Tree Homomorphisms Research Articles

A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings

This paper connects the classes of weighted alternating finite automata (WAFA), weighted finite tree automata (WFTA), and polynomial automata (PA). First, we investigate the use of trees in the run semantics for weighted alternating automata and prove that the behavior of a weighted alternating automaton can be characterized as the composition of the behavior of a weighted finite tree automaton and a specific tree homomorphism, if weights are taken from a commutative semiring. Based on this, we give a Nivat-like characterization for weighted alternating automata. Moreover, we show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. Additionally, we give a logical characterization of weighted alternating automata, which uses weighted MSO logic for trees. Finally, we investigate the strong connection between weighted alternating automata and polynomial automata. We prove: A weighted language is recognized by a weighted alternating automaton iff its reversal in recognized by a polynomial automaton. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with weights taken from the rational numbers decidable.

Weighted Tree Automata with Constraints

The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Giménez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints , can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.

Definability Results for Top-Down Tree Transducers

Top-down tree transducers are well established formalism for describing tree translations. Such transducers can be further enhanced with look-ahead allowing them to inspect input subtrees before processing them. Oftentimes it is advantageous to know when the look-ahead can be eliminated and the translation be implemented by a simpler transducer. We show that for a given top-down transducer with look-ahead it is decidable whether or not its translation is definable (1) by a linear deterministic top-down tree transducer or (2) by a tree homomorphism. We present algorithms that construct equivalent such transducers if they exist. Our results also apply to bottom-up transducers as well as compositions of transducers.

Lattice-valued tree pushdown automata: Pumping lemma and closure properties

In this paper, we introduce the concept of lattice-valued tree pushdown automata that extends the concepts of lattice-valued pushdown automata, lattice-valued tree automata and fuzzy tree pushdown automata. Also, we consider fuzzy tree languages recognized by lattice-valued (deterministic) tree pushdown automata (called lattice-valued context-free tree languages). Then, we prove the pumping lemma that can be used in proving that certain sets of tree languages are not recognizable. Also, we study closure properties of lattice-valued context-free tree languages with respect to some operations. We show that lattice-valued context-free tree languages recognized by lattice-valued deterministic tree pushdown automata are closed under union, concatenation, star, linear and alphabetic tree homomorphism and inverse tree homomorphism. By applying the pumping lemma on an example, we show that lattice-valued context-free tree languages are not closed under intersection. However, we prove that the intersection of lattice-valued context-free tree language and lattice-valued regular tree language is recognizable by a lattice-valued tree pushdown automaton. Also, we show that lattice-valued non-deterministic tree pushdown automata are only closed under union. However, if L satisfies distributivity, then non-deterministic tree pushdown automata are closed under all of these operations.

Linear weighted tree automata with storage and inverse linear tree homomorphisms

We introduce linear weighted tree automata with storage and show that this model generalizes linear pushdown tree automata. We prove that the class of weighted tree languages recognizable by our linear automaton model is closed under inverse linear tree homomorphisms.

A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings

In this paper, we give a Nivat-like characterization for weighted alternating automata over commutative semirings (WAFA). To this purpose we prove that weighted alternating can be characterized as the concatenation of weighted finite tree automata (WFTA) and a specific class of tree homomorphism. We show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. We give a logical characterization of weighted alternating automata, which uses weighted MSO logic for trees. Finally we investigate the strong connection between weighted alternating automata and polynomial automata. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with the rational numbers as weights is decidable.

Linear context-free tree languages and inverse homomorphisms

We prove that the class of linear context-free tree languages is not closed under inverse linear tree homomorphisms. In fact, we prove a stronger result: we encode Dyck words into a linear context-free tree language and prove that its preimage under a certain linear tree homomorphism cannot be generated by any context-free tree grammar. A positive result can still be obtained: the linear monadic context-free tree languages are closed under inverse linear tree homomorphisms.

