Abstract
Over-approximating the descendants (successors) of an initial set of terms under a rewrite system is used in reachability analysis. The success of such methods depends on the quality of the approximation. Regular approximations (i.e. those using finite tree automata) have been successfully applied to protocol verification and Java program analysis. In [2, 10], non-regular approximations have been shown more precise than regular ones. In [3] (fixed version of [2]), we have shown that sound over-approximations using synchronized tree languages can be computed for left-and-right-linear term rewriting systems (TRS). In this paper, we present two new contributions extending [3]. Firstly, we show how to compute at least all innermost descendants for any left-linear TRS. Secondly, a procedure is introduced for computing over-approximations independently of the applied rewrite strategy for any left-linear TRS.
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- 1.
This approximation is often exact, but not always. This is due to the fact that a tree language expressed by a copying CS-program cannot always be expressed by a non-copying one.
- 2.
I.e. clause heads are not linear.
- 3.
In former papers, synchronized tree-tuple languages were defined thanks to sorts of grammars, called constraint systems. Thus “CS” stands for Constraint System.
- 4.
In other words, the overlap of l on the clause head \(P(t_1,\ldots ,t_n)\) is done at a non-variable position.
- 5.
The number of arguments.
- 6.
From a theoretical point of view, left-linearity is sufficient when every critical pair is convergent. However, to make every critical pair convergent by completion, full linearity is necessary (see Theorem 3).
- 7.
NC stands for Non-Copying. SNC stands for Strongly Non-Copying.
- 8.
If the loop while is run several times, predicate symbols of the form \((Q_i^{x})^{y}\) may appear.
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Boichut, Y., Pelletier, V., Réty, P. (2016). Synchronized Tree Languages for Reachability in Non-right-linear Term Rewrite Systems. In: Lucanu, D. (eds) Rewriting Logic and Its Applications. WRLA 2016. Lecture Notes in Computer Science(), vol 9942. Springer, Cham. https://doi.org/10.1007/978-3-319-44802-2_4
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