Equivalence Problem For Languages Research Articles

Decision problems and projection languages for restricted variants of two-dimensional automata

A two-dimensional automaton has a read-only input head that moves in four directions on a finite array of cells labeled by symbols of the input alphabet. A three-way two-dimensional automaton is prohibited from making upward moves, while a two-way two-dimensional automaton can only move downward and rightward.We show that the language emptiness problem for unary three-way nondeterministic two-dimensional automata is NP-complete, and is in P for general-alphabet two-way nondeterministic two-dimensional automata. We also show that the language equivalence and inclusion problems for two-way deterministic two-dimensional automata are decidable, while the language universality, equivalence, and inclusion problems for two-way nondeterministic two-dimensional automata are undecidable. The deterministic case is the first known positive decidability result for a language equivalence problem on two-dimensional automata over a general alphabet.Finally, we discuss the notion of row and column projection languages. We show that the row projection language of a unary three-way nondeterministic two-dimensional automaton is always regular, and that there exists a unary three-way deterministic two-dimensional automaton with a nonregular column projection language. For two-way nondeterministic two-dimensional automata, on the other hand, both the row and column projection languages are always regular.

Model-based mutant equivalence detection using automata language equivalence and simulations

Mutation analysis is a popular technique for assessing the strength of test suites. It relies on the mutation score, which indicates their fault-revealing potential. Yet, there are mutants whose behaviour is equivalent to the original system, wasting analysis resources and preventing the satisfaction of a 100% mutation score. For finite behavioural models, the Equivalent Mutant Problem (EMP) can be transformed to the language equivalence problem of non-deterministic finite automata for which many solutions exist. However, these solutions are quite expensive, making computation unbearable when used for tackling the EMP. In this paper, we report on our assessment of a state-of-the-art exact language equivalence tool and two heuristics we proposed. We used 12 models, composed of (up to) 15,000 states, and 4710 mutants. We introduce a random and a mutation-biased simulation heuristics, used as baselines for comparison. Our results show that the exact approach is often more than ten times faster in the weak mutation scenario. For strong mutation, our biased simulations can be up to 1000 times faster for models larger than 300 states, while limiting the error of misclassifying non-equivalent mutants as equivalent to 8% on average. We therefore conclude that the approaches can be combined for improved efficiency.

A new bound for the D0L language equivalence problem

We study the language equivalence problem for smooth and loop-free D0L systems. We show that the number of initial terms in the associated D0L sequences we have to consider to decide language equivalence depends only on the cardinality of the underlying alphabet.

Quantum Markov chains: Description of hybrid systems, decidability of equivalence, and model checking linear-time properties

In this paper, we study a model of quantum Markov chains that is a quantum analogue of Markov chains and is obtained by replacing probabilities in transition matrices with quantum operations. We show that this model is very suited to describe hybrid systems that consist of a quantum component and a classical one. Indeed, hybrid systems are often encountered in quantum information processing. Thus, we further propose a model called hybrid quantum automata (HQA) that can be used to describe the hybrid systems receiving inputs (actions) from the outer world. We show the language equivalence problem of HQA is decidable in polynomial time. Furthermore, we apply this result to the trace equivalence problem of quantum Markov chains, and thus it is also decidable in polynomial time. Finally, we discuss model checking linear-time properties of quantum Markov chains, and show the quantitative analysis of regular safety properties can be addressed successfully.

Language equivalence of probabilistic pushdown automata

We study the language equivalence problem for probabilistic pushdown automata (pPDA) and their subclasses. We show that the problem is interreducible with the multiplicity equivalence problem for context-free grammars, the decidability of which has been open for several decades. Interreducibility also holds for pPDA with one control state.In contrast, for the case of a one-letter input alphabet we show that pPDA language equivalence (and hence multiplicity equivalence of context-free grammars) is in PSPACE and at least as hard as the polynomial identity testing problem.

Bad news on decision problems for patterns

We study the inclusion problem for pattern languages, which—due to Jiang et al. [T. Jiang, A. Salomaa, K. Salomaa, S. Yu, Decision problems for patterns, Journal of Computer and System Sciences 50 (1995) 53–63]—is known to be undecidable. More precisely, Jiang et al. demonstrate that there is no effective procedure deciding the inclusion for the class of all pattern languages over all alphabets. Most applications of pattern languages, however, consider classes over fixed alphabets, and therefore it is practically more relevant to ask for the existence of alphabet-specific decision procedures. Our first main result states that, for all but very particular cases, this version of the inclusion problem is also undecidable. The second main part of our paper disproves the prevalent conjecture on the inclusion of so-called similar E-pattern languages, and it explains the devastating consequences of this result for the intensive previous research on the most prominent open decision problem for pattern languages, namely the equivalence problem for general E-pattern languages.

The equivalence problem for languages defined by transductions on D0L languages

We study the equivalence problem for languages defined by various types of transducers acting on D0L languages. We show that, given ϵ-free sequential transducers S 1, …, S k , T 1, …, T k and D0L languages L 1, L 2, whether or not |S 1|(L 1)∪···∪|S k |(L 1)=|T 1|(L 2)∪···∪|T k |(L 2) is decidable.

The language equivalence problem for HD0L systems having D0L growths

We show that language equivalence is decidable for HD0L systems having D0L growths. By definition, an HD0L system H has D0L growth if the length sequence of H is a D0L length sequence.

Bounds for the D0L language equivalence problem

We have recently proved that there is a bound for the sequence equivalence problem of polynomially bounded D0L systems depending only on the cardinality of the underlying alphabet. In this paper we deduce a similar bound for the language equivalence problem of polynomially bounded D0L systems. More generally, we prove that if a given class of D0L systems (satisfying certain natural conditions) has a uniform bound for sequence equivalence then it also has a uniform bound for language equivalence.

