Recompression: a simple and powerful technique for word equations

In this paper we present an application of a simple technique of local recompression, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is based on local modification of variables (replacing X by aX or Xa) and iterative replacement of pairs of letters appearing in the equation by a `fresh' letter, which can be seen as a bottom-up compression of the solution of the given word equation, to be more specific, building an SLP (Straight-Line Programme) for the solution of the word equation. Using this technique we give a new, independent and self-contained proofs of most of the known results for word equations. To be more specific, the presented (nondeterministic) algorithm runs in O(n log n) space and in time polynomial in log N, where N is the size of the length-minimal solution of the word equation. The presented algorithm can be easily generalised to a generator of all solutions of the given word equation (without increasing the space usage). Furthermore, a further analysis of the algorithm yields a doubly exponential upper bound on the size of the length-minimal solution. The presented algorithm does not use exponential bound on the exponent of periodicity. Conversely, the analysis of the algorithm yields an independent proof of the exponential bound on exponent of periodicity. We believe that the presented algorithm, its idea and analysis are far simpler than all previously applied. Furthermore, thanks to it we can obtain a unified and simple approach to most of known results for word equations. As a small additional result we show that for O(1) variables (with arbitrary many appearances in the equation) word equations can be solved in linear space, i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were simplified, overall presentation improve

Approximation of grammar-based compression via recompression

In this paper we present a simple linear-time algorithm constructing a context-free grammar of size O(g log(N/g)) for the input string, where N is the size of the input string and g the size of the optimal grammar generating this string. The algorithm works for arbitrary size alphabets, but the running time is linear assuming that the alphabet \Sigma of the input string can be identified with numbers from {1, ..., N^c} for some constant c. Otherwise, additional cost of O(n log|\Sigma|) is needed. Algorithms with such approximation guarantees and running time are known, the novelty of this paper is a particular simplicity of the algorithm as well as the analysis of the algorithm, which uses a general technique of recompression recently introduced by the author. Furthermore, contrary to the previous results, this work does not use the LZ representation of the input string in the construction, nor in the analysis.Comment: 22 pages, some many small improvements, to be submited to a journa

Solutions of Word Equations over Partially Commutative Structures

We give NSPACE(n log n) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n log n). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups

On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

Space-efficient conversions from SLPs

We give algorithms that, given a straight-line program (SLP) with $g$ rules that generates (only) a text $T [1..n]$, builds within $O(g)$ space the Lempel-Ziv (LZ) parse of $T$ (of $z$ phrases) in time $O(n\log^2 n)$ or in time $O(gz\log^2(n/z))$. We also show how to build a locally consistent grammar (LCG) of optimal size $g_{lc} = O(\delta\log\frac{n}{\delta})$ from the SLP within $O(g+g_{lc})$ space and in $O(n\log g)$ time, where $\delta$ is the substring complexity measure of $T$. Finally, we show how to build the LZ parse of $T$ from such a LCG within $O(g_{lc})$ space and in time $O(z\log^2 n \log^2(n/z))$. All our results hold with high probability

Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)

In this paper, a compressed membership problem for finite automata, both deterministic and non-deterministic, with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. Furthermore, such recompression induces only small changes in the automaton, in particular, the size of the automaton remains polynomial. Using this technique it is shown that the compressed membership for NFA with compressed labels is in NP, thus confirming the conjecture of Plandowski and Rytter and extending the partial result of Lohrey and Mathissen; as it is already known, that this problem is NP-hard, we settle its exact computational complexity. Moreover, the same technique applied to the compressed membership for DFA with compressed labels yields that this problem is in P; for this problem, only trivial upper-bound PSPACE was known