Decidable Emptiness Research Articles

Variations of checking stack automata: Obtaining unexpected decidability properties

We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties:1.The deterministic version has a decidable membership problem but has an undecidable emptiness problem.2.The nondeterministic version has an undecidable membership problem and emptiness problem. There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, i.e., the membership problem for both versions are either decidable or undecidable, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.

Ordered multi-stack visibly pushdown automata

Visibly Pushdown Automata (VPA) are a special case of pushdown machines where the stack operations are driven by the input. In this paper, we consider VPA with multiple stacks, namely n-VPA, with n>1. These automata introduce a useful model to effectively describe concurrent pushdown systems using a simple communication mechanism between stacks. We show that n-VPA are strictly more expressive than VPA. Indeed, n-VPA accept some context-sensitive languages that are not context-free and some context-free languages that are not accepted by any VPA. Nevertheless, we show that the class of languages accepted by n-VPA is closed under union and intersection. On the contrary, this class turns out to be neither closed under complementation nor determinizable. Moreover, we show that the emptiness problem for n-VPA is undecidable. By adding an ordering constraint on stacks (n-OVPA), decidability of emptiness can be recovered, as well as the closure under complementation. Using these properties along with the automata-theoretic approach, one can prove that the model checking problem over n-OVPA models against n-OVPA specifications is decidable.

Quantified data automata for linear data structures: a register automaton model with applications to learning invariants of programs manipulating arrays and lists

We propose a new automaton model, called quantified data automata (QDA) over words, that can model quantified invariants over linear data structures, and study their theory, including closure properties, canonical minimality, and decidability of emptiness. We build poly-time active learning algorithms for them, where the learner is allowed to query the teacher with membership and equivalence queries. In order to express invariants in decidable logics, we invent a decidable subclass of QDAs, called elastic QDAs, and show translations to decidable theories of arrays and lists. We also prove that every QDA has a unique minimally-over-approximating elastic QDA, showing a robust technique for abstracting QDA-expressible properties to the decidable fragments expressed by elastic QDAs. We then give an application of these theoretically sound and efficient active learning algorithms to program verification by building a passive learning framework that efficiently learns adequate quantified linear data structure invariants from samples obtained from dynamic executions for a class of programs.

Walking on Data Words

Data words are words with additional edges that connect pairs of positions carrying the same data value. We consider a natural model of automaton walking on data words, called Data Walking Automaton, and study its closure properties, expressiveness, and the complexity of some basic decision problems. Specifically, we show that the class of deterministic Data Walking Automata is closed under all Boolean operations, and that the class of non-deterministic Data Walking Automata has decidable emptiness, universality, and containment problems. We also prove that deterministic Data Walking Automata are strictly less expressive than non-deterministic Data Walking Automata, which in turn are captured by Class Memory Automata.

Randomization in Automata on Infinite Trees

We study finite automata running over infinite binary trees. A run of such an automaton over an input tree is a tree labeled by control states of the automaton: the labeling is built in a top-down fashion and should be consistent with the transitions of the automaton. A branch in a run is accepting if the ω-word obtained by reading the states along the branch satisfies some acceptance condition (typically an ω-regular condition such as a Büchi or a parity condition). Finally, a tree is accepted by the automaton if there exists a run over this tree in which every branch is accepting. In this article, we consider two relaxations of this definition, introducing a qualitative aspect. First, we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of nonaccepting branches. In this qualitative setting, a tree is accepted by the automaton if there exists a run over this tree in which almost every branch is accepting. This leads to a new class of tree languages, qualitative tree languages . This class enjoys many good properties: closure under union and intersection (but not under complement), and emptiness is decidable in polynomial time. A dual class, positive tree languages , is defined by requiring that an accepting run contains a non-negligeable set of branches. The second relaxation is to replace the existential quantification (a tree is accepted if there exists some accepting run over the input tree) with a probabilistic quantification (a tree is accepted if almost every run over the input tree is accepting). For the run, we may use either classical acceptance or qualitative acceptance. In particular, for the latter, we exhibit a tight connection with partial observation Markov decision processes. Moreover, if we additionally restrict operation to the Büchi condition, we show that it leads to a class of probabilistic automata on infinite trees enjoying a decidable emptiness problem. To our knowledge, this is the first positive result for a class of probabilistic automaton over infinite trees.

