Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Tools and Algorithms for the Construction and Analysis of Systems, 7th International Conference, TACAS 2001, Genova, Italy, April 2-6, 2001

Finite State Morphology

The ISO standard for the Standard Generalized Markup Language (SGML) provides a syntactic meta-language for the definition of textual markup systems. In the standard, the right-hand sides of productions are based on regular expressions, although only regular expressions that denote words unambiguously, in the sense of the ISO standard, are allowed. In general, a word that is denoted by a regular expression is witnessed by a sequence of occurrences of symbols in the regular expression that match the word. In an unambiguous regular expression as defined by Book et al. (1971, IEEE Trans. Comput. C-20 (2), 149–153), each word has at most one witness. But the SGML standard also requires that a witness be computed incrementally from the word with a one-symbol lookahead; we call such regular expressions 1- unambiguous . A regular language is a 1- unambiguous language if it is denoted by some 1-unambiguous regular expression. We give a Kleene theorem for 1-unambiguous languages and characterize 1-unambiguous regular languages in terms of structural properties of the minimal deterministic automata that recognize them. As a result we are able to prove the decidability of whether a given regular expression denotes a 1-unambiguous language; if it does, then we can construct an equivalent 1-unambiguous regular expression in worst-case optimal time.

One-unambiguous regular languages

Formal concept analysis via multi-adjoint concept lattices

Canonical derivatives, partial derivatives and finite automaton constructions

We present different techniques for reducing the number of states and transitions in nondeterministic automata. These techniques are based on the two preorders over the set of states, related to the inclusion of left and right languages. Since their exact computation is NP-hard, we focus on polynomial approximations which enable a reduction of the NFA all the same. Our main algorithm relies on a first approximation, which can be easily implemented by means of matrix products with an O(mn3) time complexity, and optimized to an O(mn) time complexity, where m is the number of transitions and n is the number of states. This first algorithm appears to be more efficient than the known techniques based on equivalence relations as described by Lucian Ilie and Sheng Yu. Afterwards, we briefly describe some more accurate approximations and the exact (but exponential) calculation of these preorders by means of determinization.

NFA reduction algorithms by means of regular inequalities

The notion of expression derivative due to Brzozowski leads to the construction of a deterministic automaton from an extended regular expression, whereas the notion of partial derivative due to Antimirov leads to the construction of a non-deterministic automaton from a simple regular expression. In this paper, we generalize Antimirov partial derivatives to regular expressions extended to complementation and intersection. For a simple regular expression with n symbols, Antimirov automaton has at most n+1 states. As far as an extended regular expression is concerned, we show that the number of states can be exponential.

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Partial derivatives of an extended regular expression

Our aim is to give a proof of the fact that two notions related to regular expressions, the prebases due to Mirkin and the partial derivatives introduced by Antimirov lead to the construction of identical nondeterministic automata recognizing the language of a given regular expression.

From Mirkin's Prebases to Antimirov's Word Partial Derivatives

This document gives a generalization on the alphabet size of the method that is described in Nicaud's thesis for randomly generating complete DFAs. First, we recall some properties of m-ary trees and we give a bijection between the set of m-ary trees and the set R(m, n) of generalized tuples. We show that this bijection can be built on any total prefix order on Σ*, Then we give the relations that exist between the elements of R(m,n) and complete DFAs built on an alphabet of size greater than 2. We give algorithms that allow us to randomly generate accessible complete DFAs. Finally, we provide experimental results that show that most of the accessible complete DFAs built on an alphabet of size greater than 2 are minimal.

Jean-Marc Champarnaud

Papers

Canonical derivatives, partial derivatives and finite automaton constructions

NFA reduction algorithms by means of regular inequalities

Partial derivatives of an extended regular expression

From Mirkin's Prebases to Antimirov's Word Partial Derivatives

Random generation of DFAs