Results 1  10
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52
Linear ranking with reachability
 In CAV
, 2005
"... Abstract. We present a complete method for synthesizing lexicographic linear ranking functions supported by inductive linear invariants for loops with linear guards and transitions. Proving termination via linear ranking functions often requires invariants; yet invariant generation is expensive. Thu ..."
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Cited by 50 (9 self)
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Abstract. We present a complete method for synthesizing lexicographic linear ranking functions supported by inductive linear invariants for loops with linear guards and transitions. Proving termination via linear ranking functions often requires invariants; yet invariant generation is expensive. Thus, we describe a technique that discovers just the invariants necessary for proving termination. Finally, we describe an implementation of the method and provide extensive experimental evidence of its effectiveness for proving termination of C loops. 1 Introduction Guaranteed termination of program loops is necessary in many settings, suchas embedded systems and safety critical software. Additionally, proving general temporal properties of infinite state programs requires termination proofs, forwhich automatic methods are welcome [19, 11, 15]. We propose a termination analysis of linear loops based on the synthesis of lexicographic linear rankingfunctions supported by linear invariants.
Nonlinear Loop Invariant Generation using Gröbner Bases
, 2004
"... We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generati ..."
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Cited by 40 (4 self)
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We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generation has been focussed on the construction of linear invariants for linear programs. Consequently, there has been little progress toward nonlinear invariant generation. In this paper, we demonstrate a technique that encodes the conditions for a given template assertion being an invariant into a set of constraints, such that all the solutions to these constraints correspond to nonlinear (algebraic) loop invariants of the program. We discuss some tradeoffs between the completeness of the technique and the tractability of the constraintsolving problem generated. The application of the technique is demonstrated on a few examples.
DySy: Dynamic symbolic execution for invariant inference
 In Proc. 30th ACM/IEEE International Conference on Software Engineering (ICSE). ACM
, 2007
"... Dynamically discovering likely program invariants from concrete test executions has emerged as a highly promising software engineering technique. Dynamic invariant inference has the advantage of succinctly summarizing both “expected” program inputs and the subset of program behaviors that is normal ..."
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Cited by 39 (6 self)
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Dynamically discovering likely program invariants from concrete test executions has emerged as a highly promising software engineering technique. Dynamic invariant inference has the advantage of succinctly summarizing both “expected” program inputs and the subset of program behaviors that is normal under those inputs. In this paper, we introduce a technique that can drastically increase the relevance of inferred invariants, or reduce the size of the test suite required to obtain good invariants. Instead of falsifying invariants produced by preset patterns, we determine likely program invariants by combining the concrete execution of actual test cases with a simultaneous symbolic execution of conditions over program variables that the concrete tests satisfy during their execution. In this way, we obtain the benefits of dynamic inference tools like Daikon: the inferred invariants correspond to the observed program behaviors. At the same time, however, our inferred invariants are much more suited to the program at hand than Daikon’s hardcoded invariant patterns. The symbolic invariants are literally derived from the program text itself, with appropriate value substitutions as dictated by symbolic execution. We implemented our technique in the DySy tool, which utilizes a powerful symbolic execution and simplification engine. The results confirm the benefits of our approach. In Daikon’s prime example benchmark, we infer the majority of the interesting Daikon invariants, while eliminating invariants that a human user is likely to consider irrelevant.
Constructing Invariants for Hybrid Systems
 in Hybrid Systems: Computation and Control, LNCS 2993
, 2004
"... Abstract. An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideal ..."
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Cited by 37 (7 self)
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Abstract. An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideals over polynomial rings. We extend our previous work on the generation of algebraic invariants for discrete transition systems in order to generate algebraic invariants for hybrid systems. In doing so, we present a new technique to handle consecution across continuous differential equations. The techniques we present allow a tradeoff between the complexity of the invariant generation process and the strength of the resulting invariants. 1
Program analysis as constraint solving
 In PLDI
, 2008
"... A constraintbased approach to invariant generation in programs translates a program into constraints that are solved using offtheshelf constraint solvers to yield desired program invariants. In this paper we show how the constraintbased approach can be used to model a wide spectrum of program ana ..."
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Cited by 33 (11 self)
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A constraintbased approach to invariant generation in programs translates a program into constraints that are solved using offtheshelf constraint solvers to yield desired program invariants. In this paper we show how the constraintbased approach can be used to model a wide spectrum of program analyses in an expressive domain containing disjunctions and conjunctions of linear inequalities. In particular, we show how to model the problem of contextsensitive interprocedural program verification. We also present the first constraintbased approach to weakest precondition and strongest postcondition inference. The constraints we generate are boolean combinations of quadratic inequalities over integer variables. We reduce these constraints to SAT formulae using bitvector modeling and use offtheshelf SAT solvers to solve them. Furthermore, we present interesting applications of the above analyses, namely bounds analysis and generation of mostgeneral counterexamples for both safety and termination properties. We also present encouraging preliminary experimental results demonstrating the feasibility of our technique on a variety of challenging examples.
Verification of Java Programs using Symbolic Execution and Invariant Generation
, 2004
"... Software verification is recognized as an important and difficult problem. We present a novel framework, based on symbolic execution, for the automated verification of software. The framework uses annotations in the form of method specifications and loop invariants. We present a novel iterative... ..."
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Cited by 31 (4 self)
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Software verification is recognized as an important and difficult problem. We present a novel framework, based on symbolic execution, for the automated verification of software. The framework uses annotations in the form of method specifications and loop invariants. We present a novel iterative...
Path invariants
 In PLDI
, 2007
"... The success of software verification depends on the ability to find a suitable abstraction of a program automatically. We propose a method for automated abstraction refinement which overcomes some limitations of current predicate discovery schemes. In current schemes, the cause of a false alarm is i ..."
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Cited by 30 (3 self)
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The success of software verification depends on the ability to find a suitable abstraction of a program automatically. We propose a method for automated abstraction refinement which overcomes some limitations of current predicate discovery schemes. In current schemes, the cause of a false alarm is identified as an infeasible error path, and the abstraction is refined in order to remove that path. By contrast, we view the cause of a false alarm —the spurious counterexample — as a fullfledged program, namely, a fragment of the original program whose controlflow graph may contain loops and represent unbounded computations. There are two advantages to using such path programs as counterexamples for abstraction refinement. First, we can bring the whole machinery of program analysis to bear on path programs, which are typically small compared to the original program. Specifically, we use constraintbased invariant generation to automatically infer invariants of path programs —socalled path invariants. Second, we use path invariants for abstraction refinement in order to remove not one infeasibility at a time, but at once all (possibly infinitely many) infeasible error computations that are represented by a path program. Unlike previous predicate discovery schemes, our method handles loops without unrolling them; it infers abstractions that involve universal quantification and naturally incorporates disjunctive reasoning.
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.
Constraintbased linearrelations analysis
 In Proc. SAS, LNCS 3148
, 2004
"... 1 Introduction Linearrelations analysis discovers linear relationships among the variables of aprogram that hold in all the reachable program states. Such relationships are called linear invariants. Invariants are useful in the verification of both safetyand liveness properties. Many existing techn ..."
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Cited by 29 (2 self)
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1 Introduction Linearrelations analysis discovers linear relationships among the variables of aprogram that hold in all the reachable program states. Such relationships are called linear invariants. Invariants are useful in the verification of both safetyand liveness properties. Many existing techniques rely on the presence of these invariants to prove properties of interest. Some types of analysis, e.g., variablebounds analysis, can be viewed as specializations of linearrelations analysis. Traditionally, this analysis is framed as an abstract interpretation in the domainof polyhedra [6, 7]. The analysis is carried out using a propagationbased technique, wherein increasingly accurate polyhedral iterates, converging towards thefinal result, are computed. This convergence is ensured through the use of widening, or extrapolation, operators. Such techniques are popular in the domains ofdiscrete and hybrid programs, motivating tools like
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.