Decidability of Downward XPath

We investigate the satisfiability problem for downward-XPath, the fragment of XPath that includes the child and descendant axes, and tests for (in)equality of attributes’ values. We prove that this problem is decidable, EXPTIME-complete. These bounds also hold when path expressions allow closure under the Kleene star operator. To obtain these results, we introduce a Downward Data automata model (DD automata) over trees with data, which has a decidable emptiness problem. Satisfiability of downward-XPath can be reduced to the emptiness problem of DD automata and hence its decidability follows. Although downward-XPath does not include any horizontal axis, DD automata are more expressive and can perform some horizontal tests. Thus, we show that the satisfiability remains in EXPTIME even in the presence of the regular constraints expressible by DD automata. However, the same problem in the presence of any regular constraint is known to have a nonprimitive recursive complexity. Finally, we give the exact complexity of the satisfiability problem for several fragments of downward-XPath.

On pebble automata for data languages with decidable emptiness problem

In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we show that the emptiness problem for weak 2-pebble automata is decidable, while the same problem for weak 3-pebble automata is undecidable. We also introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is still decidable. It is also robust: alternating, non-deterministic and deterministic top view weak PA have the same recognition power; and are strong enough to accept all data languages expressible in Linear Temporal Logic with the future-time operators, augmented with one register freeze quantifier.

On the Expressive Power of 2-Stack Visibly Pushdown Automata

Visibly pushdown automata are input-driven pushdown automata that recognize some non-regular context-free languages while preserving the nice closure and decidability properties of finite automata. Visibly pushdown automata with multiple stacks have been considered recently by La Torre, Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express properties beyond the class of context-free languages. At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure. These results require a restricted version of visibly pushdown automata with multiple stacks whose behavior can be split up into a fixed number of phases. In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form. We show that they are expressively equivalent to the existential fragment of monadic second-order logic. Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt words with multiple nestings. Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation. Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown automata running on infinite (nested) words. Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic.

Reachability in Petri Nets with Inhibitor Arcs

We define 2 operators on relations over natural numbers such that they generalize the operators ‘+’ and ‘*’ and show that the membership and emptiness problem of relations constructed from finite relations with these operators and ∪ is decidable. This generalizes Presburger arithmetics and allows to decide the reachability problem for those Petri nets where inhibitor arcs occur only in some restricted way. Especially the reachability problem is decidable for Petri nets with only one inhibitor arc, which solves an open problem in [H. Kleine Büning, T. Lettmann, and E. W. Mayr. Projections of vector addition system reachability sets are semilinear. Theoret. Comput. Sci., 64:343–350, 1989]. Furthermore we describe the corresponding automaton having a decidable emptiness problem.

Alternating timed automata

A notion of alternating timed automata is proposed. It is shown that such automata with only one clock have decidable emptiness problem over finite words. This gives a new class of timed languages that is closed under boolean operations and which has an effective presentation. We prove that the complexity of the emptiness problem for alternating timed automata with one clock is nonprimitive recursive. The proof gives also the same lower bound for the universality problem for nondeterministic timed automata with one clock. We investigate extension of the model with epsilon-transitions and prove that emptiness is undecidable. Over infinite words, we show undecidability of the universality problem.

RELATING TREE SERIES TRANSDUCERS AND WEIGHTED TREE AUTOMATA

Bottom-up tree series transducers (tst) over the semiring [Formula: see text] are implemented with the help of bottom-up weighted tree automata (wta) over an extension of [Formula: see text]. Therefore bottom-up [Formula: see text]-weighted tree automata ([Formula: see text]-wta) with [Formula: see text] a distributive Ω-algebra are introduced. A [Formula: see text]-wta is essentially a wta but uses as transition weight an operation symbol of the Ω-algebra [Formula: see text] instead of a semiring element. The given tst is implemented with the help of a [Formula: see text]-wta, essentially showing that [Formula: see text]-wta are a joint generalization of tst (using IO-substitution) and wta. Then a semiring and a wta are constructed such that the wta computes a formal representation of the semantics of the [Formula: see text]-wta. The applicability of the obtained presentation result is demonstrated by deriving a pumping lemma for deterministic finite [Formula: see text]-wta from a known pumping lemma for deterministic finite wta. Finally, it is observed that the known decidability results for emptiness cannot be applied to obtain decidability of emptiness for finite [Formula: see text]-wta. Thus with help of a weaker version of the derived pumping lemma, decidability of the emptiness problem for finite [Formula: see text]-wta is shown under mild conditions on [Formula: see text].