On the Images of ℕ-Rational Sequences Counting Multiplicities

A mapping ϕ:ℤ → ℤ is called piecewise affine if there exist integers a≥ 1 and uj≥ 1, vj for 0≤ j<a such that ϕ(an+j)=ujn+vj whenever n∈ ℤ and 0≤ j<a. We prove that if s=(s(n))n≥ 0 and t=(t(n))n≥ 0 are ℕ-rational sequences such that s takes each value exactly as many times as t, then there exists a piecewise affine mapping ϕ:ℤ → ℤ such that s(n)=t(ϕ(n)) for almost all n≥ 0. As an application we solve the HD0L language equivalence problem in some cases.

Complete formal systems for equivalence problems

We describe four complete and recursively enumerable formal systems S 0, D 0, H 0, B 0 . Each one of them proves the decidability of some equivalence problem for some class of automata: namely the language equivalence problem for simple automata, the language equivalence problem for deterministic pushdown automata, the function equivalence problem for deterministic pushdown transducers with outputs in an abelian group, the bisimulation equivalence problem for loop-free pushdown automata.

On the equivalence problem for E-pattern languages

On the one hand, the inclusion problem for nonerasing and erasing pattern languages is undecidable (see Jiang et al., 1995). On the other hand, the language equivalence problem for nonerasing pattern languages is trivially decidable (see Angluin, 1980) but the question of whether the same holds for erasing pattern languages is still open. It has been conjectured by Jiang et al. that the language equivalence problem for erasing pattern languages is also decidable. In this paper, we introduce a new normal form for patterns and show, using the normal form, that the language equivalence problem for erasing pattern languages is decidable in many special cases. We conjecture that our normal form procedure decides the problem in the general case, too. If the conjecture holds true, then the normal form is the shortest pattern generating a given erasing pattern language.

On the Existence of Minimum Asynchronous Automata and on the Equivalence Problem for Unambiguous Regular Trace Languages

Given a recognizable trace language T there always exists a minimum (up to isomorphism) monoid automation which recognizes T . In this paper we consider finite state asynchronous automata, introduced by W. Zielonka ("Notes on Finite Asynchronous Automata," Technical Report, Institute of Mathematics, Warsaw Technical University, 1984) and we show that this result cannot be extended to this model. More precisely, we exhibit a recognizable trace language which does not admit a unique minimum asynchronous automaton. This result solves an open problem posed by Zielonka. We further prove that every recognizable trace language defined on a concurrent alphabet with a transitive dependency relation admits a unique minimum asynchronous automation. In the second part of this paper we investigate the equivalence problem for regular trace languages. This problem has been investigated by many authors: in particular, Aalbersberg and Hoogeboom (1987) have shown that the equivalence problem for regular trace languages defined over a concurrent alphabet 〈 Σ , C 〉 is decidable if and only if the concurrency relation C is transitive. We show that this result does not hold when we restrict our attention to unambiguous regular trace languages we exhibit a concurrent alphabet 〈 Σ ′, C ′〉 with a non transitive concurrency relation such that the equivalence problem for unambiguous regular trace languages on 〈 Σ ′, C ′〉 can be decided in polynomial time. Further we show that the problem of distinguishing whether or not a regular expression on a concurrent alphabet 〈 Σ , C 〉 defines an unambiguous regular trace language is undecidable .

Simple reductions between D0 L language and sequence equivalence problems

We show by a simple argument how any algorithm for solving the D0 L language equivalence problem can be used to solve the D0 L sequence equivalence problem, and vice versa.

Finite generation of ambiguity in context-free languages

This paper presents a pair of sufficient conditions, within an algebraic model of the generation of context-free languages, so that the intersection of two context-free languages is finitely partitioned by common sublanguages. When such a context-free algebraic system satisfies this pair of sufficient conditions, the ambiguity in deriving strings of symbols is shown to be finitely generated. These conditions cannot be necessary, because these problems are equivalent to the language equivalence problem for context-free grammars, which is undecidable.

The membership and equivalence problems for picture languages

The paper describes two principal results: (1) there is a nondeterministic polynomial-time algorithm to determine whether an arbitrary picture is a member of a context-free picture language and (2) the equivalence problem for regular picture languages is not decidable (not even partially decidable). This solves open questions described in earlier publications. Furthermore, these results are best possible. That is, the membership problem for context-free picture languages was earlier shown to be NP-hard and the complement of the equivalence problem for regular picture languages is easily seen to be partially decidable.

Infinite arrays and controlled deterministic table 0L array systems

Deterministic table 0L array systems with control are considered for the generation of infinite arrays. Rewriting of a rectangular array is done in parallel by a table of rules, the rightmost edge horizontally or the lowermost edge vertically downwards. The application of the tables is controlled by a control set. Cube-free and square-free infinite arrays are obtained as an application of this model. The adherence of the array language of a controlled deterministic table 0L array system is related to the adherence of its control set. The limit language equivalence problem and the adherence equivalence problem are shown to be undecidable for this system.

The equivalence problem for real-time strict deterministic languages

The equivalence problem is decidable for deterministic real-time pushdown store automata accepting by empty stack. Further, the equivalence problem is decidable for two deterministic pushdown store automata, one of which is real-time accepting by empty stack.

Equivalence problems for deterministic context-free languages and monadic recursion schemes

The equivalence problem for deterministic context-free languages is shown to be decidable if and only if the strong equivalence problem for monadic recursion schemes is decidable.