Paths, tree homomorphisms and disequalities for -clauses

It is well known that satisfiability is decidable for Horn clauses of the class . Since arbitrary Horn clauses can naturally be approximated by -clauses, can be used for realizing any program analysis which can be specified by means of Horn clauses. Recently, we have shown that decidability for Horn clauses from is retained if the clauses are either extended with tests for disequality between subterms identified by paths or for disequality between homomorphic images of terms. These two results refer to orthogonal extensions of -clauses. Here, we provide a generalization of both results. For that, we introduce hom-path disequalities and show that for each finite set of -clauses extended with such tests an equivalent tree automaton with hom-path disequalities can be constructed. Since emptiness for that class of automata has been shown decidable by Godoy et al. in 2010, we conclude that satisfiability is decidable for -clauses with hom-path disequalities.

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 Complexity of Approximately Counting Tree Homomorphisms

We study two computational problems, parameterised by a fixed tree H . #HOMSTO( H ) is the problem of counting homomorphisms from an input graph G to H . #WHOMSTO( H ) is the problem of counting weighted homomorphisms to H , given an input graph G and a weight function for each vertex v of G . Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in # P . We give a complete trichotomy for #WHOMSTO( H ). If H is a star, then #WHOMSTO( H ) is in FP. If H is not a star but it does not contain a certain induced subgraph J 3, then #WHOMSTO( H ) is equivalent under approximation-preserving (AP) reductions to #BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class #RHΠ1 under AP-reductions. Finally, if H contains an induced J 3 , then #WHOMSTO( H ) is equivalent under AP-reductions to #SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, #WHOMSTO( H ) is complete for #P under AP-reductions. The results are similar for #HOMSTO( H ) except that a rich structure emerges if H contains an induced J 3 . We show that there are trees H for which #HOMSTO( H ) is # SAT -equivalent (disproving a plausible conjecture of Kelk). However, it is still not known whether #HOMSTO( H ) is #SAT-hard for every tree H which contains an induced J 3. It turns out that there is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model . In particular, we show that for a family of graphs Jq , parameterised by a positive integer q , the problem #HOMSTO( Jq ) is AP-interreducible with the problem of approximating the partition function of the q -state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of #HOMSTO( Jq ).

Excessively duplicating patterns represent non-regular languages

A constrained term pattern s:φ represents the language of all instances of the term s satisfying the constraint φ. 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 with “excessively duplicating” variables. We prove that when each pattern of a non-empty set has a duplicated variable constrained to an infinite language, then the language represented by the set is necessarily non-regular. We prove this result independently of the kind of constraints used, just assuming that they are mappings from variables to arbitrary languages. Our result provides an efficient procedure for detecting, in some cases, non-regularity of images of regular languages under tree homomorphisms.

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.

On the solidity of general varieties of tree languages

For a class of hypersubstitutions K, we dene the K-solidity of general varieties of tree languages (GVTLs) that contain tree languages over all alphabets, general varieties ofnite algebras (GVFAs), and general varieties ofnite congruences (GVFCs). We show that if K is a so-called category of substitutions, a GVTL is K-solid exactly in case the corresponding GVFA, or the corresponding GVFC, is K-solid. We establish the solidity status of several known GVTLs with respect to certain categories of substitutions derived from some important classes of tree homomorphisms.

Structural properties of the reconciliation space and their applications in enumerating nearly-optimal reconciliations between a gene tree and a species tree

IntroductionA gene tree for a gene family is often discordant with the containing species tree because of its complex evolutionary course during which gene duplication, gene loss and incomplete lineage sorting events might occur. Hence, it is of great challenge to infer the containing species tree from a set of gene trees. One common approach to this inference problem is through gene tree and species tree reconciliation.ResultsIn this paper, we generalize the traditional least common ancestor (LCA) reconciliation to define a reconciliation between a gene tree and species tree under the tree homomorphism framework. We then study the structural properties of the space of all reconciliations between a gene tree and a species tree in terms of the gene duplication, gene loss or deep coalescence costs. As application, we show that the LCA reconciliation is the unique one that has the minimum deep coalescence cost, provide a novel characterization of the reconciliations with the optimal duplication cost, and present efficient algorithms for enumerating (nearly-)optimal reconciliations with respect to each cost.ConclusionsThis work provides a new graph-theoretic framework for studying gene tree and species tree reconciliations.

