Non-emptiness Problem Research Articles

Complexity of the emptiness problem for graph-walking automata and for tilings with star subgraphs

This paper proves the decidability of the emptiness problem for two models which recognize finite graphs: graph-walking automata, and tilings of graphs by star subgraphs (star automata). Furthermore, it is proved that the non-emptiness problem for graph-walking automata (that is, whether a given automaton accepts at least one graph) is NEXP-complete. For star automata, which generalize nondeterministic tree automata to the case of graphs, it is proved that their non-emptiness problem is NP-complete.

Complexity issues for timeline-based planning over dense time under future and minimal semantics

The problem of timeline-based planning (TP) over dense temporal domains is known to be undecidable in the general case. We first prove that the restriction to the future semantics does not suffice to recover decidability. Then, we introduce two semantic variants of TP, called strong minimal and weak minimal semantics, and show that they allow one to express meaningful properties. Both semantics are based on the minimality in the time distances of the existentially-quantified time events from the universally-quantified reference one, but the weak minimal variant distinguishes minimality in the past from minimality in the future. Surprisingly, we show that, despite the (apparently) small differences between the two semantics, the TP problem is still undecidable for the strong minimal one, while it is PSPACE-complete for the weak minimal one. Membership in PSPACE is determined by exploiting a strictly more expressive extension (ECA+) of the well-known robust class of Event-Clock Automata (ECA), that allows us to encode the weak minimal TP problem and to reduce it to non-emptiness of Timed Automata (TA). Finally, an extension of ECA+(ECA++) is considered, proving that its non-emptiness problem is undecidable. We believe that the two extensions of ECA (ECA+ and ECA++), introduced for technical reasons, are actually valuable per sé in the field of TA.1

Nonemptiness Problems of Wang Cubes with Two Colors

This investigation studies the nonemptiness problems of Wang cubes with two colors. Wang cubes are unit cubes with colored faces, which are generalized from Wang tiles. For a set $B$ of Wang cubs, $\Sigma(B)$ is the set of all global patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. The nonemptiness problem is to determine whether $\Sigma(B) \neq \emptyset$ or not. Denote by $\mathcal{P}(B)$ the set of all periodic patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. For Wang cubes, the corresponding Wang's conjecture is that if $\Sigma(B) \neq \emptyset$, then $\mathcal{P}(B) \neq \emptyset$. We introduce the transition matrices and trace operators to determine whether $\Sigma(B) \neq \emptyset$ and $\mathcal{P}(B) \neq \emptyset$ or not, respectively. A basic set $B$ is called a minimal cycle generator if $\mathcal{P}(B) \neq \emptyset$ but $\mathcal{P}(B') = \emptyset$ for all $B' \subsetneqq B$. By computer computation, there exist $86$ equivalence classes of minimal cycle generators with two colors. By verifying that the basic sets $B$ that contains no minimal cycle generators satisfy $\Sigma(B) = \emptyset$, we prove that the Wang's conjecture for Wang cubes with two colors is true.

Binary Reachability of Timed-register Pushdown Automata and Branching Vector Addition Systems

Timed-register pushdown automata constitute a very expressive class of automata, whose transitions may involve state, input, and top-of-stack timed registers with unbounded differences. They strictly subsume pushdown timed automata of Bouajjani et al., dense-timed pushdown automata of Abdulla et al., and orbit-finite timed-register pushdown automata of Clemente and Lasota. We give an effective logical characterisation of the reachability relation of timed-register pushdown automata. As a corollary, we obtain a doubly exponential time procedure for the non-emptiness problem. We show that the complexity reduces to singly exponential under the assumption of monotonic time. The proofs involve a novel model of one-dimensional integer branching vector addition systems with states. As a result interesting on its own, we show that reachability sets of the latter model are semilinear and computable in exponential time.

Circuits and Expressions over Finite Semirings

The computational complexity of the circuit and expression evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or a multiplicative identity. The following dichotomy is shown: If a finite semiring is such that (i) the multiplicative semigroup is solvable and (ii) it does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 ≠ 0, then the circuit evaluation problem is in DET ⊆ NC 2 , and the expression evaluation problem for the semiring is in TC 0 . For all other finite semirings, the circuit evaluation problem is P-complete and the expression evaluation problem is NC 1 -complete. As an application, we determine the complexity of intersection non-emptiness problems for given context-free grammars (regular expressions) with a fixed regular language.

