The Octahedron Abstract Domain

Clarisó, Robert; Cortadella, Jordi

doi:10.1007/978-3-540-27864-1_23

Cited by 57 publications

(62 citation statements)

References 25 publications

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“…For example, in the flip-flop case (see Sect. 4), the constraint, say Z, found in [7] is uncomparable with our constraint K 0 . We can run Imitator once more with a reference valuation π 1 ∈ Z \ K 0 .…”

Section: Final Remarksmentioning

confidence: 75%

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IMITATOR: A Tool for Synthesizing Constraints on Timing Bounds of Timed Automata

André

2009

Theoretical Aspects of Computing - ICTAC 2009

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Abstract. We present here Imitator, a tool for synthesizing constraints on timing bounds (seen as parameters) in the framework of timed automata. Unlike classical synthesis methods, we take advantage of a given reference valuation of the parameters for which the system is known to behave properly. Our aim is to generate a constraint such that, under any valuation satisfying this constraint, the system is guaranteed to behave, in terms of alternating sequences of locations and actions, as under the reference valuation. This is useful for safely relaxing some values of the reference valuation, and optimizing timing bounds of the system. We have successfully applied our tool to various examples of asynchronous circuits and protocols. ContextTimed automata [1] are finite control automata equipped with clocks, which are real-valued variables which increase uniformly. This model is useful for reasoning about real-time systems, because one can specify quantitatively the interval of time during which the transitions can occur, using timing bounds. However, the behavior of a system is very sensitive to the values of these bounds, and it is rather difficult to find their correct values. It is therefore interesting to reason parametrically, by considering that these bounds are unknown constants, or parameters, and try to synthesize a constraint (i.e., a conjunction of linear inequalities) on these parameters which will guarantee a correct behavior of the system. Such automata are called parametric timed automata (PTA) [2,11].The synthesis of constraints for PTA has been mainly done by supposing given a set of "bad states" (see, e.g., [8,9]). The goal is to find a set of parameters for which the considered timed automaton does not reach any of these bad states. We call such a method a bad-state oriented method. By contrast, we present in this paper a tool based on a good-state oriented method. Principle of ImitatorThe tool Imitator (Inverse Method for Inferring Time AbstracT behaviOR) implements the algorithm InverseMethod , described in [4]. We assume given a

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Section: Final Remarksmentioning

confidence: 75%

“…In this case, the constraint generated by our method may be the same as the constraint from the literature, but not necessarily : for example, in the case of the flip-flop circuit, K 0 is uncomparable with the original constraint of [7] (see [5] for details).…”

Section: Methodsmentioning

confidence: 99%

IMITATOR: A Tool for Synthesizing Constraints on Timing Bounds of Timed Automata

André

2009

Theoretical Aspects of Computing - ICTAC 2009

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“…Boogie was successfully able to prove that I was indeed a loop invariant and was able to show that the assertion holds. As another example, on parametrizing with the Octahedron [16] abstract domain, our technique discovers the simpler conjunctive loop invariant: i + y = x + j in 0.09s.…”

Section: Finding Invariants For the Examplementioning

confidence: 99%

Verification as Learning Geometric Concepts

et al. 2013

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Abstract. We formalize the problem of program verification as a learning problem, showing that invariants in program verification can be regarded as geometric concepts in machine learning. Safety properties define bad states: states a program should not reach. Program verification explains why a program's set of reachable states is disjoint from the set of bad states. In Hoare Logic, these explanations are predicates that form inductive assertions. Using samples for reachable and bad states and by applying well known machine learning algorithms for classification, we are able to generate inductive assertions. By relaxing the search for an exact proof to classifiers, we obtain complexity theoretic improvements. Further, we extend the learning algorithm to obtain a sound procedure that can generate proofs containing invariants that are arbitrary boolean combinations of polynomial inequalities. We have evaluated our approach on a number of challenging benchmarks and the results are promising.

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“…TVPI polyhedra form a so-called weakly relational domain and thereby constitute a proper subclass of general polyhedra. Other sub-classes include Difference Bounds Matrices (DBMs for short) [5,61,46], the Octagon domain [47] and the Octahedron domain [17]. The abstract domain of DBMs represents inequalities of the form x i −x j ≤ c ij , x i , x j ∈ X by storing c ij in an n×n matrix such that the entry at position i, j is c ij .…”

Section: Related Workmentioning

confidence: 99%

“…Thus k − 1 more steps are necessary to obtain a Z-polyhedron [31] where linear independence is detected between sets of constraints and exploited by applying meet and join on these smaller sets. The Octagon domain was generalised into the Octahedron domain [17], allowing more than two variables with zero or unary coefficients whilst maintaining a hull operation that is polynomial in the number of variables.…”

Section: Related Workmentioning

confidence: 99%

The two variable per inequality abstract domain

Simon

King

2010

Higher-Order Symb Comput

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Abstract. This article presents the Two Variable Per Inequality abstract domain (TVPI domain for short). This so-called weakly-relational domain is able to express systems of linear inequalities where each inequality has at most two variables. The domain represents a sweet-point in the performance-cost tradeoff between the faster Octagon domain and the more expressive domain of general convex polyhedra. In particular, we detail techniques to closely approximate integral TVPI systems, thereby finessing the problem of excessively growing coefficients, yielding -to our knowledge -the only relational domain that combines linear relations with arbitrary coefficients and strongly polynomial performance. Keywords: polyhedral analysis, integer programming, abstract interpretationStatic analysis methods have evolved from inferring prerequisite invariants for compiler optimisations to tools in their own right that are able to prove the absence of run-time errors of software. The abstract interpretation framework [21] provides a way of constructing and justifying program analyses. The key idea is to simulate each operation in a program with an abstract analog that operates on a description of the program state, rather than the state itself. The descriptions are chosen to trace a property of interest, for instance, an interval that encloses all possible values of a variable would be of value when reasoning about array bounds. These descriptions constitute what is known as an abstract domain. An abstract domain is usually presented as a lattice D, , , and the analysis itself is formulated as a set of recursive equations over D. The recursive equations are solved iteratively, until a fixpoint is reached which is detected using the ordering predicate . Each equation expresses how a program statement transforms the state at one program point to the state at another. Equations may be recursive because of loops in the program. The meet and join † This paper is a revised extract of the first author's PhD thesis [65], which, in turn, extends [72].

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The Octahedron Abstract Domain

Cited by 57 publications

References 25 publications

IMITATOR: A Tool for Synthesizing Constraints on Timing Bounds of Timed Automata

IMITATOR: A Tool for Synthesizing Constraints on Timing Bounds of Timed Automata

Verification as Learning Geometric Concepts

The two variable per inequality abstract domain

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