Counter Machines and Verification Problems

We study various generalizations of reversal-bounded multicounter machines and show that they have decidable emptiness, infiniteness, disjointness, containment, and equivalence problems. The extensions include allowing the machines to perform linear-relation tests among the counters and parameterized constants (e.g., “Is 3x−5y−2D 1+9D 2<12 ?”, where x,y are counters, and D 1,D 2 are parameterized constants). We believe that these machines are the most powerful machines known to date for which these decision problems are decidable. Decidability results for such machines are useful in the analysis of reachability problems and the verification/debugging of safety properties in infinite-state transition systems. For example, we show that (binary, forward, and backward) reachability and safety are solvable for these machines.

Augmenting the discrete timed automaton with other data structures

We describe a general automata-theoretic approach for analyzing the verification problems of discrete timed automata (i.e., timed automata with integer-valued clocks) augmented with various data structures. Formally, let C be a class of nondeterministic machines with reversal-bounded counters and possibly other data structures (e.g., a pushdown stack, a queue, a read-write worktape, etc.). Let A be a discrete timed automaton and M be a machine in C . Denote by A⊕M the combined automaton, i.e., A augmented with M (in some precise sense to be defined). We show that if C has a decidable emptiness problem, then the (binary, forward, backward) reachability, safety, and invariance for A⊕M are solvable. We give examples of such C 's and exhibit some new properties of discrete timed automata that can be verified. We also briefly consider reachability in discrete timed automata operating in parallel.

Systolic tree [formula omitted]-languages: the operational and the logical view

The class of ω-languages recognized by systolic (binary) tree automata is introduced. This class extends the class of Büchi ω-languages though maintaining the closure under union, intersection and complement and the decidability of emptiness. The class of systolic tree ω-languages is characterized in terms of a (suitable) concatenation of (finitary) systolic tree languages. A generalization of Buc̈hi Theorem is provided which establishes a correspondence between systolic tree ω-languages and a suitable extension of the sequential calculus S1S.

On quasi orders of words and the confluence property

We investigate the confluence property, that is, the property of a language to contain, for any two words of it, one which is bigger, with respect to a given quasi order on the respective free monoid, than each of the former two. This property is investigated mainly for regular and context-free languages. As a consequence of our study, we give an answer to an old open problem raised by Haines concerning the effective regularity of the sets of subwords. Namely, we prove that there are families with a decidable emptiness problem for which the regularity of the sets of subwords is not effective.

ON THE EQUIVALENCE OF TWO-WAY PUSHDOWN AUTOMATA AND COUNTER MACHINES OVER BOUNDED LANGUAGES

It is known that two-way pushdown automata are more powerful than two-way counter machines. The result is also true for the case when the pushdown store and counter are reversal-bounded. In contrast, we show that two-way reversal-bounded push-down automata over bounded languages (i.e., subsets of [Formula: see text] for some distinct symbols a1,…, ak) are equivalent to two-way reversal-bounded counter machines. We also show that, unlike the unbounded input case, two-way reversal-bounded pushdown automata over bounded languages have decidable emptiness, equivalence and containment problems.

R70-2 Nested Stack Automata

A nested stack automaton is a generalization of the pushdown automaton. Basically, the nested stack automaton consists of an input tape, a finite control, and a single pushdown list. However, the nested stack automaton can access symbols in the interior of the stack in a read-only mode and create new stacks nested (to arbitrary depths) within the main stack, subject to the restriction that the stack head may not move up a stack without first having destroyed all stacks created at that level. The importance of the model is that the class of languages accepted is precisely the indexed languages. The indexed languages have most of the properties of the context- free languages, i.e., derivation trees, recursiveness, decidable emptiness problem, closed under concatenation, Kleene closure, homomorphisms, inverse homomorphisms, and intersection with regular sets. Furthermore, the indexed grammars are capable of exhibiting syntactic features in algorithmic programming languages not representable by context-free grammars.

Full AFLs and nested iterated substitution

A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set. Every superAFL is a full AFL containing all context-free languages. If L is a full principal AFL, then Ŝ∞(L), the least superAFL containing L, is full principal. If L is not substitution closed, the substitution closure of L is properly contained in Ŝ∞(L). The indexed languages form a superAFL which is not the least superAFL containing the one-way stack languages. If L has a decidable emptiness problem, so does Ŝ∞(L). IfDs is an AFA, L = L(Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L(Ds) = Ŝ∞(L) if, and only if, Ds is nontrivial. If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested.