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    Modular arithmetic Decision Procedure

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    All integer data types in programs (such as int, short, byte) have an underlying finite representation in hardware. This finiteness can result in subtle integer-overflow errors that are hard to reason about both for humans and analysis tools alike. As a first step towards finding such errors automatically, we will describe two modular arithmetic decision procedures for reasoning about bounded integers. We show how to deal with modular arithmetic operations and inequalities for both linear and non-linear problems. Both procedures are suitable for integration with Nelson-Oppen framework [1, 2, 3]. The linear solver is composed of Müller-Seidl algorithm [4] and an arbitrary integer solver for solving preprocessed congruences and inequalities. For the non-linear problems we use Newton’s p-adic iteration algorithm to progressively reason about the satisfiability of the input constraints modulo 2k, for increasing k. We use a SAT solver only for the base case when k = 1. According to our knowledge, this is the first Nelson-Oppen decision procedure capable of reasoning about multiplication over bounded integers without converting the entire problem to a SAT instance.

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    Description

    Title : Modular arithmetic Decision Procedure
    Author(s) : Domagoj Babic, Madanlal Musuvathi
    Abstract : All integer data types in programs (such as int, short, byte) have an underlying finite representation in hardware. This finiteness can result in subtle integer-overflow errors that are hard to reason about both for humans and analysis tools alike. As a first step towards finding such errors automatically, we will describe two modular arithmetic decision procedures for reasoning about bounded integers. We show how to deal with modular arithmetic operations and inequalities for both linear and non-linear problems. Both procedures are suitable for integration with Nelson-Oppen framework [1, 2, 3]. The linear solver is composed of Müller-Seidl algorithm [4] and an arbitrary integer solver for solving preprocessed congruences and inequalities. For the non-linear problems we use Newton’s p-adic iteration algorithm to progressively reason about the satisfiability of the input constraints modulo 2k, for increasing k. We use a SAT solver only for the base case when k = 1. According to our knowledge, this is the first Nelson-Oppen decision procedure capable of reasoning about multiplication over bounded integers without converting the entire problem to a SAT instance.
    Keywords : modular arithmetic, bounded integers, bit-vector arithmetic, decision procedure, linear arithmetic, nonlinear arithmetic

    Subject : unspecified
    Area : Computer Science
    Language : English
    Year : 2005

    Affiliations Microsoft Research
    Institution : Microsoft Researcbh
    Issue : TR-2005-114
    Url : http://www.domagoj-babic.com/uploads/Pubs/MSRTR05/techrep05.pdf

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