Parameterized complexity of basic decision problems for tree automata

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Parameterized complexity of basicdecision problems for tree automataWitold Charatonik a & Agata Chorowska ba Institute of Computer Science , University of Wrocław , Joliot-Curie 15, 50-383 , Wrocław , Polandb Mathematical Institute , University of Wrocław , pl. Grunwaldzki2/4, 50-384 , Wrocław , PolandAccepted author version posted online: 30 Apr 2013.Publishedonline: 14 May 2013.

To cite this article: Witold Charatonik & Agata Chorowska (2013) Parameterized complexity ofbasic decision problems for tree automata, International Journal of Computer Mathematics, 90:6,1150-1170, DOI: 10.1080/00207160.2012.762451

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International Journal of Computer Mathematics, 2013Vol. 90, No. 6, 1150–1170, http://dx.doi.org/10.1080/00207160.2012.762451

Parameterized complexity of basic decision problemsfor tree automata

Witold Charatonika and Agata Chorowskab*

aInstitute of Computer Science, University of Wrocław, Joliot-Curie 15, 50-383 Wrocław, Poland;bMathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

(Received 14 August 2011; revised version received 25 June 2012; second revision received 15 October 2012;third revision received 17 December 2012; accepted 21 December 2012)

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

Keywords: classical tree automata; rigid tree automata; tree automata with global equality and disequality;t-DAG automata; parameterized complexity theory

2010 AMS Subject Classifications: 68Q15; 68Q17; 68Q45

1. Introduction

1.1 Parameterized complexity

In classical complexity theory, the complexity of a problem is measured by the amount of resources(time or space) required to solve the problem. This amount is presented as a function of the size ofthe input of the problem. There are many problems that are hard in this theory, but appear not tobe so hard under a more refined analysis of complexity that takes into account the structure of theinput data. A notable example of such a problem is LTL (Linear Temporal Logic) model checking(i.e. the problem whether a given Kripke structure satisfies a given formula of the logic LTL) inthe area of automated verification [13]. The problem is hard in classical complexity theory, wherewe consider the input as a whole, but it becomes tractable if we look into the structure of the inputand assume that the ‘hard’ part (here, the checked formula) is relatively small in comparison withthe ‘easy’ part (the checked Kripke structure).

Parameterized complexity [8,10] gives a framework for the analysis of such problems. In thistheory, an instance x of a problem comes together with a parameter k (typically, the size of somepart of the input x). If a problem is decidable in time f (k)p(|x|) for some function f and some

*Corresponding author. Email: [email protected]

© 2013 Taylor & Francis

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International Journal of Computer Mathematics 1151

polynomial p, then it is considered to be tractable (with respect to the parameter k) and is calledfixed-parameter tractable. There are two fundamental hierarchies of problems that are not knownto be fixed-parameter tractable: the W-hierarchy and the A-hierarchy. It is believed (but there isno proof of that, similarly as there is no proof of P �= NP) that they do not collapse, so if oneproves that a problem is hard for some levels of these hierarchies, then the existence of efficientalgorithms for this problem is unlikely.

1.2 Tree automata

The theory of finite tree automata (TA) [6] is a straightforward extension of the theory of finiteword automata. The main task of this theory is to provide a finite representation for infinite sets ofterms with efficient operations for manipulating these sets and decidable basic decision problems.

TA were introduced in the 1950s in the context of circuit verification. Nowadays they havenumerous applications. We refer the reader to [6] for references to applications of TA in constraintsolving, semantic unification, term rewriting and automated deduction. This does not cover allareas of applications. For example, TA have been found useful in static program analysis, inparticular in analysis techniques based on set constraints [7] and in the analysis of cryptographicprotocols [15]. Moreover, they are widely used in the context of XML. They are, for instance,employed in the analysis of XML documents, in defining XML schemas and queries, in navigationand in XML transformations. XML applications are described with details in [16].

However, TA have some limitations; perhaps the best known of them is the inability to testfor equality of trees. For example, the set {f (t, t) | t ∈ T�}, where T� is the set of all trees, isnot recognizable by a TA. When the expressive power of standard TA turns out not to be strongenough, different types of extensions are applied. Typical examples of such extensions are rigid treeautomata (RTA), automata with global constraints (TAGED) or automata on DAG representationsof trees (t-DAG). Unfortunately, many decision problems for these classes of automata are knownto be NP-hard in classical complexity theory, and as a consequence, assuming that P �= NP, theydo not have polynomial time algorithms.

1.3 Our contribution

In this paper, we investigate the parameterized complexity of basic decision problems (mem-bership, nonemptiness, k-emptiness, inclusion and universality) for the four above-mentionedclasses of TA (classical, RTA, TAGED and t-DAG automata). As parameters we consider all pos-sible combinations of the number of states, the cardinality of the signature, the size of the termor the t-dag, the number k and the size of an automaton (if it is relevant, i.e. in the case of theinclusion problem).

Our results are summarized in Table 2 at the end of the paper. In particular, we present manyhardness results in classes of the W-hierarchy.A consequence of all these hardness (which includescompleteness) results is that these problems are not fixed-parameter tractable (unless the W-hierarchy is not strict). This is rather bad news for the theory of TA, and it was quite surprisingfor us – we expected more of these problems to be fixed-parameter tractable.

1.4 Related work

We are not aware of any work on parameterized complexity of decision problems for TA. Theclosest results [17] concern automata on words. This paper is an extended and revised versionof [2].

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1152 W. Charatonik and A. Chorowska

2. Parameterized complexity theory

Below we recall the most important concepts from the parameterized complexity theory that areused in this paper [8,10].

Definition 2.1 A parameterized problem over an alphabet � is a pair (A, κ) consisting of a setA ⊆ �∗ and a function κ : �∗ → N. The function κ is called a parameterization of the problem.

Definition 2.2 Let (A, κA) and (B, κB) be parameterized problems over � and �, respectively.An FPT (fixed parameter tractable)-reduction of (A, κA) to (B, κB) is a mapping F : �∗ → �∗ suchthat for every input x ∈ �∗, we have x ∈ A ⇔ F(x) ∈ B, there exist a computable function h and apolynomial p such that for every x ∈ �∗, the result F(x) is computable in time h(κA(x))p(|x|) andthere exists a computable function g such that for every x ∈ �∗, we have κB(F(x)) � g(κA(x)).

We write (A, κA) ≤fpt (B, κB) if there is an FPT-reduction from (A, κA) to (B, κB). If a class Cof parameterized problems is closed under FPT-reductions, then the notions of C-hardness andC-completeness are as in classical complexity theory, but they refer to FPT-reductions. Moreprecisely, (B, κB) is C-hard if (A, κA) ≤fpt (B, κB) for all parameterized problems (A, κA) ∈ C. Ifadditionally (B, κB) belongs to C, then it is C-complete. Therefore, to show that (A, κA) ∈ C, it isenough to show that (A, κA) ≤fpt (B, κB) for some (B, κB) ∈ C, and to show that (A, κA) is C-hard,it is enough to show that (B, κB) ≤fpt (A, κA) for some C-hard problem (B, κB).

We will often use the following observation.

Observation 2.3 Let C be a class of parameterized problems closed under FPT-reductions andlet κ and κ ′ be two parameterizations of the same problem A. Then, (A, κ + κ ′) ≤fpt (A, κ). Inconsequence,

• if (A, κ + κ ′) is C-hard then (A, κ) is C-hard and• if (A, κ) ∈ C then (A, κ + κ ′) ∈ C.