Weighted Extended Tree Transducers

Weighted extended tree transducers (wxtts) over countably complete semirings are systematically explored. It is proved that the extension in the left-hand sides of a wxtt can be simulated by the inverse of a linear and nondeleting tree homomorphism. In addition, a characterization of the class of weighted tree transformations computable by bottom-up wxtts in terms of bimorphisms is provided. Backward and forward application to recognizable weighted tree languages are standard operations for wxtts. It is shown that the backward application of a linear wxtt preserves recognizability and that the domain of an arbitrary bottom-up wxtt is recognizable. Examples demonstrate that neither backward nor forward application of arbitrary wxtts preserves recognizability. Finally, a HASSE diagram relates most of the important subclasses of weighted tree transformations computable by wxtts.

PROPERTIES OF QUASI-RELABELING TREE BIMORPHISMS

The fundamental properties of the class QUASI of quasi-relabeling relations are investigated. A quasi-relabeling relation is a tree relation that is defined by a tree bimorphism (φ, L, ψ), where φ and ψ are quasi-relabeling tree homomorphisms and L is a regular tree language. Such relations admit a canonical representation, which immediately also yields that QUASI is closed under finite union. However, QUASI is not closed under intersection and complement. In addition, many standard relations on trees (e.g., branches, subtrees, v-product, v-quotient, and f-top-catenation) are not quasi-relabeling relations. If quasi-relabeling relations are considered as string relations (by taking the yields of the trees), then every Cartesian product of two context-free string languages is a quasi-relabeling relation. Finally, the connections between quasi-relabeling relations, alphabetic relations, and classes of tree relations defined by several types of top-down tree transducers are presented. These connections yield that quasi-relabeling relations preserve the regular and algebraic tree languages.

Fuzzy tree language recognizability

A fuzzy tree language with membership grades in an arbitrary set is syntactically recognizable (s-recognizable) if its syntactic algebra is finite. The equality problem for such languages is decidable and their syntactic algebra can be effectively constructed provided that they are s-recognizable. Linear (but non arbitrary) tree homomorphisms preserve s-recognizability. Tree automata whose transitions are weighted over the unit interval and whose behavior is computed with respect to a pair made of a t-norm distributive over a t-conorm have the syntactic recognition power and thus their equivalence problem is decidable. However, s-recognizability is more powerful when dealing with non-distributive pairs of such operations.

Storage-to-tree transducers with look-ahead

We generalize Engelfriet's decomposition result stating that the class of transformations induced by top–down tree transducers with regular look-ahead is equal to the composition of the class of top–down tree transformations and the class of linear tree homomorphisms. Replacing the input trees with an arbitrary storage type, the top–down tree transducers are turned into regular storage-to-tree transducers. We show that the class of transformations induced by regular storage-to-tree transducers with positive look-ahead is equal to the composition of the class of transformations induced by regular storage-to-tree transducers with the class of linear tree homomorphisms. We also show that the classes of transformations induced by both IO and OI context-free storage-to-tree transducers are closed under positive look-ahead, and are closed under composition with linear tree homomorphisms.

Complexity of tree homomorphisms

For a fixed oriented tree T, we consider the complexity of deciding whether or not a given digraph G is homomorphic to T. It was shown by Gutjahr, Woeginger and Welzl that there exist trees T for which this homomorphism problem is NP-complete. However, it seems difficult to decide just which trees T yield NP-complete homomorphism problems. In this paper, we first identify a class of simple trees with NP-complete homomorphism problems; these trees have exactly one vertex of degree 3 and all other vertices of degree 1 or 2. Our smallest tree has only 45 vertices. (The previous known smallest NP-complete tree has 81 vertices.) In order to gain insight into the structure of oriented trees T which have NP-complete homomorphism problems, we list all subtrees that are necessary in such oriented trees.

Duality and Polynomial Testing of Tree Homomorphisms

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.