Artur Jeż

Conjunctive Grammars Can Generate Non-regular Unary Languages

Conjunctive grammars were introduced by A. Okhotin in [1] as a natural extension of context-free grammars with an additional operation of intersection in the body of any production of the grammar. Several theorems and algorithms for context-free grammars generalize to the conjunctive case. Still some questions remained open. A. Okhotin posed nine problems concerning those grammars. One of them was a question, whether a conjunctive grammar over unary alphabet can generate only regular languages. We give a negative answer, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language \(\{ a^{4^{n}} : n \in \mathbb{N} \}\). We then generalise this result—for every set of numbers L such that their representation in some k-ary system is regular set we show that \(\{ a^{k^{n}} : n \in L \}\) is generated by some conjunctive grammar over unary alphabet.

Conjunctive grammars over a unary alphabet: undecidability and unbounded growth

It has recently been proved (Jez, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some nonregular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as nonexistence of an r.e. bound on the growth rate of generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.

Recompression: A Simple and Powerful Technique for Word Equations

In this article, we present an application of a simple technique of local recompression, previously developed by the author in the context algorithms for compressed strings [Jez 2014a, 2015b, 2015a], 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 occurring in the equation by a “fresh” letter, which can be seen as a bottom-up compression of the solution of the given word equation, or, to be more specific, building a Straight-Line Programme for the solution of the word equation. Using this technique, we give new, independent, and self-contained proofs of many 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 n and log N, where n is the size of the input equation and N the size of the length-minimal solution of the word equation. Furthermore, for O(1) variables, the bound on the space consumption is in fact linear, that is, O(m), where m is the size of the space used by the input. This yields that for each k the set of satisfiable word equations with k variables is context sensitive. 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 an independent proof of 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.

Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

It has recently been proved (Jeż, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some non-regular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as non-existence of a recursive function bounding the growth rate of the generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.

Faster fully compressed pattern matching by recompression

In this paper, a fully compressed pattern matching problem is studied. The compression is represented by straight-line programs (SLPs), i.e. a context-free grammar generating exactly one string; the term fully means that both the pattern and the text are given in the compressed form. The problem is approached using a recently developed technique of local recompression: the SLPs are refactored, so that substrings of the pattern and text are encoded in both SLPs in the same way. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. This technique yields an

Complexity of Equations over Sets of Natural Numbers

\mathcal{O}((n+m)\log M \log(n+m))

On the Computational Completeness of Equations over Sets of Natural Numbers

algorithm for compressed pattern matching, where n (m) is the size of the compressed representation of the text (pattern, respectively), while M is the size of the decompressed pattern. Since M≤2m, this substantially improves the previously best

Approximation of Grammar-based Compression via Recompression

\mathcal{O}(m^2n)

One-Nonterminal Conjunctive Grammars over a Unary Alphabet

algorithm. Since LZ compression standard reduces to SLP with log( N / n) overhead and in

A really simple approximation of smallest grammar

\mathcal{O}(n \log(N/n))