We say that a parameterized problem (A, κ) is fixed-parameter tractable or that it is in theclass FPT (respectively, it is in the class para-NP) if there exists a deterministic (respectively,nondeterministic) algorithm that for all x ∈ �∗ decides whether x ∈ A in time f (κ(x))p(|x|),where f is a computable function and p is a polynomial. A parameterized problem (A, κ) is inthe class para-co-NP if its complement (�∗\A, κ) is in the para-NP class. W[P] is the class ofparameterized problems (A, κ) such that there exists a nondeterministic algorithm that for allx ∈ �∗ decides whether x ∈ A in time f (κ(x))p(|x|) and uses g(κ(x)) log |x| nondeterministicsteps, where f , g are computable functions and p is a polynomial.

The W-hierarchy is a fundamental hierarchy of problems that are not known to be fixed-parameter tractable. For the purpose of the current paper, it is not important how exactly the levelsof this hierarchy are defined; it is more important that they are closed under FPT-reductions andthe inclusions

FPT ⊆ W[1] ⊆ W[2] ⊆ · · · W[P] ⊆ para-NP

are known to be true (and despite a lot of research already done, they are not known to be strict orto collapse). The following problems are known to be complete for the respective levels of thesehierarchies (see [10] for proofs). To simplify the notation, we write values of parameter functionsnext to inputs of problems instead of treating parameters as separate functions.

The problem of parameterized model checking for �1-formulas and the parameterized cliqueproblem are W[1]-complete. A clique for a graph G = (V , E) is a set X ⊆ V such that for everyv, v′ ∈ X there is an edge {v, v′} ∈ E.

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Problem 2.4 (p-MC(�1)) Instance: A structure A and a first-order formula φ of the form∃x1 . . . ∃xn ψ , where ψ does not contain quantifiers. Parameter: |φ|. Question: A |= φ?

Problem 2.5 (p-CLIQUE) Instance: A graph G and k ∈ N. Parameter: k. Question: Is there aclique of k elements in G?

The problem p-DOMINATING-SET is W[2]-complete. A dominating set for a graph G =(V , E) is a set X ⊆ V such that for every vertex v ∈ V it belongs to X or there exists v′ ∈ X suchthat {v, v′} ∈ E.

Problem 2.6 (p-DOMINATING-SET) Instance: A graph G and k ∈ N. Parameter: k. Ques-tion: Is there a dominating set of k elements in G?

The parameterized colourability problem is para-NP-complete. Its complement is para-co-NP-complete. We say that a graph G = (V , E) is k-colourable if there exists a function α : V →{1, . . . , k} such that for every v, v′ ∈ V if α(v) = α(v′) then {v, v′} /∈ E.

Problem 2.7 (p-COLORABILITY) Instance: A graph G and k ∈ N. Parameter: k. Ques-tion: Is G k-colourable?

3. Finite automata

3.1 Preliminaries

Before discussing different types of finite automata on trees, we introduce the basic terminologyconnected with this subject. This is standard terminology when dealing with TA [6].

A signature � is a finite set of function symbols with their arity. We use the notation |�| todenote the cardinality of �. A term over a signature � is either a constant symbol from � orhas the form f (t1, t2, . . . , tn), where f ∈ � is an n-ary function symbol and t1, t2, . . . , tn are terms.We identify terms with their tree representations, so we use both notions of a tree and a terminterchangeably. A language over a signature � is a set (not necessarily finite) of terms over �.The positions Pos(t) in a term t are sequences of positive integers (ε, the empty sequence, is theroot position; generally, a number sequence represents a node as the path of argument-order edgesfollowed from the root to that node). Hence, t can be seen as a function from its set of positionsPos(t) into its set of function symbols �. By t|p, where t is a term and p ∈ Pos(t), we denotethe subterm of t at the position p. The size of a term t (in this paper denoted by |t|) is equal to|Pos(t)| (or, equivalently, the number of vertices in its tree presentation). A DAG representationof a term (in short, a t-dag) over a signature � is a directed, acyclic, ordered graph with verticeslabelled with symbols from � such that if a vertex is labelled with an n-ary symbol then it hasn immediate successors. Moreover, it cannot contain two different isomorphic closed subgraphs.A subgraph H of a directed graph G is closed if for every vertex of H all its successors in G alsobelong to H . The size of a t-dag is the number of its vertices. For n ∈ N, we will frequently usethe notation n to denote the set {1, 2, . . . , n}.

Example 3.1 Let � = {a, e, f , g} for a constant symbol a, a unary function symbol e and binaryfunction symbols f and g. Consider an example term t over the signature �

t = g(f (e(a), e(a)), g(f (e(a), e(a)), a)).

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Below we present the tree and the t-dag representations of the term t:

The set of positions Pos(t) in the term t can be illustrated as follows:

3.2 Classes of automata

In this section, we present the models of automata we are interested in. All considered automataare nondeterministic.

3.2.1 Classical TA

This class of automata is described with details in [6].

Definition 3.2 A TA is a 4-tuple 〈�, Q, F, δ〉, where � is a signature, Q is a finite set of states,F ⊆ Q is a set of final states and δ is a set of transition rules of the form f (q1, q2 . . . , qn) → qwith q, q1, q2 . . . , qn ∈ Q and f ∈ � of arity n.

An automaton starts a computation at the leaves of a tree and moves upward to the root associ-ating inductively states with subtrees in such a way that its transition rules are fulfilled. The size

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of the TA A = 〈�, Q, F, δ〉 is equal to |�| + |Q| + |δ| and is denoted by |A |. Here, |δ| denotesthe sum of sizes of all transition rules in δ and not just the number of the rules. We define the sizeof the rule f (q1, q2 . . . , qn) → q to be n + 1.

Definition 3.3 A run of a TA 〈�, Q, F, δ〉 on a term t is a mapping r : Pos(t) → Q such that forevery position p ∈ Pos(t), if t(p) = f , where f is an n-ary symbol in �, r(p) = q and r(p · i) = qi

for all i ∈ n, then the transition f (q1, q2, . . . , qn) → q belongs to δ. A run is successful if it mapsthe root of t to a final state.

An automaton A accepts a tree t if there exists a successful run of A on t. The set of all treesaccepted by A is called the language of A and is denoted by L (A ).

Example 3.4 Let A = 〈�, Q, F, δ〉 be a TA with � = {a, f } for a constant symbol a and a binaryfunction symbol f , Q = {q, q̄, qF}, F = {qF} and δ consisting of the following transition rules:

(1) a → q,(2) f (q, q) → q | q̄,(3) f (q, q̄) → q | qF ,(4) f (q̄, q) → q | qF .

The automaton A recognizes the language of trees of height at least 3. An accepting run r on theterm f (f (f (a, a), a), a) can be defined as follows:

3.2.2 TA with global equality and disequality constraints (TAGED)

The TAGED class is described with details in [9].

Definition 3.5 A TA with global constraints (TAGED) is a 6-tuple 〈�, Q, R=, R�=, F, δ〉, where〈�, Q, F, δ〉 is a TA and R=, R�= are binary relations on Q.

Definition 3.6 A run of a TAGED 〈�, Q, R=, R�=, F, δ〉 on a term t is a mapping r that is a runof the TA 〈�, Q, F, δ〉 on t satisfying additional conditions for all p1, p2 ∈ Pos(t):

(r(p1), r(p2)) ∈ R= ⇒ t|p1 = t|p2 and (r(p1), r(p2)) ∈ R�= ⇒ t|p1 �= t|p2 .

A TAGED A accepts a tree t if there exists a successful run of A on t. The set of all trees acceptedby the TAGED A is called the language of A and is denoted by L (A ).

3.2.3 Rigid tree automata

The RTA class is presented in [12]. It is a restriction of the TAGED class where the R= relationis a subset of the identity on Q and the R�= relation is empty. The states occurring in R= are thencalled rigid. To simplify the notation, we write an RTA as a 5-tuple 〈�, Q, R, F, δ〉, where 〈�, Q,F, δ〉 is a TA and R ⊆ Q is a set of rigid states.

