133 research outputs found
Propagating Regular Counting Constraints
Constraints over finite sequences of variables are ubiquitous in sequencing
and timetabling. Moreover, the wide variety of such constraints in practical
applications led to general modelling techniques and generic propagation
algorithms, often based on deterministic finite automata (DFA) and their
extensions. We consider counter-DFAs (cDFA), which provide concise models for
regular counting constraints, that is constraints over the number of times a
regular-language pattern occurs in a sequence. We show how to enforce domain
consistency in polynomial time for atmost and atleast regular counting
constraints based on the frequent case of a cDFA with only accepting states and
a single counter that can be incremented by transitions. We also prove that the
satisfaction of exact regular counting constraints is NP-hard and indicate that
an incomplete algorithm for exact regular counting constraints is faster and
provides more pruning than the existing propagator from [3]. Regular counting
constraints are closely related to the CostRegular constraint but contribute
both a natural abstraction and some computational advantages.Comment: Includes a SICStus Prolog source file with the propagato
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Remarks on separating words
The separating words problem asks for the size of the smallest DFA needed to
distinguish between two words of length <= n (by accepting one and rejecting
the other). In this paper we survey what is known and unknown about the
problem, consider some variations, and prove several new results
Testing the Equivalence of Regular Languages
The minimal deterministic finite automaton is generally used to determine
regular languages equality. Antimirov and Mosses proposed a rewrite system for
deciding regular expressions equivalence of which Almeida et al. presented an
improved variant. Hopcroft and Karp proposed an almost linear algorithm for
testing the equivalence of two deterministic finite automata that avoids
minimisation. In this paper we improve the best-case running time, present an
extension of this algorithm to non-deterministic finite automata, and establish
a relationship between this algorithm and the one proposed in Almeida et al. We
also present some experimental comparative results. All these algorithms are
closely related with the recent coalgebraic approach to automata proposed by
Rutten
Incomplete operational transition complexity of regular languages
The state complexity of basic operations on regular languages considering complete deterministic finite automata (DFA) has been extensively studied in the literature. But, if incomplete DFAs are considered, transition complexity is also a significant measure. In this paper we study the incomplete (deterministic) state and transition complexity of some operations for regular and finite languages. For regular languages we give a new tight upper bound for the transition complexity of the union, which refutes the conjecture presented by Y. Gao et al. For finite languages, we correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right operand is larger than the left one. We also present some experimental results to test the behavior of those operations on the average case, and we conjecture that for many operations and in practical applications the worst-case complexity is seldom reached
Order-Related Problems Parameterized by Width
In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints.
While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems.
In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory.
Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.
In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin
Latvian Quantum Finite State Automata for Unary Languages
We design Latvian quantum finite state automata (LQFAs for short) recognizing
unary regular languages with isolated cut point 1/2. From an architectural
point of view, we combine two LQFAs recognizing with isolated cut point,
respectively, the finite part and the ultimately periodic part of any given
unary regular language L. In both modules, we use a component addressed in the
literature and here suitably adapted to the unary case, to discriminate strings
on the basis of their length. The number of basis states and the isolation
around the cut point of the resulting LQFA for L exponentially depends on the
size of the minimal deterministic finite state automaton for L.Comment: In Proceedings NCMA 2023, arXiv:2309.0733
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