Solving the word problem in real time

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems

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

Word equations are a crucial element in the theoretical foundation of constraint solving over strings, which have received a lot of attention in recent years. 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. In this paper, 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 theory of existential Presburger Arithmetic with divisibility (PAD). Since PAD is decidable, we get a decision procedure for quadratic words equations with length constraints for which the associated counter system is \emph{flat} (i.e., all nodes belong to at most one cycle). 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, together with length constraints. Decidability holds when the constraints are additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page

Eksperimentasi Model Pembelajaran Kooperatif Tipe Jigsaw Dengan Pendekatan Problem Posing Dan Tipe Jigsaw Terhadap Kemampuan Menyelesaikan Soal Cerita Pada Pecahan Ditinjau Dari Tingkat Percaya Diri Siswa Kelas VII Smp/mts Di Kota Metro Lampung

This research aimed to find out: (1) which one is better in giving the mathematical word problem solving ability among learning models of jigsaw with problem posing approach, learning models of jigsaw, or direct instructional models, (2) which one is better in giving the mathematical word problem solving ability among students' level of confidence, students having high, medium or low level, (3) in each level of confidence, which one is better in giving the mathematical word problem solving ability among learning models of jigsaw with problem posing approach, learning models of jigsaw, or direct instructional models and (4) in each learning models, which one is better in giving the mathematical word problem solving ability among students' level of confidence, students having high, medium or low level. This research was a quasi-experimental research with 3×3 factorial design. The population of the research was all grade VII students of Junior High School (SMP) in Metro, Lampung in academic year 2013/2014. The samples were chosen by using stratified cluster random sampling. The instruments that were used to collect the data were the test and questionnaire of student's level of confidence. The technique of analyzing the data was two-ways ANOVA with unbalanced cells. The results of research showed as follows. (1) Learning models of jigsaw with problem posing approach had mathematical word problem solving ability as good as learning models of jigsaw, learning models of jigsaw with problem posing approach had better mathematical word problem solving ability than direct instructional models, and learning models of jigsaw had mathematical word problem solving ability as good as direct instructional models. (2) The students having high level of confidence had mathematical word problem solving ability as good as those having medium level of confidence. The students having high level of confidence had better mathematical word problem solving ability than those having low level of confidence. The students having medium level of confidence had mathematical word problem solving ability as good as those having low level of confidence. (3) In each level of confidence, learning models of jigsaw with problem posing approach had better mathematical word problem solving ability than direct instructional models. (4) In each learning models, the students having high level of confidence had better mathematical word problem solving ability than those having low level of confidence

Mathematical Word Problem Solving of Students with Autism Spectrum Disorders and Students with Typical Development

Mathematical Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development - Young Seh Bae - This study investigated mathematical word problem solving and the factors associated with the solution paths adopted by two groups of participants (N=40), students with autism spectrum disorders (ASDs) and typically developing students in fourth and fifth grade, who were comparable on age and IQ (greater than 80). The factors examined in the study were: word problem solving accuracy; word reading/decoding; sentence comprehension; math vocabulary; arithmetic computation; everyday math knowledge; attitude toward math; identification of problem type schemas; and visual representation. Results indicated that the students with typical development significantly outperformed the students with ASDs on word problem solving and everyday math knowledge. Correlation analysis showed that word problem solving performance of the students with ASDs was significantly associated with sentence comprehension, math vocabulary, computation and everyday math knowledge, but that these relationships were strongest and most consistent in the students with ASDs. No significant associations were found between word problem solving and attitude toward math, identification of schema knowledge, or visual representation for either diagnostic group. Additional analyses suggested that everyday math knowledge may account for the differences in word problem solving performance between the two diagnostic groups. Furthermore, the students with ASDs had qualitatively and quantitatively weaker structure of everyday math knowledge compared to the typical students. The theoretical models of the linguistic approach and the schema approach offered some possible explanations for the word problem solving difficulties of the students with ASDs in light of the current findings. That is, if a student does not have an adequate level of everyday math knowledge about the situation described in the word problem, he or she may have difficulties in constructing a situation model as a basis for problem comprehension and solutions. It was suggested that the observed difficulties in math word problem solving may have been strongly associated with the quantity and quality of everyday math knowledge as well as difficulties with integrating specific math-related everyday knowledge with the global text of word problems. Implications for this study include a need to develop mathematics instructional approaches that can teach students to integrate and extend their everyday knowledge from real-life contexts into their math problem-solving process. Further research is needed to confirm the relationships found in this study, and to examine other areas that may affect the word problem solving processes of students with ASDs

A New Algorithm for Solving the Word Problem in Braid Groups

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometrical one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition of the braid group in order to give a new approach for solving its word problem. Our algorithm is faster, in comparison with known algorithms, for short braid words with respect to the number of generators combining the braid, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms.Comment: 24 pages, 13 figure