Grammatical compression: compressed equivalence and other problems

Alberto Bertoni,Roberto Radicioni

doi:10.46298/dmtcs.526

Alberto Bertoni, Roberto Radicioni

Open Access

https://doi.org/10.46298/dmtcs.526

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Abstract

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications In this work, we focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs on a given algebra. In our exposition, we try to survey general results by presenting some meaningful examples; moreover, where possible, we outline the proofs in order to give an insight of the methods and the techniques. We recall some recent results for the problem PosSLP, consisting of deciding if the integer defined by a straight-line program on the ring Z is greater than zero; we discuss some implications in the areas of numerical analysis and strategic games. Furthermore, we propose some methods for reducing Compressed Word Problem from an algebra to another; reductions from trace monoids to the semiring of nonnegative integers are exhibited and polynomial time algorithms for compressed equivalence in monoids related to Dyck reductions are shown. Finally, we consider inclusion problems for context-free languages, proving how in some cases efficient algorithms for these problems benefit from the ability to work with compressed data.

Highlights

Consider the problem SQRT-Sum of deciding whether n i=1 √ ai >t, for integers a1,, an, t, and the problem Incl(DA) of deciding whether L ⊆ DA for context-free languages L, where DA is the Dyck language on the alphabet A ∪ A
An, t, and the problem Incl(DA) of deciding whether L ⊆ DA for context-free languages L, where DA is the Dyck language on the alphabet A ∪ A. Have these problems something in common? In this paper, we point out that efficient algorithms for these problems take advantage from the ability to work with compressed data
We focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs (SLP) on a given algebra

Summary

Introduction

An, t, and the problem Incl(DA) of deciding whether L ⊆ DA for context-free languages L, where DA is the Dyck language on the alphabet A ∪ A. We consider the problem Incl(L0) of deciding whether a context-free language is contained in a fixed language L0, and we prove that, in some cases, it can be reduced to Compressed Equivalence(A) for some monoid A (Bertoni et al (2009)). We point out that Incl(DA) can be solved in polynomial time for every Dyck language DA

Straight-line Programs and Grammatical Compression

Problems on Compressed Terms

Algebra representations

Representations

Composition systems

CompEq and Inclusion Problem for CF languages