scholarly journals KNUTH–BENDIX FOR GROUPS WITH INFINITELY MANY RULES

2000 ◽  
Vol 10 (05) ◽  
pp. 539-589 ◽  
Author(s):  
D. B. A. EPSTEIN ◽  
P. J. SANDERS
Keyword(s):  
Computer Program ◽  
Regular Language ◽  
The Body ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Central Feature ◽  
Rapid Reduction ◽  
Automatic Structures ◽  
Small Constant ◽  
Finite State

We introduce a new class of groups with solvable word problem, namely groups specified by a confluent set of short-lex-reducing Knuth–Bendix rules which form a regular language. This simultaneously generalizes short-lex-automatic groups and groups with a finite confluent set of short-lex-reducing rules. We describe a computer program which looks for such a set of rules in an arbitrary finitely presented group. Our main theorem is that our computer program finds the set of rules, if it exists, given enough time and space. (This is an optimistic description of our result — for the more pessimistic details, see the body of the paper.) The set of rules is embodied in a finite state automaton in two variables. A central feature of our program is an operation, which we call welding, used to combine existing rules with new rules as they are found. Welding can be defined on arbitrary finite state automata, and we investigate this operation in abstract, proving that it can be considered as a process which takes as input one regular language and outputs another regular language. In our programs we need to convert several nondeterministic finite state automata to deterministic versions accepting the same language. We show how to improve somewhat on the standard subset construction, due to special features in our case. We axiomatize these special features, in the hope that these improvements can be used in other applications. The Knuth–Bendix process normally spends most of its time in reduction, so its efficiency depends on doing reduction quickly. Standard data structures for doing this can become very large, ultimately limiting the set of presentations of groups which can be so analyzed. We are able to give a method for rapid reduction using our much smaller two variable automaton, encoding the (usually infinite) regular language of rules found so far. Time taken for reduction in a given group is a small constant times the time taken for reduction in the best schemes known (see [5]), which is not too bad since we are reducing with respect to an infinite set of rules, whereas known schemes use a finite set of rules. We hope that the method described here might lead to the computation of automatic structures in groups for which this is currently infeasible. Some proofs have been omitted from this paper in the interests of brevity. Full details are provided in [4].

Finite-State Technology

2012 ◽  
Author(s):  
Lauri Karttunen
Keyword(s):  
Language Processing ◽  
Regular Language ◽  
Morphological Analysis ◽  
Finite State Automata ◽  
Regular Expressions ◽  
Finite State Transducers ◽  
Finite State ◽  
Shallow Parsing ◽  
Basic Concepts ◽  
State Language

The article introduces the basic concepts of finite-state language processing: regular languages and relations, finite-state automata, and regular expressions. Many basic steps in language processing, ranging from tokenization, to phonological and morphological analysis, disambiguation, spelling correction, and shallow parsing, can be performed efficiently by means of finite-state transducers. The article discusses examples of finite-state languages and relations. Finite-state networks can represent only a subset of all possible languages and relations; that is, only some languages are finite-state languages. Furthermore, this article introduces two types of complex regular expressions that have many linguistic applications, restriction and replacement. Finally, the article discusses the properties of finite-state automata. The three important properties of networks are: that they are epsilon free, deterministic, and minimal. If a network encodes a regular language and if it is epsilon free, deterministic, and minimal, the network is guaranteed to be the best encoding for that language.

scholarly journals Non Homogeneous Rough Finite State Automaton

2021 ◽  
Vol 11 (2) ◽  
pp. 629-641
Author(s):  
B. Praba ◽  
R. Saranya
Keyword(s):  
Artificial Intelligence ◽  
Machine Learning ◽  
Dynamical Behaviour ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Finite State ◽  
Transition Map

Objective: The study of finite state automaton is an essential tool in machine learning and artificial intelligence. The class of rough finite state automaton captures the uncertainty using the rough transition map. The need to generalize this concept arises to adhere the dynamical behaviour of the system. Hence this paper focuses on defining non-homogeneous rough finite state automaton. Methodology: With the aid of Rough finite state automata we define the concept of non-homogeneous rough finite state automata. Findings: Non homogeneous Rough Finite State Automata (NRFSA) Mt is defined by a tuple (Q,Σ,δt,q0 (t),F(t)) The dynamical behaviour of any system can be expressed in terms of an information system at time t. This leads us to define non-homogeneous rough finite state automaton. For each time ‘t’ we generate lower approximation rough finite state automaton Mt_ and the upper approximation rough finite state automaton Mt- and the defined concepts are elaborated with suitable examples. The ordered pair , Mt=(M(t)-,M(t)-) is called as the non-homogeneous rough finite state automaton. Conclusion: Over all our study reveals the characterization of the system which changes its behaviour dynamically over a time ‘t’. Novelty: The novelty of the proposed article is that it clearly immense the system behaviour over a time ‘t’. Using this concept the possible and the definite transitions in the system can be calculated in any given time ‘t’.

