Abstract
We investigate the problem of deciding first-order theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automata-based solution for the case where the different equational axiom systems are linear and variable-disjoint (this includes the case where all axioms are ground), and where the logic does not permit to express tree relations x = f(y,z). We show that the problem is undecidable when these restrictions are relaxed. As motivation and application, we show how to translate the model-checking problem of \(\mathcal{A}\pi \mathcal{L}\), a spatial equational logic for the applied pi-calculus, to the validity of first-order formulas in term algebras with multiple congruence relations.
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Jacquemard, F., Lozes, É., Treinen, R., Villard, J. (2012). Multiple Congruence Relations, First-Order Theories on Terms, and the Frames of the Applied Pi-Calculus. In: Mödersheim, S., Palamidessi, C. (eds) Theory of Security and Applications. TOSCA 2011. Lecture Notes in Computer Science, vol 6993. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27375-9_10
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