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Par(p) , and a finiteness constraint (constraint for short) of p is an expression of the form x y, where x, y ⊆ pos(p). 38 S. ) Gil, and E. Zarivach A set of fact F satisfies constraint x y if the search for p-facts in F with some fixed assignment to positions x, yields only a finite variety of combinations of values for positions y. More formally, Definition 3. Let σ = x y be a constraint on predicate p and F be a set of facts. Then, F |= σ (read F satisfies σ) if the set {b[y] | b ∈ F and pred(b) = p and b[x] = a[x]} is finite for every p-fact a ∈ F.

But, since weak safety EXP-time complete [13], it is no wonder that our algorithm for computing CΠ is exponential. 7 Deciding Termination If a program is weakly safe, then any finite number of rule applications contributes a finite number of facts to the program semantics. However, the semantics of weakly safe program may include an unbounded number of facts, since in general, the number of rule applications is unbounded. Indeed, Sagiv and Vardi [13] showed that the independent problem of termination is undecidable, without being able to produce an algorithm for determining termination in the case that weak safety is known, or conversely, to prove that no such algorithm exists.

This can be seen by observing Algorithm 2: every recursive call to L-Match is a return-statement, so the algorithm does not change when the recursion stack is not used at all. 24 M. G¨ otz, C. Koch, and W. t. the depth-first left-to-right ordering. We illustrate the remainder of q in Q in Figure 1. Given a data node d and a query node q, the algorithm first tries to match the remainder of q in Q consistently with what has already been matched in D (lines 2–11). If this fails, it either returns false (line 13), or enters a backtracking procedure (lines 15–21).

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