Simple picture processing based on finite automata and regular grammars

We are introducing and discussing finite automata working on rectangular-shaped arrays (i.e., pictures) in a boustrophedon reading mode. We prove close relationships with the well-established class of regular matrix (picture) languages. We derive several combinatorial, algebraic and decidability results for the corresponding class of picture languages. For instance, we show pumping and interchange lemmas for our picture language class. We also explain similarities and differences to the status of decidability questions for classical finite string automata. For instance, the non-emptiness problem for our picture-processing automaton model turns out to be NP-complete. Finally, we sketch possible applications to character recognition.

Finite Automata Over Infinite Alphabets: Two Models with Transitions for Local Change

Two models of automata over infinite alphabets are presented, mainly with a focus on the alphabet [Formula: see text]. In the first model, transitions can refer to logic formulas that connect properties of successive letters. In the second, the letters are considered as columns of a labeled grid which an automaton traverses column by column. Thus, both models focus on the comparison of successive letters, i.e. “local changes”. We prove closure (and non-closure) properties, show the decidability of the respective non-emptiness problems, prove limits on decidability results for extended models, and discuss open issues in the development of a generalized theory.

Parametrized automata simulation and application to service composition

The service composition problem asks whether, given a client and a community of available services, there exists an agent (called the mediator) that suitably delegates the actions requested by the client to the available community of services. We address this problem in a general setting where the agents communication actions are parametrized by data from an infinite domain and possibly subject to constraints. For this purpose, we define parametrized automata (PAs), where transitions are guarded by conjunction of equalities and disequalities. We solve the service composition problem by showing that the simulation preorder of PAs is decidable and devising a procedure to synthesize a mediator out of a simulation preorder. We also show that the Nonemptiness problem of PAs is PSPACE-complete.

The complexity of intersecting finite automata having few final states

The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be $${\oplus}$$źL-complete or NP-complete according to whether a nontrivial monoid other than a direct product of cyclic groups of order 2 is allowed in the automata. We further consider idempotent commutative automata and (Abelian, mainly) group automata with one, two, or three final states over a singleton or larger alphabet, elucidating (under the usual hypotheses on complexity classes) the complexity of the intersection nonemptiness and related problems in each case.

Nonemptiness problems of Wang tiles with three colors

This investigation studies nonemptiness problems of plane edge coloring with three colors. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges that have one of p colors are arranged side by side such that the touching edges of the adjacent tiles have the same colors. Given a basic set B of Wang tiles, the nonemptiness problem is to determine whether or not Σ(B)≠∅, where Σ(B) is the set of all global patterns on Z2 that can be constructed from the Wang tiles in B. Wang's conjecture is that for any B of Wang tiles, Σ(B)≠∅ if and only if P(B)≠∅, where P(B) is the set of all periodic patterns on Z2 that can be generated by the tiles in B.When p≥5, Wang's conjecture is known to be wrong. When p=2, the conjecture is true. This study proves that when p=3, the conjecture is also true. If P(B)≠∅, then B has a subset B′ of minimal cycle generators such that P(B′)≠∅ and P(B″)=∅ for B″⫋B′. This study demonstrates that the set C(3) of all minimal cycle generators contains 787,605 members that can be classified into 2,906 equivalence classes. N(3) is the set of all maximal non-cycle generators: if B∈N(3), then P(B)=∅ and P(B˜)≠∅ for B˜⫌B. Wang's conjecture is shown to be true by proving that B∈N(3) implies Σ(B)=∅.

Extending two-variable logic on data trees with order on data values and its automata

Data trees are trees in which each node, besides carrying a label from a finite alphabet, also carries a data value from an infinite domain. They have been used as an abstraction model for reasoning tasks on XML and verification. However, most existing approaches consider the case where only equality test can be performed on the data values. In this article we study data trees in which the data values come from a linearly ordered domain, and in addition to equality test, we can test whether the data value in a node is greater than the one in another node. We introduce an automata model for them which we call ordered-data tree automata (ODTA), provide its logical characterisation, and prove that its non-emptiness problem is decidable in 3-NE xp T ime . We also show that the two-variable logic on unranked data trees, studied by Bojanczyk et al. [2009], corresponds precisely to a special subclass of this automata model. Then we define a slightly weaker version of ODTA, which we call weak ODTA , and provide its logical characterisation. The complexity of the non-emptiness problem drops to NP. However, a number of existing formalisms and models studied in the literature can be captured already by weak ODTA. We also show that the definition of ODTA can be easily modified, to the case where the data values come from a tree-like partially ordered domain, such as strings.