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3.2.4 Automata on dag representations of trees (t-DAG automata)

The t-DAG class is introduced in [4] and used in [5] for solving set constraints. It is a class ofautomata running on dag representations of terms. In other words, a t-DAG automaton is a TAthat associates equal states to equal subtrees of a tree. Using the notation from Definition 3.6,it means that t|p1 = t|p2 ⇒ r(p1) = r(p2). In this sense, it is an automata class dual to the RTAclass.

We do not give the definition of a t-DAG automaton as it is syntactically the same as thedefinition of a TA. The two classes of automata differ in the definition of a run.

Definition 3.7 A run of a t-DAG automaton 〈�, Q, F, δ〉 on a t-dag G is a mapping r from theset of nodes of G to the set Q such that for every node v of G labelled with an n-ary symbol f ∈ �,if v1, v2, . . . , vn are successors of v, then f (r(v1), r(v2), . . . , r(vn)) → r(v) belongs to δ. A run issuccessful if it maps the root of G to a final state.

Note that a t-DAG automaton 〈�, Q, F, δ〉 considered as an automaton on trees corresponds toa TAGED 〈�, Q, R=, R�=, F, δ〉 with R= = ∅ and R �= = {(q, q′) | q, q′ ∈ Q, q �= q′}.

3.3 Parameterized decision problems for finite automata on trees

We introduce some parameterized decision problems for finite automata on trees. By Q we denotethe set of states of an automaton and by � its signature. Numbers k are in a binary representation.For each problem, different parameterizations can be considered.

Problem 3.8 (NONEMPTINESS) Instance: An automaton A . Parameter: |Q|, |�| or a sumof them. Question: Is the language recognized by A nonempty?

Problem 3.9 (k-EMPTINESS) Instance: An automaton A and k ∈ N. Parameter: |Q|, |�|, kor a sum of some of them. Question: Does there exist a tree/t-dag of size k accepted by A ?

Problem 3.10 (MEMBERSHIP) Instance: An automaton A and a tree/t-dag t. Parame-ter: |Q|, |�|, |t| or a sum of some of them. Question: Is t accepted by A ?

Problem 3.11 (UNIVERSALITY) Instance: An automaton A . Parameter: |Q|, |�| or a sumof them. Question: Is the language recognized by A total?

Problem 3.12 (INCLUSION) Instance: Two automata A1 and A2 defined over the same signa-ture �. Parameter: |Q1|, |Q2|, |�|, |A1|, |A2| or a sum of some of them. Question: Is L (A1) ⊆L (A2)?

3.4 Known results

Table 1 presents the known classical complexity theory results for the described decision problems.For proofs see [1,3,4,6,9,12,14].

4. Results

Most of the theorems in this section are proved by FPT-reductions. A definition of FPT-reductioncan be found in Section 2 (see Definition 2.2).

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Table 1. Summary of known results.

TA RTA TAGED t-DAG

EMPTINESS PTIME PTIME ExpTime-hard decidable NP-completeMEMBERSHIP LOGCFL-complete NP-complete NP-complete NP-completeUNIVERSALITY ExpTime-complete Undecidable Undecidable UndecidableINCLUSION ExpTime-complete Undecidable Undecidable Undecidable

4.1 Classical TA

4.1.1 k-EMPTINESS

Proposition 4.1 For TA, the k-EMPTINESS problem parameterized by k, k + |�|, k + |Q| orby k + |�| + |Q| belongs to FPT.

Proof This problem can be solved by the following algorithm. For a given TA, A = 〈�, Q, F, δ〉,and a number k constructs a TA Ak = 〈�, Q × k, F × {k}, δk〉, where δk consists of transitions

f ((q1, i1), (q2, i2), . . . , (qn, in)) →⎛⎝q, 1 +

n∑j=1

ij

⎞⎠ , if 1 +

n∑j=1

ij ≤ k,

where f (q1, q2, . . . , qn) → q ∈ δ. Then, |Ak| ≤ |A |kψ(k), where ψ(k) is the number of all rep-resentations of k as a sum of positive integers. This is because the number of transitions in δk

is bounded by the number of transitions in δ times ψ(k) and the size of each transition in δk isequal to the size of the corresponding transition in δ. The number of states of Ak is equal to k|Q|.Since nonemptiness of Ak can be checked in time bounded by a polynomial in |Ak|, the presentedalgorithm is an FPT algorithm. �

4.1.2 UNIVERSALITY

Theorem 4.2 For TA, the UNIVERSALITY problem parameterized by |�| + |Q| is para-co-NP-hard.

Proof We propose a reduction of p-NON-COLORABILITY, which is a para-co-NP-completeproblem. Consider an instance of the p-NON-COLORABILITY problem: a graph G = (V , E) of nvertices v1, v2, . . . , vn and a number k ∈ N. We construct a TA A = 〈�, Q, F, δ〉 such that G is notk-colourable if and only if A accepts all terms over the signature �. Let � = {a1, a2, . . . , ak , f },where ai is a constant symbol and f is an n-ary function symbol.

Constant symbols {a1, a2, . . . , ak} ⊆ � represent k colours, while n arguments of the functionsymbol f correspond to vertices of the graph G. A term over � of the form f (x1, x2, . . . , xn) forxi ∈ {a1, a2, . . . , ak} represents a colouring of G with a set of k-colours (xi = aj means that thevertex vi has the colour j). Let Q = {q1, q2, . . . , qk , q, q̄, qF} and F = {qF}. Let δ consist of thefollowing transitions:

(1) ai → qi | q | qF for i ∈ k,(2) f (p1, p2, . . . , pn) → qF if there exists an edge {vi, vj} ∈ E such that pi = pj = ql for some l

and pm = qF for all m �= i, m �= j,(3) f (q, . . . , q) → q | q̄,(4) f (p1, p2, . . . , pn) → q | qF if pi = q̄ for some i and pj = q for all j �= i.

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The automaton A accepts a term t if and only if it does not represent a good colouring. Indeed,all terms of height greater than 2 are accepted using transition rules

ai → q,

f (q, . . . , q) → q | q̄,

f (q, . . . , q̄, . . . , q) → q | qF .

Clearly, all terms of height 1 are also accepted.Finally, a term of height 2, that is, a term representing a colouring, is accepted if and only if

the colouring is bad. Indeed, it can be accepted only using the transition rule of type (2) and thecondition in (2) means that vertices vi and vj are joined by an edge and coloured with the samecolour l.

Thus, A accepts all terms over the signature � if and only if there are no good k-colouringsof G; that is, G is not k-colourable. Moreover, |Q| = k + 3, |�| = k + 1 (so the value of theparameter |Q| + |�| is bounded by a function of the parameter k) and the size of A dependspolynomially on the number k and the size of G: there are 3k transitions of type (1), at most

( n2k

)transitions of type (2), 2 transitions of type (3) and 2n transitions of type (4); transitions of types(1), (2), (3) and (4) have sizes 1, n + 1, n + 1 and n + 1, respectively. �

Corollary 4.3 For TA, the UNIVERSALITY problem parameterized by |�| or by |Q| is para-co-NP-hard.

The result above shows that universality is a very hard problem. Even if we fix the numberof states in the input automaton, it remains co-NP-hard. In fact, since 3-colourability is an NP-complete problem, we can observe that already universality of automata with six states is co-NP-hard.

Note that there is another subtle point concerning the signature. The theorem above states para-co-NP-hardness of the universality problem parameterized by the cardinality of the signature,but not by its size. The difference is that formally a signature is a set of pairs consisting of afunction symbol and its arity, so a function symbol of arity n contributes 1 to the cardinality of thesignature and log n to its size. In the proof of the theorem, we used k + 1 function symbols, whichis a number depending on the number k of colours and not depending on the size of the input graph– this is allowed by FPT-reductions. However, one of the symbols has arity n, which is the numberof vertices in the graph; since the value log n does not depend on the value k of the parameter, thisis not allowed by FPT-reductions. Therefore, Theorem 4.2 and Corollary 4.3 say nothing about thecomplexity of the universality problem parameterized by the size of the signature. The hardnessresult for the parameterization by the size of the signature can be achieved by using a standardapproach of binarization, used, for example, to convert an arbitrary context-free grammar intoa grammar in Chomsky normal form [11].

Theorem 4.4 For TA, the UNIVERSALITY problem parameterized by the size of � is para-co-NP-hard.

Sketch of proof We propose a reduction of p-NON-COLORABILITY similar to the one fromTheorem 4.2. The main idea is to encode the n-ary function symbol f as n − 1 compositionsof the binary symbol s. Then, the signature � = {a1, a2, . . . , ak , s} has size O(k) and termsrepresenting colourings have the form

s(. . . s(s(x1, x2), x3) . . . , xn),

where xi ∈ {a1, a2, . . . , ak}.

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As in Theorem 4.2, the automaton that recognizes terms not representing good colourings isa union of two automata: one that recognizes terms that do not represent any colouring and anotherone that recognizes bad colourings. The first one can be easily constructed using the fact that a termdoes not represent a colouring if and only if it contains a subterm with the right child labelled bys or if it has a height different from n. This can be done using O(n) states. The second automaton,as in the proof of Theorem 4.2, guesses a pair of vertices and their colours and propagates thisinformation to the root. There are n2k2 possible guesses, so by using additional O(n3k2) statesthe automaton can propagate a guess from the leaves to the root and recognize a bad colouring.Thus, the size of the resulting automaton is polynomial in k and n. �

Remark 1 Another decision problem that is frequently considered in the context of automatais the equality of recognized languages. Given two automata A1 and A2 defined over the samesignature �, one can ask whether L (A1) = L (A2).

Obviously this problem can be solved by checking two instances of the INCLUSION problem,so for TA it is in EXPTIME. It is also easy to observe that the problem is at least as hard asthe UNIVERSALITY problem: to solve UNIVERSALITY, it is enough to check the equality ofa given automaton and an automaton recognizing all terms over �. Therefore, the problem isundecidable for RTA, TAGED and DAG automata, and it is EXPTIME-hard for TA. Theorem 4.2implies that the problem parameterized by |�|, |�| + |Q1|, |�| + |Q2|, |Q1| + |Q2| and |�| +|Q1| + |Q2| is para-co-NP-hard.

4.1.3 INCLUSION

Proposition 4.5 For TA, the INCLUSION problem parameterized by |�|, |Q1|, |Q2|, |Q1| + |Q2|or by |A1| is para-co-NP-hard.

Proof There is an automaton A1 = 〈�, {q}, {q}, {f (q, . . . , q) → q | f ∈ �}〉 with one state thataccepts all terms over a given signature �. The theorem is a consequence of this fact and Theo-rems 4.2 and 4.4. Indeed, for an automaton A = 〈�, Q, F, δ〉, the language recognized by A istotal if and only if L (A1) ⊆ L (A ).

Note that without loss of generality we can assume that for every f ∈ � there is a transition ruleof the form f (q1, . . . , qn) → q in δ. Otherwise A obviously is not total and the UNIVERSALITYproblem for A is trivial. Similarly, we can assume that F �= ∅. Under these assumptions, wehave that |A1| ≤ |A |, so the size of the INCLUSION instance is polynomial in the size of theUNIVERSALITY instance.

We still have to show that the value of the parameter in the INCLUSION instance is boundedby a function of the parameter in the UNIVERSALITY instance. For parameterizations with|�|, |Q1|, |Q2|, |Q1| + |Q2|, we use Theorem 4.2: since |Q1| = 1, each of these parameterizationsis bounded by |�| + |Q|. For the parameterization with |A1|, the result follows from Theorem 4.4:a function symbol of arity n contributes log n to the size of the signature and additionally O(n)

to the size of the transition table, so the size of the automaton A1 is exponential in the size of thesignature. �

Proposition 4.6 For TA, the INCLUSION problem parameterized by the size of A2 is in FPT.

Proof This problem for given automata A1, A2 can be solved by the algorithm that first constructsa complementation TA A ′

2 of A2, then constructs an intersection TA of A1 and A ′2 and checks

nonemptiness of the received automaton. The automaton constructed in the first step has sizebounded by a function of |A2|. The second and the third step of the algorithm can be implementedby polynomial time algorithms. �

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4.2 Rigid tree automata

4.2.1 k-EMPTINESS.

As a consequence of Theorem 4.14 and Observation 2.3, we obtain the following.

Proposition 4.7 For RTA, the k-EMPTINESS problem parameterized by k, k + |�|, k + |Q| orby k + |�| + |Q| belongs to W[1].

4.2.2 MEMBERSHIP

Proposition 4.8 The MEMBERSHIP problem for RTA parameterized by |t|, |t| + |�|, |t| + |Q|or by |t| + |�| + |Q| is in W[1].

Proof This is a direct consequence of Proposition 4.18 and Observation 2.3. �

Theorem 4.9 For RTA, the MEMBERSHIP problem parameterized by |Q| is W[2]-hard.

Proof We propose a reduction of p-DOMINATING-SET, which is a W[2]-complete prob-lem. Consider an instance of the p-DOMINATING-SET problem: a graph G = (V , E) withV = {v1, v2, . . . , vn} and a number k ∈ N. We construct an RTA A = 〈�, Q, R, F, δ〉 and a termt over the signature � such that in G there is a dominating set of k elements if and only if Aaccepts the term t.

Let � = {xv | v ∈ V} ∪ {fv,w | v, w ∈ V} ∪ {f , g}, where xv are constant symbols, fv,w aresymbols of binary functions and f and g are symbols of n-ary functions. Let Q ={q1, . . . , qk , q, q0, qF}, R = {q1, . . . , qk} and F = {qF}. States qi for i ∈ {1, . . . , k} represent achoice of vertices of a dominating set. Let δ consist of the following transitions:

(1) xv → qi for v ∈ V and i ∈ {0, 1, . . . , k},(2) fv,w(qi, qj) → q0 for v, w ∈ V and i, j ∈ {0, 1, . . . , k},(3) fv,w(q0, qi) → q for v, w ∈ V such that {v, w} ∈ E and i ∈ k,(4) fv,w(qi, qj) → q for v, w ∈ V , i ∈ k and j ∈ {0, 1, . . . , k},(5) g(p1, p2, . . . , pn) → q, if there exists i ∈ n such that pi = q and pj = q0 for all j �= i,(6) f (q, q, . . . , q) → qF .

Let t be the term

f (g(fv1,v1(xv1 , xv1), . . . , fv1,vn(xv1 , xvn)), . . . , g(fvn,v1(xvn , xv1), . . . , fvn,vn(xvn , xvn))).

Intuitively, the automaton guesses a dominating set of k vertices and assigns the state qi to thesubterm xvi . While evaluating the subterm g(fvi ,v1(xvi , xv1), . . . , fvi ,vn(xvi , xvn)), the automaton provesthat the vertex vi is dominated, and finally, at the root, it requires that all the vertices are dominated.

More precisely, suppose that X = {a1, a2, . . . , ak} is a dominating set in G. For j ∈ n, let ij ∈{1, 2 . . . , k} be such that {vj, aij } ∈ E or vj = aij . The index ij is chosen such that aij is a witness forthe domination of the vertex vj. Intuitively, a subterm g(fvi ,v1(xvi , xv1), . . . , fvi ,vn(xvi , xvn)) is used tocheck that the vertex vi is dominated by some vertex in X. An accepting run r can be defined asfollows:

• r(xai) = qi,• r(xv) = q0 for v /∈ X ,• r(fvj ,aij

(xvj , xaij)) = q,

• r(fvj ,vi(xvj , xvi)) = q0 for vi �= aij ,

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• r(g(fvi ,v1(xvi , xv1), . . . , fvi ,vn(xvi , xvn))) = q for i ∈ {1, 2 . . . , n},• r(t) = qF .

