Class Of Tree Automata Research Articles

On the Regular Emptiness Problem of Subzero Automata

Subzero automata is a class of tree automata whose acceptance condition can express probabilistic constraints. Our main result is that the problem of determining if a subzero automaton accepts some regular tree is decidable.

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.

The HOM problem is decidable

We close affirmatively a question that has been open for long time: decidability of the HOM problem. The HOM problem consists in determining, given a tree homomorphism H and a regular tree language L represented by a tree automaton, whether H ( L ) is regular. In order to decide the HOM problem, we develop new constructions and techniques that are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular languages through tree homomorphisms. Our contributions are based on the following new constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with disequality constraints recognizing the complementary language. We also define a new class of tree automata with arbitrary disequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of a tree automaton with disequality constraints and the image of a regular language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism. We combine the above constructions adequately to provide an algorithm deciding the HOM problem. This is the journal version of a paper presented in the 42nd ACM Symposium on Theory of Computing (STOC 2010). Here, we provide all proofs and examples. Moreover, we obtain better complexity results via the modification of some proofs and a careful complexity analysis. In particular, the obtained time complexity for the decision of HOM is a tower of three exponentials.

Parameterized complexity of basic decision problems for tree automata

There are many decision problems in automata theory (including membership, emptiness, inclusion and universality problems) that are NP-hard for some classes of tree automata (TA). The study of their parameterized complexity allows us to find new bounds of their nonpolynomial time algorithmic behaviours. We present results of such a study for classical TA, rigid tree automata, TA with global equality and disequality and t-DAG automata. As parameters we consider the number of states, the cardinality of the signature, the size of the term or the t-dag and the size of the automaton.

Rigid tree automata and applications

We introduce the class of rigid tree automata (RTA), an extension of standard bottom-up automata on ranked trees with distinguished states called rigid. Rigid states define a restriction on the computation of RTA on trees: RTA can test for equality in subtrees reaching the same rigid state. RTA are able to perform local and global tests of equality between subtrees, non-linear tree pattern matching, and some inequality and disequality tests as well. Properties like determinism, pumping lemma, Boolean closure, and several decision problems are studied in detail. In particular, the emptiness problem is shown decidable in linear time for RTA whereas membership of a given tree to the language of a given RTA is NP-complete. Our main result is the decidability of whether a given tree belongs to the rewrite closure of an RTA language under a restricted family of term rewriting systems, whereas this closure is not an RTA language. This result, one of the first on rewrite closure of languages of tree automata with constraints, is enabling the extension of model checking procedures based on finite tree automata techniques, in particular for the verification of communicating processes with several local non-rewritable memories, like security protocols. Finally, a comparison of RTA with several classes of tree automata with local and global equality tests, with dag automata and Horn clause formalisms is also provided.

Automata-based verification of programs with tree updates

This paper describes a verification framework for Hoare-style pre- and post-conditions of programs manipulating balanced tree-like data structures. Since the considered verification problem is undecidable, we appeal to the standard semi-algorithmic approach in which the user has to provide loop invariants, which are then automatically checked, together with the program pre- and post-conditions. We specify sets of program states, representing tree-like memory configurations, using Tree Automata with Size Constraints (TASC). The main advantage of this new class of tree automata is that they recognise tree languages based on arithmetic reasoning about the lengths of various (possibly all) paths in trees, like, e.g., in AVL trees or red–black trees. TASCs are closed under union, intersection, and complement, and their emptiness problem is decidable. Thus we obtain a class of automata which are an interesting theoretical contribution by itself. Further, we show that, under few restrictions, one can automatically compute the effect of tree-updating program statements on the set of configurations represented by a TASC, which makes TASC a practical verification tool. We tried out our approach on the insertion procedure for red–black trees, for which we verified that the output on an arbitrary balanced red–black tree is also a balanced red–black tree.

Interactive learning of node selecting tree transducer

We develop new algorithms for learning monadic node selection queries in unranked trees from annotated examples, and apply them to visually interactive Web information extraction. We propose to represent monadic queries by bottom-up deterministic Node Selecting Tree Transducers (NSTTs), a particular class of tree automata that we introduce. We prove that deterministic NSTTs capture the class of queries definable in monadic second order logic (MSO) in trees, which Gottlob and Koch (2002) argue to have the right expressiveness for Web information extraction, and prove that monadic queries defined by NSTTs can be answered efficiently. We present a new polynomial time algorithm in RPNI-style that learns monadic queries defined by deterministic NSTTs from completely annotated examples, where all selected nodes are distinguished. In practice, users prefer to provide partial annotations. We propose to account for partial annotations by intelligent tree pruning heuristics. We introduce pruning NSTTs—a formalism that shares many advantages of NSTTs. This leads us to an interactive learning algorithm for monadic queries defined by pruning NSTTs, which satisfies a new formal active learning model in the style of Angluin (1987). We have implemented our interactive learning algorithm integrated it into a visually interactive Web information extraction system—called SQUIRREL—by plugging it into the Mozilla Web browser. Experiments on realistic Web documents confirm excellent quality with very few user interactions during wrapper induction.

Distinguishability, Simulation and Universality of Moore Tree Automata

We extend the notions of distinguishability of states, simulation and universality of string automata to tree automata of Moore type. In particular, we prove that for each finite class of tree automata satisfying certain conditions, there is a unique minimal universal tree automaton. The transfer from strings to trees brings new aspects but also adds difficulties in generalizing classical Moore results.

On the equivalence problem for letter-to-letter top-down tree transducers

Letter to letter top-down tree transducers are investigated in this paper. Informally, trees which appear in the rules of such transducers are reduced to one letter in the right-hand side as in the left one. With an encoding of the tree transformations induced by such transducers into recognizable forests, we recently established the decidability of equivalence for linear top-down transducers. Here, in order to capture the non-linearity of top-down transducers, we introduce new classes of tree automata with equivalence constraints between direct subterms for which equivalence is decidable. We then show that the equivalence problem for non-linear top-down transducers can be reduced to the equivalence problem of automata with equivalence constraints.

Automata for Reduction Properties Solving

We introduce a new class of tree automata, which we call Reduction Automata (RA), and we use it to prove the decidability of the whole first order of theory of reduction (the theory of reduction is the set of true sentences built up with unary predicates red t ( x) " x is reducible by t", i.e. "some instance of t is a subterm of x"). So we link rewriting, logic and formal languages.

Tree Automata Help One To Solve Equational Formulae In AC-Theories

In this paper we consider particular equational formulae where equality = AC is the congruence induced by a set of associative-commutative axioms. The formulae we are interested in have the form t ≠ ACt1 ∧…t ≠ ACtn and are usually known as complement problems. To solve a complement problem is to find an instance of t which is not an instance of any ti modulo associativity-commutativity. We give a decision procedure based on tree automata which solves these formulae when all the ti1s are linear. We show that this solution also gives a decision of inductive reducibility modulo associativity and commutativity in the linear case and we give several extensions of this approach. Then, we define a new class of tree automata, called conditional tree automata which recognize sets of generalized terms, i.e terms written as multisets, and where the application of a rule depends on the satisfiability of a formula of Presburger's arithmetic. This class of tree automata allows one to solve non-linear complement problems when all occurrences of a non-linear variable occur under the same node (in flattened terms). This solution also provides a procedure to decide inductive reducibility modulo associativity-commutativity in the same case.