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We show that satisfiability for CTL * with equality-, order-, and modulo-constraints over Z is decidable. Previously, decidability was only known for certain fragments of CTL * , e.g., the existential and positive fragments and EF. IntroductionTemporal logics like LTL, CTL or CTL * are nowadays standard languages for specifying system properties in model-checking. They are interpreted over node labeled graphs (Kripke structures), where the node labels (also called atomic propositions) represent abstract properties of a system. Clearly, such an abstracted system state does in general not contain all the information of the original system state. Consider for instance a program that manipulates two integer variables x and y. A useful abstraction might be to introduce atomic propositions v −2 32 , . . . , v 2 32 for v ∈ {x, y}, where the meaning of v k for −2 32 < k < 2 32 is that the variable v ∈ {x, y} currently holds the value k, and v −2 32 (resp., v 2 32 ) means that the current value of v is at most −2 32 (resp., at least 2 32 ). It is evident that such an abstraction might lead to incorrect results in model-checking.To overcome these problems, extensions of temporal logics with constraints have been studied. Let us explain the idea in the context of LTL. For a fixed relational structure A (typical examples for A are number domains like the integers or rationals extended with certain relations) one adds atomic formulas of the form r(X i1 x 1 , . . . , X i k x k ) (so called constraints) to standard LTL. Here, r is (a name of) one of the relations of the structure A, i 1 , . . . , i k ≥ 0, and x 1 , . . . , x k are variables that range over the universe of A. An LTL-formula containing such constraints is interpreted over (infinite) paths of a standard Kripke structure, where in addition every node (state) associates with each of the variables x 1 , . . . , x k an element of A (one can think of A-registers attached to the system states). A constraint r(X i1 x 1 , . . . , X i k x k ) holds in a path s 0 → s 1 → s 2 → · · · if the tuple (a 1 , . . . , a k ), where a j is the value of variable x j at state s ij , belongs to the A-relation r. In this way, the values of variables at different system states can be compared. In our example from the first paragraph, one might choose for A the structure (Z, <, =, (= a ) a∈Z ), where = a is the unary predicate that only holds for a. This structure has infinitely many predicates, which is not a problem; our main result will actually talk about an expansion of (Z, <, =, (= a ) a∈Z ). Then, one might for instance write down a formula (<(x, X 1 y))U(= 100 (y)) which holds on a path if and only if there is a point of time where variable y holds the value 100 and for all previous points of time t, the value of x at time t is strictly smaller than the value of y at time t + 1.

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Reasoning for Description logics with concrete domains and w.r.t. general TBoxes easily becomes undecidable. For particular, restricted concrete domains decidablity can be regained. We introduce a novel way to integrate a concrete domain D into the well-known description logic ALC, we call the resulting logic ALCP(D). We then identify sufficient conditions on D that guarantee decidability of the satisfiability problem, even in the presence of general TBoxes. In particular, we show decidability of ALCP(D) for several domains over the integers, for which decidability was open. More generally, this result holds for all negation-closed concrete domains with the EHD-property, which stands for the existence of a homomorphism is definable. Such technique has recently been used to show decidability of CTL with local constraints over the integers.

Recently, we have shown that satisfiability for ECTL * with constraints over Z is decidable using a new technique. This approach reduces the satisfiability problem of ECTL * with constraints over some structure A (or class of structures) to the problem whether A has a certain model theoretic property that we called EHD (for "existence of homomorphisms is decidable"). Here we apply this approach to concrete domains that are tree-like and obtain several results. We show that satisfiability of ECTL * with constraints is decidable over (i) semi-linear orders (i.e., tree-like structures where branches form arbitrary linear orders), (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching trees of height h for each fixed h ∈ N. We prove that all these classes of structures have the property EHD. In contrast, we introduce Ehrenfeucht-Fraïssé-games for WMSO+B (weak MSO with the bounding quantifier) and use them to show that the infinite (order) tree does not have property EHD. As a consequence, a different approach has to be taken in order to settle the question whether satisfiability of ECTL * (or even LTL) with constraints over the infinite (order) tree is decidable.