scholarly journals Applications of the finite state automata for counting restricted permutations and variations

2012 ◽  
Vol 22 (2) ◽  
pp. 183-198
Author(s):  
Vladimir Baltic
Keyword(s):  
The Other ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Combinatorial Structures ◽  
Restricted Permutations ◽  
Finite State

In this paper, we use the finite state automata to count the number of restricted permutations and the number of restricted variations. For each type of restricted permutations, we construct a finite state automaton able to recognize and enumerate them. We, also, discuss how it encompasses the other known methods for enumerating permutations with restricted position, and in one case, we establish connections with some other combinatorial structures, such as subsets and compositions.

INFIX-FREE REGULAR EXPRESSIONS AND LANGUAGES

2006 ◽  
Vol 17 (02) ◽  
pp. 379-393 ◽  
Author(s):  
YO-SUB HAN ◽  
YAJUN WANG ◽  
DERICK WOOD
Keyword(s):  
Polynomial Time ◽  
Regular Language ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Regular Languages ◽  
Finite State Automaton ◽  
Primality Test ◽  
Finite State ◽  
State Pair ◽  
Free Decomposition

We study infix-free regular languages. We observe the structural properties of finite-state automata for infix-free languages and develop a polynomial-time algorithm to determine infix-freeness of a regular language using state-pair graphs. We consider two cases: 1) A language is specified by a nondeterministic finite-state automaton and 2) a language is specified by a regular expression. Furthermore, we examine the prime infix-free decomposition of infix-free regular languages and design an algorithm for the infix-free primality test of an infix-free regular language. Moreover, we show that we can compute the prime infix-free decomposition in polynomial time. We also demonstrate that the prime infix-free decomposition is not unique.

scholarly journals A Transformation-Based Approach to Implication of GSTE Assertion Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Guowu Yang ◽  
William N. N. Hung ◽  
Xiaoyu Song ◽  
Wensheng Guo
Keyword(s):  
Model Checking ◽  
Directed Graphs ◽  
Classical Model ◽  
Regular Languages ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Symbolic Trajectory Evaluation ◽  
Finite State ◽  
Symbolic Trajectory ◽  
Language Containment

Generalized symbolic trajectory evaluation (GSTE) is a model checking approach and has successfully demonstrated its powerful capacity in formal verification of VLSI systems. GSTE is an extension of symbolic trajectory evaluation (STE) to the model checking ofω-regular properties. It is an alternative to classical model checking algorithms where properties are specified as finite-state automata. In GSTE, properties are specified as assertion graphs, which are labeled directed graphs where each edge is labeled with two labeling functions: antecedent and consequent. In this paper, we show the complement relation between GSTE assertion graphs and finite-state automata with the expressiveness of regular languages andω-regular languages. We present an algorithm that transforms a GSTE assertion graph to a finite-state automaton and vice versa. By applying this algorithm, we transform the problem of GSTE assertion graphs implication to the problem of automata language containment. We demonstrate our approach with its application to verification of an FIFO circuit.

scholarly journals Incremental Construction and Maintenance of Minimal Finite-State Automata

2002 ◽  
Vol 28 (2) ◽  
pp. 207-216 ◽  
Author(s):  
Rafael C. Carrasco ◽  
Mikel L. Forcada
Keyword(s):  
Computational Linguistics ◽  
Efficient Method ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Construction Problem ◽  
Finite State ◽  
Single String ◽  
Intersection Operation ◽  
Incremental Construction ◽  
Special Case

Daciuk et al. [Computational Linguistics 26(1):3–16 (2000)] describe a method for constructing incrementally minimal, deterministic, acyclic finite-state automata (dictionaries) from sets of strings. But acyclic finite-state automata have limitations: For instance, if one wants a linguistic application to accept all possible integer numbers or Internet addresses, the corresponding finite-state automaton has to be cyclic. In this article, we describe a simple and equally efficient method for modifying any minimal finite-state automaton (be it acyclic or not) so that a string is added to or removed from the language it accepts; both operations are very important when dictionary maintenance is performed and solve the dictionary construction problem addressed by Daciuk et al. as a special case. The algorithms proposed here may be straightforwardly derived from the customary textbook constructions for the intersection and the complementation of finite-state automata; the algorithms exploit the special properties of the automata resulting from the intersection operation when one of the finite-state automata accepts a single string.