Note that we consider here the run r as if it was a function defined on the set of subterms of aterm instead of on the set of its positions. It is just to simplify the notation, and we believe that itwill not lead to any confusion due to the fact that all equal subtrees are recognized in the samestate anyway.

Conversely, let r be a successful run of A on t. Let X = {v ∈ V | ∃l ∈ tvr(l) �= q0}, where tv isthe set of all positions of t labelled by xv. The rigidity of states qi for i > 0 implies that |X| � k.The run r is successful, so for every i ∈ n

r(g(fvi ,v1(xvi , xv1), . . . , fvi ,vn(xvi , xvn))) = q.

This means that for exactly one j ∈ n, we have r(fvi ,vj (xvi , xvj )) = q. Then, either r(xvi) �= q0 andvi ∈ X or r(xvj ) �= q0, vj ∈ X and {vi, vj} ∈ E. So, vi is dominated by vj. Hence, the set X is adominating set in G.

Moreover, the size of the automaton A and the length of the term t depend polynomially on thesize of the graph G. Indeed, there are n(k + 1) transitions of the form (1), n2(k + 1)2 transitionsof the form (2), n2k transitions of the form (3), n2k(k + 1) transitions of the form (4), n transitionsof the form (5) and one transition of the form (6). Transitions of the forms (1), (2), (3), (4), (5)and (6) have sizes 1, 3, 3, 3, n + 1 and n + 1, respectively. Moreover, |Q| = k + 3, so the value ofthe parameter |Q| is bounded by a function of the parameter k. �

4.3 TA with equalities and disequalities (TAGED)

4.3.1 NONEMPTINESS

Theorem 4.10 For TAGED, the NONEMPTINESS problem parameterized by |�| is W[2]-hard.

Proof We propose a reduction of p-DOMINATING-SET, which is a W[2]-complete prob-lem. Consider an instance of the p-DOMINATING-SET problem: a graph G = (V , E) withV = {v1, v2, . . . , vn} and a number k ∈ N. We construct a TAGED A = 〈�, Q, R=, ∅, F, δ〉 suchthat in G there is a dominating set of k elements if and only if the language recognized by A isnonempty.

Let � = {x0, x1, . . . , xk , f , g}, where xi are constant symbols and f and g are symbols of n-aryfunctions. Let Q = {r1, . . . , rn, q1, . . . , qk , p1, . . . , pk , qF}, R= = {(ri, rj) | i, j ∈ n} and F = {qF}.Let δ consist of the following transitions:

(1) x0 → pj for j ∈ k,(2) xi → pj | qi for i, j ∈ k, i �= j,(3) f (pl, . . . , pl, ql, pl . . . , pl) → rj for l ∈ k, j ∈ n, if the position i of the state ql fulfils i = j or

{vi, vj} ∈ E,(4) g(r1, . . . , rn) → qF .

The size of the automaton A depends polynomially on the size of the graph G and the numberk. Moreover, |�| = k + 3.

Thus, it is enough to show that in G there is a dominating set of k elements if and only if Aaccepts some term.

Suppose that X = {vi1 , vi2 , . . . , vik } is a dominating set in G. Let χ : n → {0, . . . , k} be a functionsuch that χ(i) = 0 if vi �∈ X and χ(i) = j if vi = vij . Intuitively, the function χ gives the numberof a given vertex in the dominating set. We will represent the set X by the term f (xχ(1), . . . , xχ(n)).

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To check that this term indeed represents a dominating set, the automaton takes n copies t1, . . . , tnof this term and accepts g(t1, . . . , tn). The copy ti is used to check that the vertex vi is dominatedby some element in X .

Formally, for i, j ∈ n, we define yi,j to be xχ(j). Then, the term

t = g(f (y1,1, . . . , y1,n), . . . , f (yn,1, . . . , yn,n))

is accepted by A and an accepting run r on t can be defined as follows:

•r(i · j) =

{qφ(i), if yi,j = xφ(i),pφ(i), otherwise,

where φ : n → k is such that {vj, viφ(j)} ∈ E or vj = viφ(j) ; that is, viφ(j) is a witness for the vertexvj that X is a dominating set:

• r(i) = ri,• r(ε) = qF .

Conversely, let t be a term accepted by A . Note that, from the definition of δ, the term tmust have the form t = g(f (y1,1, . . . , y1,n), . . . , f (yn,1, . . . , yn,n)) for some yi,j ∈ {x0, . . . , xk}. Since(ri, rj) ∈ R=, we have that yi,l = yj,l for all i, j, l ∈ n. It allows us to rewrite t in a simplified formt = g(f (y1, . . . , yn), . . . , f (y1, . . . , yn)).

Let r be a successful run of A on t. Let ψ(i) ∈ n be such that r(i · ψ(i)) = ql for some l ∈ k.Then, X = {vψ(i) | i ∈ n} is a dominating set in G with |X| ≤ k. Indeed, if r(i · ψ(i)) = ql, thenyi,ψ(i) = xl. Moreover, from the definition of the transition rule (3), r(i · j) = pl for all j �= ψ(i). Inconsequence, since pl can be reached only by a transition of type (1) or (2) yi,j �= xl for j �= ψ(i).It implies that for each l the symbol xl occurs at most once among y1, . . . , yn and thus r(i′ · j′) �= ql

for all pairs (i′, j′) �= (i, ψ(i)). Hence, vψ(i) is the only vertex brought into X by the state qi. Sincethere are k states qi, |X| ≤ k. �

The following proposition is a corollary of Theorem 4.12.

Proposition 4.11 For TAGED, the NONEMPTINESS problem parameterized by |Q| is W[1]-hard.

Theorem 4.12 For TAGED, the NONEMPTINESS problem parameterized by |�| + |Q| is W[1]-hard.

Proof We propose a reduction of p-CLIQUE, which is a W[1]-complete problem. Consider aninstance of the p-CLIQUE problem: a graph G = (V , E) with V = {v1, v2, . . . , vn} and a numberk ∈ N. We construct a TAGED A = 〈�, Q, R=, ∅, F, δ〉 such that in G there is a clique of kelements if and only if the language recognized by A is nonempty.

Let � = {x0, x1, . . . , xk} ∪ {f , g}, where xi are constant symbols, f is a symbol of ann-ary function and g is a symbol of a 1

2 k(k − 1)-ary function. Let Q = {q1, . . . , qk ,q1,2, q1,3, . . . , qk−1,k , p1,2, p1,3, . . . , pk−1,k , qF}, R= = {(qi,j, qi′,j′) | i, i′, j, j′ ∈ k, i < j, i′ < j′} andF = {qF}. Let δ consist of the following transitions:

(1) x0 → pi,j for i, j ∈ k, i < j,(2) xl → ql | pi,j for i, j, l ∈ k, i < j, i �= l �= j,(3) f (pi,j, . . . , pi,j, qi, pi,j . . . , pi,j, qj, pi,j . . . , pi,j) → qi,j for i, j ∈ k, i < j if the positions l, m of

the states qi and qj, respectively, fulfil {vl, vm} ∈ E,(4) g(q1,2, q1,3, . . . , qk−1,k) → q.

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The size of the automaton A depends polynomially on the size of the graph G and the number k.Moreover, |�| = k + 3 and |Q| = k + k(k − 1) + 1. Thus, it is enough to show that in G there isa clique of k elements if and only if the language recognized by A is nonempty.

Consider a term of the form f (y1, . . . , yn), where yi ∈ {x0, x1, . . . , xk} and every symbol xi for1 ≤ i ≤ k appears exactly once among y1, . . . , yn. Such a term represents a choice of k verticesof the graph G corresponding to positions of symbols {x1, . . . , xk} as arguments of f .