On expressive power of regular realizability problems

A regular realizability (RR) problem is a problem of testing nonemptiness of intersection of some fixed language (filter) with a regular language. We show that RR problems are universal in the following sense. For any language L there exists an RR problem equivalent to L under disjunctive reductions in nondeterministic log space. From this result, we derive existence of complete problems under polynomial reductions for many complexity classes, including all classes of the polynomial hierarchy.

Efficient emptiness check for timed Büchi automata

The B\"uchi non-emptiness problem for timed automata refers to deciding if a given automaton has an infinite non-Zeno run satisfying the B\"uchi accepting condition. The standard solution to this problem involves adding an auxiliary clock to take care of the non-Zenoness. In this paper, it is shown that this simple transformation may sometimes result in an exponential blowup. A construction avoiding this blowup is proposed. It is also shown that in many cases, non-Zenoness can be ascertained without extra construction. An on-the-fly algorithm for the non-emptiness problem, using non-Zenoness construction only when required, is proposed. Experiments carried out with a prototype implementation of the algorithm are reported.

Nonemptiness problems of plane square tiling with two colors

This investigation studies nonemptiness problems of plane square tiling. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges of p colors are arranged side by side such that adjacent tiles have the same colors. Given a set of Wang tiles B, the nonemptiness problem is to determine whether or not Σ(B) ≠ 0, where Σ(B) is the set of all global patterns on ℤ 2 that can be constructed from the Wang tiles in B. When p > 5, the problem is well known to be undecidable. This work proves that when p = 2, the problem is decidable. is the set of all periodic patterns on ℤ 2 that can be generated by B. If ≠ 0, then has a subset B' of minimal cycle generator such that P(B') ≠ 0 and P(B) = 0 for B ⊂ ≠ B'. This study demonstrates that the set of all minimal cycle generators C(2) contains 38 elements. N(2) is the set of all maximal noncycle generators: if ∈ N(2), then = 0 and ⊃ ≠ implies ≠ O. N(2) has eight elements. That Σ (B) = 0 for any ∈ N(2) is proven, implying that if Σ (B) ≠ 0, then ≠ 0. The problem is decidable for p = 2: Σ(B) ≠ 0 if and only if has a subset of minimal cycle generators. The approach can be applied to corner coloring with a slight modification, and similar results hold.

The Effects of Bounding Syntactic Resources on Presburger LTL

LTL over Presburger constraints is the extension of LTL where the atomic formulae are quantifier-free Presburger formulae having as free variables the counters at different states of the model. This logic is known to admit undecidable satisfiability and model-checking problems. We study decidability and complexity issues for fragments of LTL with Presburger constraints obtained by restricting the syntactic resources of the formulae (the number of variables, the maximal distance between two states for which counters can be compared and, to a smaller extent, the set of Presburger constraints), while preserving the strength of the logical operators. We provide a complete picture refining known results from the literature. We show that model-checking and satisfiability problems for the fragments of LTL with difference constraints restricted to two variables and distance one and to one variable and distance two are highly undecidable, enlarging significantly the class of known undecidable fragments. On the positive side, we prove that the fragment restricted to one variable and to distance one augmented with propositional variables is pspace-complete. Since the atomic formulae can state quantitative properties on the counters, this extends some results about model-checking pushdown systems and one-counter automata. In order to establish the pspace upper bound, we show that the non-emptiness problem for Büchi one-counter automata taking values in ℤ and allowing zero tests and sign tests, is only nlogspace-complete. Finally, we establish that model-checking one-counter automata with complete quantifier-free Presburger LTL restricted to one variable is also pspace-complete, whereas the satisfiability problem is undecidable.

