorder theory of polynomial equations as well as Presburger arithmetic.
Efficient Algorithms and Bounds for WuRitt Characteristic Sets
 In Proceedings of MEGA 90: Meeting on Effective Methods in Algebraic Geometry
, 1990
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The BoyerMoore Theorem Prover and Its Interactive Enhancement
, 1995
"... . The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. F ..."
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Cited by 34 (0 self)
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. The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in which theorems are proved. Then we briefly describe the two mechanized theorem proving systems. Next, we present a simple but illustrative example in some detail in order to give an impression of how these systems may be used successfully. Finally, we give extremely short descriptions of a large number of applications of these systems, in order to give an idea of the breadth of their uses. This paper is intended as an informal introduction to systems that have been described in detail and similarly summarized in many other books and papers; no new results are reported here. Our intention here is merely to present Nqthm to a new audience. This research was supported in part by ONR Contract N...
A Theorem Prover for a Computational Logic
, 1990
"... We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of line ..."
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We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.
Computational Real Algebraic Geometry
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY, CHAPTER 29
, 1997
"... Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad ..."
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Cited by 26 (5 self)
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Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer aided design, geometric theorem proving, etc. The algorithmic problems that arise in this context are formulated as decision problems for the firstorder theory of reals and the related problems of quantifier elimination (Section 1). The associated geometric structures are then examined via an exploration of the semialgebraic sets (Section 2). Algorithmic problems for semialgebraic sets are considered next. In particular, there is a discussion of real algebraic numbers and their representation which relies on such classical theorems as Stu
Implementing NonLinear Constraints With Cooperative Solvers
, 1995
"... We investigate the use of cooperation between solvers in the scheme of constraint logic programming languages over the domain of nonlinear polynomial constraints. Instead of using a general and often inefficient decision procedure we propose a new approach for handling these constraints by cooperat ..."
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Cited by 24 (12 self)
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We investigate the use of cooperation between solvers in the scheme of constraint logic programming languages over the domain of nonlinear polynomial constraints. Instead of using a general and often inefficient decision procedure we propose a new approach for handling these constraints by cooperating specialised solvers. Our approach requires the design of a client/server architecture to enable communication between the various components. The main modules are a linear solver, a nonlinear solver, a constraint manager, a communication protocol component and an answer processor module. This work is motivated by the need for a declarative system for robot motion planning and geometric problem solving. We have implemented a prototype called CoSAc (Constraint System Architecture) to validate our approach using cooperating solvers for nonlinear constraints over the real numbers. Our language is illustrated by an example that also shows the advantages of cooperation.
Geometric constraint satisfaction using optimization methods
, 1999
"... The numerical approach to solving geometric constraint problems is indispensable for building a practical CAD system. The most commonlyused numerical method is the Newton–Raphson method. It is fast, but has the instability problem: the method requires good initial values. To overcome this problem, ..."
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Cited by 21 (6 self)
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The numerical approach to solving geometric constraint problems is indispensable for building a practical CAD system. The most commonlyused numerical method is the Newton–Raphson method. It is fast, but has the instability problem: the method requires good initial values. To overcome this problem, recently the homotopy method has been proposed and experimented with. According to the report, the homotopy method generally works much better in terms of stability. In this paper we use the numerical optimization method to deal with the geometric constraint solving problem. The experimental results based on our implementation of the method show that this method is also much less sensitive to the initial value. Further, a distinctive advantage of the method is that under and overconstrained problems can be handled naturally and efficiently. We also give many instructive examples to illustrate the above advantages.