Undecidability of the word problem for Yamamura’s HNN-extension under nice conditions

2016 ◽  
Vol 93 (1) ◽  
pp. 86-96
Author(s):  
Mohammed Abu Ayyash ◽  
Emanuele Rodaro
Keyword(s):  
Word Problem ◽  
Hnn Extension

scholarly journals Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

2016 ◽  
Vol 99 (113) ◽  
pp. 177-191
Author(s):  
Mohammed Ayyash ◽  
Emanuele Rodaro
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Polynomial Time ◽  
Inverse Semigroups ◽  
Free Graph ◽  
Hnn Extensions ◽  
Language L ◽  
Hnn Extension ◽  
Context Free ◽  
Standard Presentation

We prove that the Sch?tzenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems.

HNN EXTENSIONS OF SEMILATTICES

1999 ◽  
Vol 09 (05) ◽  
pp. 555-596 ◽  
Author(s):  
AKIHIRO YAMAMURA
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Inverse Semigroups ◽  
The Third ◽  
Maximal Group ◽  
Hnn Extensions ◽  
Solvable Word Problem ◽  
Hnn Extension ◽  
Subgroup Theorem ◽  
Solvable Word

The main purpose of this paper is to investigate properties of an HNN extension of a semilattice, to give its equivalent characterizations and to discuss similarities with free groups. An HNN extension of a semilattice is shown to be a universal object in a certain category and an F-inverse cover over a free group for every inverse semigroup in the category. We also show that a graph with respect to a certain subset of an HNN extension of a semilattice is a tree and that this property characterizes an HNN extension of a semilattice. Moreover, we look into three subclasses: the class of full HNN extensions of semilattices with an identity, the class of universally E-unitary inverse semigroups and the class of HNN extensions of finite semilattices. The first class consists of factorizable E-unitary inverse semigroups whose maximal group homomorphic images are free. We obtain a generalization of the Nielsen–Schreier subgroup theorem to this class. The second consists of inverse semigroups presented by relations on Dyck words. An inverse semigroup in the third class has a relatively easy finite presentation using a Dyck language and has solvable word problem.

scholarly journals Improving Language Comprehension to Enhance Word-Problem Solving

2019 ◽  
Vol 36 (2) ◽  
pp. 142-156
Author(s):  
Lynn S. Fuchs ◽  
Douglas Fuchs ◽  
Pamela M. Seethaler ◽  
Caitlin Craddock
Keyword(s):  
Problem Solving ◽  
Word Problem ◽  
Language Comprehension ◽  
Word Problem Solving

scholarly journals Students’ suspension of sense making in problem solving

ZDM ◽  
2021 ◽  
Author(s):  
Gemma Carotenuto ◽  
Pietro Di Martino ◽  
Marta Lemmi
Keyword(s):  
Problem Solving ◽  
Qualitative Study ◽  
Word Problem ◽  
Word Problems ◽  
Mathematical Problem ◽  
Critical Issue ◽  
Mathematical Problem Solving ◽  
Sense Making ◽  
Mathematical Problems

AbstractResearch on mathematical problem solving has a long tradition: retracing its fascinating story sheds light on its intricacies and, therefore, on its needs. When we analyze this impressive literature, a critical issue emerges clearly, namely, the presence of words and expressions having many and sometimes opposite meanings. Significant examples are the terms ‘realistic’ and ‘modeling’ associated with word problems in school. Understanding how these terms are used is important in research, because this issue relates to the design of several studies and to the interpretation of a large number of phenomena, such as the well-known phenomenon of students’ suspension of sense making when they solve mathematical problems. In order to deepen our understanding of this phenomenon, we describe a large empirical and qualitative study focused on the effects of variations in the presentation (text, picture, format) of word problems on students’ approaches to these problems. The results of our study show that the phenomenon of suspension of sense making is more precisely a phenomenon of activation of alternative kinds of sense making: the different kinds of active sense making appear to be strongly affected by the presentation of the word problem.

scholarly journals Strategic competence for multistep fraction word problems: an overlooked aspect of mathematical knowledge for teaching

2021 ◽  
Author(s):  
Yasemin Copur-Gencturk ◽  
Tenzin Doleck
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Mathematical Knowledge ◽  
Teaching And Learning ◽  
Mathematics Learning ◽  
Mathematical Knowledge For Teaching ◽  
Important Indicator ◽  
Strategic Competence ◽  
The Usa ◽  
Valid Solution

AbstractPrior work on teachers’ mathematical knowledge has contributed to our understanding of the important role of teachers’ knowledge in teaching and learning. However, one aspect of teachers’ mathematical knowledge has received little attention: strategic competence for word problems. Adapting from one of the most comprehensive characterizations of mathematics learning (NRC, 2001), we argue that teachers’ mathematical knowledge also includes strategic competence, which consists of devising a valid solution strategy, mathematizing the problem (i.e., choosing particular strategies and presentations to translate the word problem into mathematical expressions), and arriving at a correct answer (executing a solution) for a word problem. By examining the responses of 350 fourth- and fifth-grade teachers in the USA to four multistep fraction word problems, we were able to explore manifestations of teachers’ strategic competence for word problems. Findings indicate that teachers’ strategic competence was closely related to whether they devised a valid strategy. Further, how teachers dealt with known and unknown quantities in their mathematization of word problems was an important indicator of their strategic competence. Teachers with strong strategic competence used algebraic notations or pictorial representations and dealt with unknown quantities more frequently in their solution methods than did teachers with weak strategic competence. The results of this study provide evidence for the critical nature of strategic competence as another dimension needed to understand and describe teachers’ mathematical knowledge.

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