Strongly extreme points in Orlicz spaces equipped with the [formula omitted]-Amemiya norm

This paper is a continuation of the paper [Y. Cui, L. Duan, H. Hudzik, M. Wisła, Basic theory of p -Amemiya norm in Orlicz spaces ( 1 ≤ p ≤ ∞ ) : Extreme points and rotundity in Orlicz spaces equipped with these norms, Nonlinear Anal. 69 (2008) 1796–1816] and an extension of the paper [Y. Cui, T. Wang, Strongly extreme points of Orlicz space, J. Math. 4 (1987) 335-340]. Criteria for the Kadec–Klee property with respect to the convergence in measure and for strongly extreme points in Orlicz spaces L Φ , p equipped with the p -Amemiya norm ( 1 ≤ p ≤ ∞ ) are given. Orlicz spaces with nonempty (and empty) set of strongly extreme points of the unit ball B ( L Φ , p ) are characterized. It is interesting to note that for some Orlicz functions Φ there is a critical number p 0 such that the problem of emptiness or nonemptiness of strongly extreme points of the unit ball B ( L Φ , p ) looks different for 1 ≤ p ≤ p 0 and for p 0 < p ≤ ∞ . Criteria for the midpoint local uniform rotundity of the spaces L Φ , p are also deduced. Theorems stated in this paper, even restricted to the case p ∈ { 1 , ∞ } , are more general than the results of the latter reference mentioned above because the assumption lim u → ∞ Φ ( u ) / u = ∞ was not used. The methods used in the proofs are universal for all 1 ≤ p ≤ ∞ . New notions of p -spherical convergence and H 2 , μ -points are introduced and considered. Moreover, Orlicz spaces L Φ , p which are linearly isomorphic or isometric to L ∞ are characterized as auxiliary techniques and facts for some proofs.

Branching-Time Logics Repeatedly Referring to States

While classical temporal logics lose track of a state as soon as a temporal operator is applied, several branching-time logics able to repeatedly refer to a state have been introduced in the literature. We study such logics by introducing a new formalism, hybrid branching-time logics, subsuming the other approaches and making the ability to refer to a state more explicit by assigning a name to it. We analyze the expressive power of hybrid branching-time logics and the complexity of their satisfiability problem. As main result, the satisfiability problem for the hybrid versions of several branching-time logics is proved to be 2EXPTIME-complete. To prove the upper bound, the automata-theoretic approach to branching-time logics is extended to hybrid logics. As a result of independent interest, the nonemptiness problem for alternating one-pebble Buchi tree automata is shown to be 2EXPTIME-complete. A common property of the logics studied is that they refer to only one state. This restriction is crucial: The ability to refer to more than one state causes a nonelementary blow-up in complexity. In particular, we prove that satisfiability for NCTL* has nonelementary complexity.

The Complexity of Datalog on Linear Orders

We study the program complexity of datalog on both finite and infinite linear orders. Our main result states that on all linear orders with at least two elements, the nonemptiness problem for datalog is EXPTIME-complete. While containment of the nonemptiness problem in EXPTIME is known for finite linear orders and actually for arbitrary finite structures, it is not obvious for infinite linear orders. It sharply contrasts the situation on other infinite structures; for example, the datalog nonemptiness problem on an infinite successor structure is undecidable. We extend our upper bound results to infinite linear orders with constants. As an application, we show that the datalog nonemptiness problem on Allen's interval algebra is EXPTIME-complete.

Network Formulations of Mixed-Integer Programs

We consider mixed-integer sets described by system of linear inequalities in which the constraint matrix A is totally unimodular; the right-hand side is arbitrary vector; and a subset of the variables is required to be integer. We show that the problem of checking nonemptiness of a set of this type is NP-complete, even in the case in which the linear system describes mixed-integer network flows with half-integral requirement on the nodes. This is in contrast to the case in which A is totally unimodular and contains at most two nonzeros per row. In this case, we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual-network matrix with an integral right-hand-side vector. The size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set. When this number is polynomial in the dimension of A, the extended formulation is of polynomial size. If, in addition, the corresponding list of fractional parts can be computed efficiently, then our result provides a polynomial algorithm for the optimization problem over these sets. We show that there are instances for which this list is of exponential size, and we also give conditions under which it is short and can be efficiently computed. Finally, we show that these results provide a unified framework leading to polynomial-size extended formulations for several generalizations of mixing sets and lot-sizing sets studied in the last few years.

A continuousdiscontinuous second-order transition in the satisfiability of random Horn-SAT formulas

We compute the probability of satisfiability of a class of random Horn-SAT formulae, motivated by a connection with the nonemptiness problem of finite tree automata. In particular, when the maximum...