scholarly journals On systems of word equations with simple loop sets

2007 ◽  
Vol 380 (3) ◽  
pp. 363-372 ◽  
Author(s):  
Štěpán Holub ◽  
Juha Kortelainen
Keyword(s):  
Simple Loop ◽  
Word Equations

scholarly journals Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Anthony W. Lin ◽  
Rupak Majumdar
Keyword(s):  
Theoretical Foundation ◽  
Constraint Solving ◽  
Simple Loop ◽  
Decidability Result ◽  
Presburger Arithmetic ◽  
Existential Theory ◽  
Word Equation ◽  
Word Equations ◽  
Left And Right ◽  
Np Complete

Word equations are a crucial element in the theoretical foundation of constraint solving over strings. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. We focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the existential theory of Presburger Arithmetic with divisibility (PAD). Since PAD is decidable (NP-hard and is in NEXP), we obtain a decision procedure for quadratic words equations with length constraints for which the associated counter system is flat (i.e., all nodes belong to at most one cycle). In particular, we show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, when augmented with length constraints. We extend this decidability result (in fact, with a complexity upper bound of PSPACE with a PAD oracle) in the presence of regular constraints.

Studying Word Equations by a Method of Weighted Frequencies

2018 ◽  
Vol 162 (2-3) ◽  
pp. 223-235 ◽  
Author(s):  
Aleksi Saarela
Keyword(s):  
Word Equations

A Method for Type Synthesis of Mechanisms Using Phase Diagrams

1994 ◽  
Author(s):  
Sio-Hou Lei ◽  
Ying-Chien Tsai
Keyword(s):  
Phase Diagram ◽  
Degrees Of Freedom ◽  
Practical Application ◽  
Type Synthesis ◽  
Planar Mechanisms ◽  
Simple Loop ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Spatial Mechanisms ◽  
Synthesis Of Mechanisms

Abstract A method for synthesizing the types of spatial as well as planar mechanisms is expressed in this paper by using the concept of phase diagram in metallurgy. The concept represented as a type synthesis technique is applied to (a) planar mechanisms with n degrees of freedom and simple loop, (b) spatial mechanisms with single degree of freedom and simple loop, to enumerate all the possible mechanisms with physically realizable kinematic pairs. Based on the technique described, a set of new reciprocating mechanisms is generated as a practical application.

scholarly journals Unification Modulo Lists with Reverse Relation with Certain Word Equations

2019 ◽  
pp. 1-17
Author(s):  
Siva Anantharaman ◽  
Peter Hibbs ◽  
Paliath Narendran ◽  
Michael Rusinowitch
Keyword(s):  
Word Equations

scholarly journals Solutions to twisted word equations and equations in virtually free groups

2020 ◽  
Vol 30 (04) ◽  
pp. 731-819
Author(s):  
Volker Diekert ◽  
Murray Elder
Keyword(s):  
Free Groups ◽  
Expressive Power ◽  
Solution Set ◽  
Complexity Bound ◽  
Free Monoids ◽  
Word Equation ◽  
All Solutions ◽  
Standard Normal ◽  
Word Equations ◽  
Solving Equations

It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper, we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in PSPACE . Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in PSPACE an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or “small enough”) finite monoid, then our algorithms can be implemented in [Formula: see text], that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.

scholarly journals The complete mitochondrial genome of Microphysogobio elongatus (Teleostei, Cyprinidae) and its phylogenetic implications

ZooKeys ◽  
2021 ◽  
Vol 1061 ◽  
pp. 57-73
Author(s):  
Renyi Zhang ◽  
Qian Tang ◽  
Lei Deng
Keyword(s):  
Complete Mitochondrial Genome ◽  
Genetic Material ◽  
Start Codon ◽  
Trna Genes ◽  
Protein Coding ◽  
Simple Loop ◽  
Stop Codons ◽  
Phylogenetic Implications ◽  
Complete Mitogenome ◽  
Rna Genes

Mitochondria are important organelles with independent genetic material of eukaryotic organisms. In this study, we sequenced and analyzed the complete mitogenome of a small cyprinid fish, Microphysogobio elongatus (Yao & Yang, 1977). The mitogenome of M. elongatus is a typical circular molecule of 16,612 bp in length containing 13 protein-coding genes (PCGs), 22 transfer RNA genes, two ribosomal RNA genes, and a 930 bp control region. The base composition of the M. elongatus mitogenome is 30.8% A, 26.1% T, 16.7% G, and 26.4% C. All PCGs used the standard ATG start codon with the exception of COI. Six PCGs terminate with complete stop codons, whereas seven PCGs (ND2, COII, ATPase 6, COIII, ND3, ND4, and Cyt b) terminate with incomplete (T or TA) stop codons. All tRNA genes exhibited typical cloverleaf secondary structures with the exception of tRNASer(AGY), for which the dihydrouridine arm forms a simple loop. The phylogenetic analysis divided the subfamily Gobioninae in three clades with relatively robust support, and that Microphysogobio is not a monophyletic group. The complete mitogenome of M. elongatus provides a valuable resource for future studies about molecular phylogeny and/or population genetics of Microphysogobio.

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