Suppose that t is a term accepted by A . Note that, from the definition of δ, the term t musthave a form t = g(f (y1

1,2, . . . , yn1,2), . . . , f (y1

k−1,k , . . . , ynk−1,k)) for some yl

i,j ∈ {x0, . . . , xk}. We willshow that all the terms f (y1

1,2, . . . , yn1,2), . . . , f (y1

k−1,k , . . . , ynk−1,k) are equal, that they all represent

the same subset of k vertices of G and that this subset is a clique.First observe that for all i, j ∈ k such that i < j, every successful run of A must assign the state

qi,j to the position of the subterm f (y1i,j, . . . , yn

i,j). Since (qi,j, qi′,j′) ∈ R= for all i, j, i′, j′ ∈ k suchthat i < j and i′ < j′, the terms f (y1

i,j, . . . , yni,j) and f (y1

i′,j′ , . . . , yni′,j′) are equal and we can denote

them by f (y1, . . . , yn).Second, to show that these terms represent a set of k vertices, we have to show that every

symbol xi for i ∈ k appears exactly once among y1, . . . , yn. Take arbitrary i ∈ k. Without lossof generality, we can assume that i < k (the proof for i = k is analogous). Consider the termf (y1

i,i+1, . . . , yni,i+1). A successful run of A on t must assign the state qi,i+1 to this term. The state

qi,i+1 is reachable only by the rule (3), which means that the state qi is assigned to one elementof the set {y1

i,i+1, . . . , yni,i+1} (so at least one of these elements is xi), qi+1 is assigned to another

element of this set and the state pi,i+1 is assigned to all remaining elements of this set. Becauseof rules (1) and (2) the states qi+1 and pi,i+1 may be assigned only to constant symbols differentfrom xi. Thus, xi occurs exactly once among y1, . . . , yn.

Finally, to see that the set represented by the term f (y1, . . . , yn) is a clique, note that the rule (3)ensures that there is an edge between (the nodes represented by) xi and xj and the rule (4) checksthat this is the case for all i, j ∈ k such that i < j.

Conversely, suppose that X = {vl1 , vl2 , . . . , vlk } is a clique in G. Let χ : n → {0, . . . , k} be thefunction such that χ(i) = 0 if vi �∈ X and χ(i) = j if vi = vlj . We will represent the set X by theterm f (xχ(1), . . . , xχ(n)). To check that this term indeed represents a clique, the automaton takes12 (k − 1)k copies t1,1, . . . , tk−1,k of this term and accepts g(t1,1, . . . , tk−1,k). The copy ti,j is used tocheck that the vertices vli and vlj are connected by an edge in G.

Formally, for i, j ∈ k such that i < j and for l ∈ n, we define yli,j to be xχ(l). Then, the term

t = g(f (y11,2, . . . , yn

1,2), . . . , f (y1k−1,k , . . . , yn

k−1,k)) is accepted by A and an accepting run r on t canbe defined as follows:

•

r(ψ(i, j) · l) =⎧⎨⎩

qi, if yli,j = xi,

qj, if yli,j = xj,

pi,j, otherwise,

• r(ψ(i, j)) = qi,j,• r(ε) = qF ,

where ψ(i, j) = ((k − i + 1)(i − 1) + 12 (i − 2)(i − 1)) + (j − i) is the position of the pair (i, j)

among all the pairs (s, t) with 1 ≤ s < t ≤ k sorted in lexicographic order. �

4.3.2 k-EMPTINESS

The proposition below follows directly from Theorem 4.16.

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Proposition 4.13 For TAGED, the k-EMPTINESS problem parameterized by k is W[1]-hard.

Theorem 4.14 For TAGED, the k-EMPTINESS problem parameterized by k belongs to W[1].

Proof It is enough to reduce the problem to p-MC(�1). Consider a TAGED 〈�, Q, R=, R�=, F, δ〉and a number k ∈ N. We construct a structure A and a formula φ such that the formula φ is truein A if and only if the automaton 〈�, Q, R=, R�=, F, δ〉 accepts some term of size k. Let A bedefined over the set A = � ∪ Q ∪ � ∪ {⊥}, where � is the set of all unlabelled trees of a size notgreater than k.

Let there be the following relations in A :

D := {(q0, f , q1, . . . , qk−1) | f (q1, . . . , qar(f )) → q0 ∈ δ and qi =⊥ for ar(f ) < i � k − 1},T := {(t0, t1, . . . , tk−1) | t1, . . . , tl are children of t0 and ti =⊥ for l < i � k − 1},

T= := {(t, t) | t ∈ �}, T�= := (� × �)\T=RA= := R=,

RA �= := R �= and FA := F.

Now we define the formula φ = ∃t1, . . . , tk , q1, . . . , qk , f1, . . . , fkφ′, where φ′ is the disjunction

∨t∈�k

( ∧1≤i≤k

(T(ti, tct(i), . . . , tct(i)+art(i)−1, ⊥, . . . , ⊥)

∧ D(qi, fi, qct(i), . . . , qct(i)+art(i)−1, ⊥, . . . , ⊥)

)

∧∧

i1,i2∈k

⎛⎝

⎛⎝RA=(qi1 , qi2) ⇒

⎛⎝T=(ti1 , ti2) ∧

∧1≤j≤art(i1,i2)

fτt(i1,j) = fτt(i2,j)

⎞⎠

⎞⎠

∧⎛⎝RA �=(qi1 , qi2) ⇒

⎛⎝T�=(ti1 , ti2) ∨

∨1≤j≤art(i1,i2)

fτt(i1,j) �= fτt(i2,j)

⎞⎠

⎞⎠

⎞⎠

∧ FA(q1)),

where �k is the set of all unlabelled trees of the size k, ct(i) is the index of the first child of thevertex of the index i in the tree t (we are numbering from the top to the bottom and from the leftto the right), τt(i, j) is the index of the jth descendant of the vertex of the index i in the tree t andart(i) is the arity of the vertex of the index i. Note that for t ∈ �k the function art is bounded byk. We use the notation art(i, j) = min(art(i), art(j)).

The size of the structure A depends on k and polynomially on the size of the automaton. Thelength of φ depends only on k.

The formula φ expresses that there exists a successful run of the automaton on someterm of the size k. The variable t represents a tree of the size k which, together withlabels of its vertices represented by the variables f1, . . . , fk , forms a term recognized by theautomaton. The variables t1, . . . , tk are subtrees of t, which is expressed by the subformulasT(ti, tct(i), . . . , tct(i)+art(i)−1, ⊥, . . . , ⊥). The variables q1, . . . , qk are states that are images of theterms t1, . . . , tk (with labels f1, . . . , fk) via a successful run of the automaton on t. The con-junction of subformulas D(qi, fi, qct(i), . . . , qct(i)+art(i)−1, ⊥, . . . , ⊥) guarantees the correctness ofthe transitions of the automaton; the conditions for a run of a TAGED are satisfied becauseof the subformulas RA=(qi1 , qi2) ⇒ (T=(ti1 , ti2) ∧ ∧

1≤j≤k fτt(i1,j) = fτt(i2,j)) and RA �=(qi1 , qi2) ⇒

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(T �=(ti1 , ti2) ∨ ∨1≤j≤k fτt(i1,j) �= fτt(i2,j)). Finally, the subformula FA(q1) expresses that the run is

successful. �

Corollary 4.15 For TAGED, the k-EMPTINESS problem parameterized by k + |Q| or by k +|�| + |Q| belongs to W[1].Theorem 4.16 For TAGED, the k-EMPTINESS problem parameterized by k + |�| is W[1]-complete.

