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**1 - 3**of**3**### This deliverable describes the scientific contributions of the third year for WP2

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### Short Description: Automata with substitutions, Fusion and Bindings, Models for Open π-calculus and distinctions, Models of Spatial Logics. Symbolic Verification of Security Protocols.

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### On the Computational Limits of Infinite Satisfaction ABSTRACT

"... We study the computational limits of Constraint Satisfaction Problems (CSP’s) allowing infinitely, or unboundedly, many indexed variables as in, e.g., xi> xi+2 for each i = 1, 2,.... We refer to these CSP’s as Infinite CSP’s (ICSP’s). These problems arise in contexts in which the number of variab ..."

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We study the computational limits of Constraint Satisfaction Problems (CSP’s) allowing infinitely, or unboundedly, many indexed variables as in, e.g., xi> xi+2 for each i = 1, 2,.... We refer to these CSP’s as Infinite CSP’s (ICSP’s). These problems arise in contexts in which the number of variables is unknown a priori as well as in optimization problems wrt the number of variables satisfying a given finite set of constraints. In particular, we investigate the decidability of the satisfiability problem for ICSP’s wrt (a) the first-order theory specifying the indices of variables and (b) the dimension of the indices. We first show that (1) if the indices are one-dimensional and specified in the theory of the natural numbers with linear order (the theory of (N, 0, succ, <)) then the satisfiability problem is decidable. We then prove that, in contrast to (1), (2) if we move to the twodimensional case then the satisfiability problem is undecidable for indices specified in (N, 0, succ, <) and even in (N, 0, succ). Finally, we show that, in contrast to (1) and (2), already in the onedimensional case (3) if we also allow addition, we get undecidability. I.e., if the one-dimensional indices are specified in Presburger arithmetic (i.e., the theory of (N, 0, succ, <, +)) then satisfiability is undecidable.