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Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations
 In International Symposium on Symbolic and Algebraic Computation 2004 (ISSAC04
, 2004
"... This paper presents the algebraic foundation for an approach for generating polynomial loop invariants in imperative programs. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Using this connection, a procedure for finding loop invaria ..."
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Cited by 29 (4 self)
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This paper presents the algebraic foundation for an approach for generating polynomial loop invariants in imperative programs. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Using this connection, a procedure for finding loop invariants is given in terms of operations on ideals, for which Gröbner basis constructions can be employed. Most importantly, it is proved that if the assignment statements in a loop are solvable (in particular, affine) mappings with positive eigenvalues, then the procedure terminates in at most 2m + 1 iterations, where m is the number of variables in the loop. The proof is done by showing that the irreducible subvarieties of the variety associated with the polynomial ideal approximating the invariant polynomial ideal of the loop either stay the same or increase their dimension in every iteration. This yields a correct and complete algorithm for inferring conjunctions of polynomial equations as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover nontrivial invariants for several examples to illustrate the power of the techniques.
Automatically generating loop invariants using quantifier elimination
 In Deduction and Applications
, 2005
"... Abstract. 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 indee ..."
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Cited by 27 (0 self)
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Abstract. 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. 1.
Combining Logic and Algebraic Techniques for Program Verification in Theorema
 SECOND INTERNATIONAL SYMPOSIUM ON LEVERAGING APPLICATIONS OF FORMAL METHODS, VERIFICATION AND VALIDATION
, 2007
"... We study and implement concrete methods for the verification of both imperative as well as functional programs in the frame of the Theorema system. The distinctive features of our approach consist in the automatic generation of loop invariants (by using combinatorial and algebraic techniques), and ..."
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Cited by 8 (8 self)
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We study and implement concrete methods for the verification of both imperative as well as functional programs in the frame of the Theorema system. The distinctive features of our approach consist in the automatic generation of loop invariants (by using combinatorial and algebraic techniques), and the generation of verification conditions as first–order logical formulae which do not refer to a specific model of computation.
Automation for Exception Freedom Proofs
, 2003
"... Runtime errors are typically seen as unacceptable within safety and security critical software. The SPARK approach to the development of high integrity software addresses the problem of runtime errors through the use of formal verification. Proofs are constructed to show that each runtime check w ..."
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Cited by 6 (2 self)
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Runtime errors are typically seen as unacceptable within safety and security critical software. The SPARK approach to the development of high integrity software addresses the problem of runtime errors through the use of formal verification. Proofs are constructed to show that each runtime check will never raise an error, thus proving freedom from runtime exceptions. Here we build upon the success of the SPARK approach by increasing the level of automation that can be achieved in proving freedom from exceptions. Our approach is based upon proof planning and a form of abstract interpretation.
An Integrated Approach to High . . .
"... Using automated reasoning techniques, we tackle the niche activity of proving that a program is free from runtime exceptions. Such a property is particularly valuable in high integrity software, e.g. safety or security critical applications. The context for our work is the SPARK Approach for the d ..."
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Using automated reasoning techniques, we tackle the niche activity of proving that a program is free from runtime exceptions. Such a property is particularly valuable in high integrity software, e.g. safety or security critical applications. The context for our work is the SPARK Approach for the development of high integrity software. The SPARK Approach provides a significant degree of automation in proving exception freedom. However, where this automation fails, the programmer is burdened with the task of interactively constructing a proof and possibly also having to supply auxiliary program annotations. We minimise this burden by increasing the automation, via an integration of proof planning and a program analysis oracle. We advocate a “cooperative” integration, where prooffailure analysis directly constrains the search for auxiliary program annotations. The approach has been successfully tested on industrial data.
Combining Logic and Algebraic Techniques for Program Verification in Theorema
"... Abstract — We study and implement concrete methods for the verification of both imperative as well as functional programs in the frame of the Theorema system. The distinctive features of our approach consist in the automatic generation of loop invariants (by using combinatorial and algebraic techniq ..."
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Abstract — We study and implement concrete methods for the verification of both imperative as well as functional programs in the frame of the Theorema system. The distinctive features of our approach consist in the automatic generation of loop invariants (by using combinatorial and algebraic techniques), and the generation of verification conditions as first–order logical formulae which do not refer to a specific model of computation. I.
Applications of Computer Algebra (ACA2004) Automatically Generating Loop Invariants Using Quantifier Elimination ∗† –Preliminary Report–
"... Abstract. 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 indee ..."
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Abstract. 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. 1.
An Integrated Approach to Program Reasoning
"... Finding tractable methods for program reasoning remains a major research challenge. Here we address this challenge using an integrated approach to tackle a niche program reasoning application. The application ..."
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Finding tractable methods for program reasoning remains a major research challenge. Here we address this challenge using an integrated approach to tackle a niche program reasoning application. The application