Proof The fact that the problem belongs to the W[1] class is a consequence of Theorem 4.14.For the proof of W[1]-hardness, we propose a reduction of p-CLIQUE. Consider an instanceof the p-CLIQUE problem: a graph G = (V , E) and a number k ∈ N. We construct a TAGEDA = (�, Q, ∅, R�=, F, δ) and a number k′ such that in G there is a clique of k elements if andonly if A accepts a tree of the size k′. Let � = {x1, x2, . . . , xk} ∪ {f , g}, where xi is a constantsymbol, f is a symbol of a 1

2 k(k − 1)-ary function and g is a symbol of a binary function.Let Q = {q1,2, q1,3, . . . , qk−1,k , qF} ∪ {qv,i | v ∈ V , 1 ≤ i ≤ k}, F = {qF}. Let R �= = {(qv,i, qv′,i) |i ∈ k, v, v′ ∈ V , v �= v′} and let δ consist of the following transitions:

(1) xi → qv,i for v ∈ V and 1 ≤ i ≤ k,(2) g(qv1,i, qv2,j) → qi,j for i, j ∈ k, i < j and v1, v2 ∈ V such that {v1, v2} ∈ E,(3) f (q1,2, q1,3, . . . , qk−1,k) → qF .

Note that t = f (g(x1, x2), g(x1, x3), . . . , g(xk−1, xk)) is the only term that can be accepted by A .Let k′ = |t| = 1 + 3

2 k(k − 1). The size of the automaton A depends polynomially on the sizeof the graph G. Moreover, since |�| = k + 2, the value of the parameter k′ + |�| is bounded bya function of the parameter k. Thus, it is enough to show that in G there is a clique of k elementsif and only if A accepts the term t.

Suppose that X = {a1, a2, . . . , ak} is a clique in G. An accepting run r of A on t can beconstructed as follows:

• r(p) = qai ,i for p ∈ Pi,• r(ψ(i, j)) = qi,j (X is a clique so there is an edge between ai and aj),• r(ε) = qF ,

where Pi is the set of positions in t with the symbol xi and ψ(i, j) = ((k − i + 1)(i − 1) + 12 (i −

2)(i − 1)) + (j − i) is the position of the pair (i, j) among all the pairs (s, t) with 1 ≤ s < t ≤ ksorted in lexicographic order.

Conversely, let r be a successful run of A on t. Let X = {v | r(xi) = qv,i for some 1 ≤ i ≤ k}.Note that by the use of relation R�=, all occurrences of a symbol xi in t are assigned the same state, sothe cardinality of X is exactly k. Moreover, for all i, j ∈ k such that i < j, we have r(ψ(i, j)) = qi,j,which implies that {vi, vj} ∈ E. Hence, the set X is a clique in G. �

Proposition 4.17 For TAGED, the k-EMPTINESS problem parameterized by |�| + |Q| or by|Q| is W[1]-hard, while it is W[2]-hard when parameterized by |�|.Proof The statement follows from the proofs of Theorems 4.12 and 4.10. It is enough to observethat the automata constructed there accept terms of only one particular size. �

4.3.3 MEMBERSHIP

Proposition 4.18 For TAGED, the MEMBERSHIP problem parameterized by |t| or by |t| + |�|is W[1]-complete.

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Proof The statement follows from the proofs of Theorems 4.14 (upper bound) and 4.16 (lowerbound). �

Corollary 4.19 For TAGED, the MEMBERSHIP problem parameterized by |t|, |t| + |Q| or by|t| + |�| + |Q| belongs to W[1].Theorem 4.20 For TAGED, the MEMBERSHIP problem parameterized by |�| is W[2]-hard.

Proof In order to show W[2]-hardness of the problem, we propose a reduction of p-DOMINATING-SET, which is a W[2]-complete problem. Consider an instance of the p-DOMINATING-SET problem: a graph G = (V , E) of n vertices v1, v2, . . . , vn and a number k ∈ N.We construct a TAGED A = 〈�, Q, R=, R�=, F, δ〉 and a term t over the signature � such that inG there is a k-dominating set if and only if A accepts the term t. Let � = {x1, x2, . . . , xk} ∪ {f , g},where xi are constant symbols, f is a k-ary function symbol and g is an n-ary function sym-bol. Let Q = {q1, q2, . . . , qn, p1, p2, . . . , pn, q, qF}, F = {qF}, R= = {(q1, q1), . . . , (qn, qn)} andR�= = {(qi, qj) | i, j ∈ n, i �= j}. Let δ consist of the following transitions:

(1) xi → q | qj for i ∈ k and j ∈ n,(2) f (q, . . . , q, qi, q, . . . , q) → pj for i, j ∈ n, if j = i or {vi, vj} ∈ E,(3) g(p1, . . . , pn) → qF .

Let t = g(f (x1, . . . , xk), . . . , f (x1, . . . , xk)). The size of the automaton A and the size of theterm t depend polynomially on k and on the size of the graph G. Moreover, |�| = k + 2.

It is enough to show that in G there is k-dominating set if and only if A accepts the term t.From the definition of the relations R= and R �=, we have that for each i all subterms labelled by xi

are mapped on the same state qj by a run of A . Moreover, for different i, states qj are different.Thus, a run of A on the term t determines a set X of k-vertices of G corresponding to k states qj

that are images of terms xi. Now, the application of the rule (3) to the term t checks if all verticeshave their witnesses in the set X . Thus, X is a dominating set. Hence, runs of A on t correspondto dominating sets in G. �

Theorem 4.21 For TAGED, the MEMBERSHIP problem parameterized by |Q| is para-NP-complete.

Proof The fact that the examined problem belongs to the para-NP class is quite obvious. Indeed,it is solvable by the algorithm that for each vertex of a given term t guesses a state from Q andthen checks in polynomial time if this mapping is an accepting run of the automaton on t.

In order to show para-NP-hardness of the problem, we propose a reduction of p-COLORABILITY, which is a para-NP-complete problem. Consider an instance of this problem:a graph G = (V , E) with n vertices v1, v2, . . . , vn and m edges e1, e2, . . . , em and a number k ∈ N.We construct a TAGED A = 〈�, Q, ∅, R�=, F, δ〉 and a term t over the signature � such that Gis k-colourable if and only if A accepts t. Let � = {v1, v2, . . . , vn, e1, e2, . . . , em, f }, where vi

is a constant symbol, ei is a symbol of a binary function and f is an m-ary function symbol.Let Q = {q1, q2, . . . , qk , q, qF}, R �= = {(qi, qj) | i, j ∈ k, i �= j} and F = {qF}. States q1, q2, . . . , qk

symbolize k colours. Let δ consist of the following transitions:

(1) v → qi for i ∈ k and v ∈ V ,(2) el(qi, qj) → q for l ∈ m and 1 ≤ i �= j ≤ k,(3) f (q, q, . . . , q) → qF .

Let t = f (e1(v11, v12), e2(v21, v22), . . . , em(vm1, vm2)), where vi1 and vi2 are the vertices incidentto the edge ei. The size of the automaton A and the size of the term t depend polynomially on kand on the size of the graph G. Moreover |Q| = k + 2.

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It is enough to show that G is k-colourable if and only if A accepts the term t. Suppose thatα : V → k is a colouring of G. We construct an accepting run r of A on t as follows:

• r(p) = qα(v) for p ∈ Pv,• r(i) = q (as α is a colouring of G, neighbouring vertices must have different colours),• r(ε) = qF ,

where Pv is the set of positions in the term t taken by the symbol v.Conversely, let r be a successful run of A on t. Let ji be such that r(vi) = qji and let α(vi) = ji.

Note that by the use of relation R�=, all occurrences of a symbol vi in t are assigned the same state,so the function α is well defined. We show that α is a colouring of G. The run r is successful, sofor every ei ∈ E, we have r(ei(vi1, vi2)) = q. This means that r(vi1) �= r(vi2), so α(vi1) �= α(vi2).Thus, the function α is a colouring of G. �

4.4 Automata on DAG representations of finite trees (t-DAG automata)

4.4.1 NONEMPTINESS

Theorem 4.22 For t-DAG automata, the NONEMPTINESS problem parameterized by |�| isW[1]-hard.

