Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity

We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of $2^{\mathcal{O}(n)}$ on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size $n$. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of $2^{2^{\mathcal{O}(n)}}$ following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub- or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs

On the Minimal Uncompletable Word Problem

Let S be a finite set of words over an alphabet Sigma. The set S is said to be complete if every word w over the alphabet Sigma is a factor of some element of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are interested in finding bounds on the minimal length of words in Sigma* which are not elements of Fact(S*) in terms of the maximal length of words in S.Comment: 5 pages; added references, corrected typo

Matching Patterns with Variables Under Simon's Congruence

We introduce and investigate a series of matching problems for patterns with variables under Simon's congruence. Our results provide a thorough picture of these problems' computational complexity

The Edit Distance to k-Subsequence Universality

A word u is a subsequence of another word w if u can be obtained from w by deleting some of its letters. In the early 1970s, Imre Simon defined the relation ?_k (called now Simon-Congruence) as follows: two words having exactly the same set of subsequences of length at most k are ?_k-congruent. This relation was central in defining and analysing piecewise testable languages, but has found many applications in areas such as algorithmic learning theory, databases theory, or computational linguistics. Recently, it was shown that testing whether two words are ?_k-congruent can be done in optimal linear time. Thus, it is a natural next step to ask, for two words w and u which are not ?_k-equivalent, what is the minimal number of edit operations that we need to perform on w in order to obtain a word which is ?_k-equivalent to u. In this paper, we consider this problem in a setting which seems interesting: when u is a k-subsequence universal word. A word u with alph(u) = ? is called k-subsequence universal if the set of subsequences of length k of u contains all possible words of length k over ?. As such, our results are a series of efficient algorithms computing the edit distance from w to the language of k-subsequence universal words

On the Complexity of Universality for Partially Ordered NFAs

International audiencePartially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper

On the Complexity of Universality for Partially Ordered NFAs

Partially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSPACE-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper