Automated deduction with built-in theories: completeness results and constraint solving techniques

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The HOM problem is EXPTIME-complete

We define a new class of tree automata with constraints and prove decidability of the emptiness problem for this class in exponential time. As a consequence, we obtain several EXPTIME-completeness results for problems on images of regular tree languages under tree homomorphisms, like set inclusion, regularity (HOM problem), and finiteness of set difference. Our result also has implications in term rewriting, since the set of reducible terms of a term rewrite system can be described as the image of a tree homomorphism. In particular, we prove that inclusion of sets of normal forms of term rewrite systems can be decided in exponential time. Analogous consequences arise in the context of XML typechecking, since types are defined by tree automata and some type transformations are homomorphic.Peer ReviewedPostprint (published version

Automatic evaluation of context-free grammars (system description)

We implement an online judge for context-free grammars. Our system contains a list of problems describing formal languages, and asking for grammars generating them. A submitted proposal grammar receives a verdict of acceptance or rejection depending on whether the judge determines that it is equivalent to the reference solution grammar provided by the problem setter. Since equivalence of context-free grammars is an undecidable problem, we consider a maximum length l and only test equivalence of the generated languages up to words of length l. This length restriction is very often sufficient for the well-meant submissions. Since this restricted problem is still NP-complete, we design and implement methods based on hashing, SAT, and automata that perform well in practice. © 2014 Springer International Publishing Switzerland.Peer Reviewe

Maximal strategies for paramodulation with non-monotonic orderings

A west ordering is a well-founded (strict partial) ordering on terms that satisfies the subterm property. In [Bofill, Godoy, Nieuwenhuis, Rubio, (BGNR-LICS99)] the completeness of an ordered paramodulation inference system w.r.t. west orderings was proved, thus dropping for the first time the monotonicity requirements on the ordering. However, the inference system still required the eager selection of negative equations. Here we improve upon [BGNR-LICS99] in two directions. On the one hand, we show that the results are compatible with constraint inheritance and the so-called basic strategy [(Bachmair, Ganzinger, Lynch, Snyder (1995IC)), (Nieuwenhuis, Rubio (1995JSC))], thus further restricting the search space. On the other hand, we introduce an inference system where also the positive equations of non-unit clauses can be selected, provided that they are maximal

Classes of term rewrite systems with polynomial confluence problems

The confluence property of ground (i.e., variable-free) term rewrite systems (GTRS) is well-known to be decidable. This was proved independently in [DTHL87,DHLT90] and in [Oya87] using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, it has been a well-known longstanding open question whether this bound is optimal (see, e.g., [RTA01]). In [CGN01] we gave the first polynomial-time algorithm for deciding the confluence of GTRS. Later in [Tiw02] this result was extended, using abstract congruent closure techniques, to linear-shallow TRS. Here, we give a new and much simpler proof of the latter result

Classes of term rewrite systems with polynomial confluence problems

The confluence property of ground (i.e., variable-free) term rewrite systems (GTRS) is well-known to be decidable. This was proved independently in [DTHL87,DHLT90] and in [Oya87] using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, it has been a well-known longstanding open question whether this bound is optimal (see, e.g., [RTA01]). In [CGN01] we gave the first polynomial-time algorithm for deciding the confluence of GTRS. Later in [Tiw02] this result was extended, using abstract congruent closure techniques, to linear-shallow TRS. Here, we give a new and much simpler proof of the latter result

Superplastic behaviour of sheet materials under biaxial stress

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Excessively duplicating patterns represent non-regular languages

A constrained term pattern s:¿ represents the language of all instances of the term s satisfying the constraint ¿. For each variable in s, this constraint specifies the language of its allowed substitutions. Regularity of languages represented by sets of patterns has been studied for a long time. This problem is known to be co-NP-complete when the constraints allow each variable to be replaced by any term over a fixed signature, and EXPTIME-complete when the constraints restrict each variable to a regular set. In both cases, duplication of variables in the terms of the patterns is a necessary condition for non-regularity. This is because duplications force the recognizer to test equality between subterms. Hence, for the specific classes of constraints mentioned above, if all patterns are linear, then the represented language is necessarily regular. In this paper we focus on the opposite case, that is when there are patterns withPeer Reviewe