Proof We propose a reduction of p-CLIQUE. Consider an instance of the p-CLIQUE problem: agraph G = (V , E) with V = {v1, v2, . . . , vn} and a number k ∈ N. We construct a t-DAG automatonA = 〈�, Q, F, δ〉 such that in G there is a clique of k elements if and only if the languagerecognized by A is nonempty.

Let � = {x1, . . . , xk} ∪ {h, f , g}, where xi are constant symbols, h is a symbol of a unary func-tion, f is a symbol of a k-ary function and g is a symbol of a 1

2 (k − 1)k-ary function. LetQ = {q1, q1

1, . . . , q1n, . . . , qk , qk

1, . . . , qkn, qF}, F = {qF}. Let δ consist of the following transitions:

(1) xi → qij for i ∈ k, j ∈ n,

(2) h(qij) → qi for i ∈ k, j ∈ n,

(3) f (q1, . . . , qi−1, qiu, qi+1 . . . , qj−1, qj

w, qj+1 . . . , qk) → qi,j for i, j ∈ k, i < j, u, w ∈ n, if {vu,vw} ∈ E,

(4) g(q1,2, q1,3, . . . , qk−1,k) → qF .

The size of the automaton A depends polynomially on the size of the graph G and the numberk. Moreover, |�| = k + 2. Thus, it is enough to show that in G there is a clique of k elementsif and only if the language recognized by A is nonempty. Note that the only t-dag that can beaccepted by A is

t = g(f (x1, x2, h(x3), . . . , h(xk)), . . . , f (h(x1), . . . , h(xk−2), xk−1, xk)).

Intuitively, the automaton guesses a set of k vertices in G and checks that this set forms a clique. Astate qi

j is used to encode the choice of the vertex vj as the ith element of the clique. The applicationof the rule (3) to the term t checks if there are edges between all chosen vertices. Hence, runs ofA on t correspond to cliques in G. �

Note that the construction above does not work for classical TA: a run of a TA could assigndifferent states to different occurrences of xi in t which would make the guess of the cliqueinconsistent.

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Theorem 4.23 For t-DAG automata, the NONEMPTINESS problem parameterized by |Q|belongs to W[1].

Proof In [4], it was proved that if a language recognized by a t-DAG automaton A is nonempty,then A accepts a t-dag of size not greater than 2|Q|3, where Q is the set of states of A . Then, thetheorem can be proved using the same method as in the proof of Theorem 4.14. �

Proposition 4.24 For t-DAG automata, the NONEMPTINESS problem parameterized by |Q| +|�| belongs to FPT.

Proof It is enough to check, for every t-dag of size not greater than 2|Q|3, if it is accepted by A .One can do this by analysing all possible runs of the automaton on this term. There are O(|�|2|Q|3)such terms and there are O(|Q|2|Q|3) possible runs on each of them. �

4.4.2 k-EMPTINESS

Theorem 4.25 For t-DAG automata, the k-EMPTINESS problem parameterized by k is W[1]-complete.

Proof The hardness part of the theorem is a corollary from Theorem 4.28. The fact that theproblem belongs to W[1] can be proved similarly as in Theorem 4.14. �

Corollary 4.26 For t-DAG automata, the k-EMPTINESS problem parameterized by k + |Q|or by k + |�| + |Q| is in W[1].

The following proposition is a consequence of the proof of Theorem 4.22.

Proposition 4.27 For t-DAG automata, the k-EMPTINESS problem parameterized by |�| isW[1]-hard.

Theorem 4.28 For t-DAG automata, the k-EMPTINESS problem parameterized by k + |�| isW[1]-complete.

Proof The proof of the fact that the examined problem is in W[1] is a consequence ofTheorem 4.25. The proof of W[1]-hardness follows the lines of the proof of Theorem 4.16. Fora given instance of the p-CLIQUE problem, we construct a t-DAG automaton A = 〈�, Q, F, δ〉,where �, Q, F, δ are as in the proof of Theorem 4.16. A minor difference is that the size off (g(x1, x2), g(x1, x3), . . . , g(xk−1, xk)) as a t-dag is 1 + 1

2 k(k + 1) (there is one node labelled f ,12 k(k − 1) nodes labelled g and k variables), so we define k′ = 1 + 1

2 k(k + 1). The only essentialdifference is at the end of the proof: the condition that all occurrences of a symbol xi in t areassigned the same state is satisfied just by the definition of a run on a t-dag and not by using therelation R�= as it was in the case of TAGED. �

4.4.3 MEMBERSHIP

Theorem 4.29 For t-DAG automata, the MEMBERSHIP problem parameterized by |�| is W[1]-hard.

Proof The fact follows directly from the proof of Theorem 4.22. �

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Theorem 4.30 For t-DAG automata, the MEMBERSHIP problem parameterized by |t| or by|t| + |�| is W[1]-complete.

Proof The proof of W[1]-hardness of the problem is the same as the proof of W[1]-hardness ofthe k-EMPTINESS problem (see Theorem 4.28). The proof of the membership in W[1] is similarto the proof of Theorem 4.16. �

Corollary 4.31 For t-DAG automata, the MEMBERSHIP problem parameterized by |t| + |Q|or by |t| + |�| + |Q| is in W[1].

Theorem 4.32 For t-DAG automata, the MEMBERSHIP problem parameterized by |Q| is para-NP-complete.

Proof The proof follows the lines of the proof of Theorem 4.21. The essential difference betweenthis proof and the proof of Theorem 4.21 is the same as between the proofs of Theorems 4.28and 4.16. �

5. Conclusion

We have studied parameterized complexity of several decision problems for the following classesof automata on finite trees: TA, RTA, TAGED and t-DAG automata. The results of our study

Table 2. Summary of our results.

Problem Parameter TA RTA TAGED t-DAG

NONEMPTINESS |�| PTIME PTIME W[2]-hard W[1]-hard|Q| W[1]-hard W[1]|�| + |Q| W[1]-hard FPT

k-EMPTINESS k FPT W[1] W[1]-complete W[1]-complete|�| W[2]-hard W[1]-hard|Q| W[1]-hardk + |�| FPT W[1] W[1]-complete W[1]-completek + |Q| FPT W[1] W[1] W[1]|�| + |Q| W[1]-hardk + |�| + |Q| FPT W[1] W[1] W[1]

MEMBERSHIP |t| PTIME W[1] W[1]-complete W[1]-complete|�| W[2]-hard W[1]-hard|Q| W[2]-hard Para-NP-complete Para-NP-complete|t| + |�| W[1] W[1]-complete W[1]-complete|t| + |Q| W[1] W[1] W[1]|�| + |Q||t| + |�| + |Q| W[1] W[1] W[1]

UNIVERSALITY |�| Para-co-NP-hard Undecidable Undecidable Undecidable|Q| Para-co-NP-hard|�| + |Q| Para-co-NP-hard

INCLUSION |�| Para-co-NP-hard Undecidable Undecidable Undecidable|Q1| Para-co-NP-hard|Q2| Para-co-NP-hard|Q1| + |Q2| Para-co-NP-hard|A1| Para-co-NP-hard|A2| FPT

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are presented in Table 2 (some of the results are taken from Table 1). These results were quitesurprising for us – since for TA the EMPTINESS problem and the MEMBERSHIP problem canbe solved in polynomial time, we had expected to obtain more FPT results for other classes ofTA. However, a lot of the examined problems turn to be hard.

As one can see there are still gaps in the presented table. Moreover, some cases are only partiallyexamined as we have only proved that they belong to some complexity class or that they are hardin that class. In consequence, there are open questions that we hope to examine in the future.

Acknowledgements

Research partially supported by Polish NCN grant number DEC-2011/03/B/ST6/00346.

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