scholarly journals On the Derivational Entropy of Left-to-Right Probabilistic Finite-State Automata and Hidden Markov Models

2018 ◽  
Vol 44 (1) ◽  
pp. 17-37 ◽  
Author(s):  
Joan Andreu Sánchez ◽  
Martha Alicia Rocha ◽  
Verónica Romero ◽  
Mauricio Villegas
Keyword(s):  
Hidden Markov Models ◽  
Markov Models ◽  
Hidden Markov ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Efficient Computation ◽  
Finite State ◽  
Probabilistic Finite State Automata ◽  
State Models ◽  
Finite State Models

Probabilistic finite-state automata are a formalism that is widely used in many problems of automatic speech recognition and natural language processing. Probabilistic finite-state automata are closely related to other finite-state models as weighted finite-state automata, word lattices, and hidden Markov models. Therefore, they share many similar properties and problems. Entropy measures of finite-state models have been investigated in the past in order to study the information capacity of these models. The derivational entropy quantifies the uncertainty that the model has about the probability distribution it represents. The derivational entropy in a finite-state automaton is computed from the probability that is accumulated in all of its individual state sequences. The computation of the entropy from a weighted finite-state automaton requires a normalized model. This article studies an efficient computation of the derivational entropy of left-to-right probabilistic finite-state automata, and it introduces an efficient algorithm for normalizing weighted finite-state automata. The efficient computation of the derivational entropy is also extended to continuous hidden Markov models.

scholarly journals Functional approach to deterministic finite-state automata systems dynamic modeling

2021 ◽  
Vol 1 ◽  
pp. 113-122
Author(s):  
Eduard S. Lapin ◽  
◽  
Marat I. Abdrakhmanov ◽  
Keyword(s):  
Dynamic Modeling ◽  
Mining Industry ◽  
Good Alternative ◽  
Functional Approach ◽  
Algorithm Analysis ◽  
Substantial Part ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Advantages And Disadvantages ◽  
Finite State

Research aim is to study the functional approach to modeling the deterministic finite-state automata system which is not confined to the elements communication topology and the heterogeneity of the algorithm types. Relevance. The substantial part of engineering systems applied in the mining industry may be described through the finite-state automaton model. They include the mine conveyor systems, shaft signal systems, processing facilities control systems, etc. Such model makes it possible to shorten the time spent on control software development and carry out algorithm analysis, debug, and testing effectively. There are a lot of effective approaches and tools to solve the problem of finite-state automata dynamic modeling, each of which has its own advantages and disadvantages. Methodology. In this article, the methodology of finite-state automata systems modeling is considered as applied to mine conveyor systems. Results. Final-state automata (FSA) models have been developed together with the conditions for FSA systems dynamic modeling as applied to mine conveyor systems. Conclusions. The considered approach to modeling, which involves functors and applicative functors 122 "Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal". No. 2. 2021 ISSN 0536-1028 for structure composition and its operational dynamics study, as well as the possibility to mathematically prove the model’s properties, makes the approach a good alternative when choosing tools for systems models development.

scholarly journals An Interaction Between User and an Augmented Reality System using A Generalized Finite State Automata and A Universal Turing Machine

2019 ◽  
Vol 8 (12) ◽  
pp. 1084-1090
Keyword(s):  
Augmented Reality ◽  
Turing Machine ◽  
Regular Language ◽  
Formal Language ◽  
Linear Complexity ◽  
Finite State Automata ◽  
Augmented Reality System ◽  
Universal Turing Machine ◽  
Finite State ◽  
Single Response

Most interactions between users and augmented reality system (ARS) are that user assigns a marker to ARS, and the ARS responds the marker. In this context, a marker is mapped to an ARS's response, or in general, an array of markers is mapped to an array of ARS's responses. This interaction is a constant or linear complexity interaction since there is only a bijective mapping between a set of markers and a set of ARS's responses. In this research, we propose the expansion of user - ARS complexity into the polynomial. It is an interaction in which not only one marker for a single response (or an array of markers for an array of ARS's responses), but the interaction by which user provides a string of markers as a word of markers (i.e., a combination of multiple markers as a word) for a single ARS's response. The set of strings of markers to the ARS provided by users built a regular language. So that, the complexity of the user-ARS interaction became polynomial. This interaction was implemented by stating the user's language by means of a generalization of finite state automata (gFSA) and placing a universal Turing machine (UTM) between user and ARS, where the UTM as an interpreter translating or mapping the user language to ARS. To summarize our research, overall we apply the idea of a formal language into the interaction between the user and ARS, thereby changing the complexity of the interaction to polynomial even expandable to nondeterministic polynomials.

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