Cezar Câmpeanu

Pattern expressions and pattern automata

We define the pattern expressions as an extension of both regular expressions and patterns. We prove several properties of the new family of languages, similar to those of extended regex languages [Campeanu et al., Int. J. Found. Comput. Sci. 14 (6) (2003) 1007-1018]. We also define an automata system that recognizes these languages. Differences between regex and pattern expressions are also discussed.

On the intersection of regex languages with regular languages

In this paper we revisit the semantics of extended regular expressions (regex), defined succinctly in the 90s [A.V. Aho, Algorithms for finding patterns in strings, in: Jan van Leeuwen (Ed.), Handbook of Theoretical Computer Science, in: Algorithms and Complexity, vol. A, Elsevier and MIT Press, 1990, pp. 255-300] and rigorously in 2003 by Campeanu, Salomaa and Yu [C. Campeanu, K. Salomaa, S. Yu, A formal study of practical regular expressions, IJFCS 14 (6) (2003) 1007-1018], when the authors reported an open problem, namely whether regex languages are closed under the intersection with regular languages. We give a positive answer; and for doing so, we propose a new class of machines - regex automata systems (RAS) - which are equivalent to regex. Among others, these machines provide a consistent and convenient method of implementing regex in practice. We also prove, as a consequence of this closure property, that several languages, such as the mirror language, the language of palindromes, and the language of balanced words are not regex languages.

Incremental construction of minimal deterministic finite cover automata

We present a fast incremental algorithm for constructing minimal Deterministic Finite Cover Automata (DFCA) for a given language. Since it was shown that the minimal DFCA for a language L has less states than the minimal Deterministic Finite Automata (DFA) for the same language L, this technique seems to be the best choice for incrementally building the automaton for a large language, especially when the number of states in the DFCA is significantly less than the number of states in the corresponding minimal DFA. We have implemented the proposed algorithm and have tested it against the best-known DFCA minimization technique.

State Complexity of Regular Languages: Finite versus Infinite

We consider the state complexity of regular languages and their operations. Especially, we compare the state complexity results on finite languages and general regular languages. The similarity relation ~ L and the equivalence relation ≡ L over ∑* are also compared. Their applications on minimization of deterministic finite cover automata and deterministic finite automata, respectively, are investigated.

Are binary codings universal

It is common sense to notice that one needs fewer digits to code numbers in ternary than in binary; new names are about log32 times shorter. Is this trade-off a consequence of the special coding scheme? The answer is negative. More generally, we argue that the answer to the question, stated in the title and formulated to the first author by C. Rackhoff, is in fact negative. The conclusion will be achieved by studying the role of the size of the alphabet in constructing instantaneous codes for all natural numbers, and defining random strings and sequences. We show that there is no optimal instantaneous code for all positive integers, and the binary is the worst possible. Codes over a fixed alphabet can be indefinitely improved themselves, but only “slightly”; in contrast, changing the size of the alphabet determines a significant, not linear, improvement. The key relation describing the above phenomenon can be expressed in terms of Chaitin complexity: changing the size of the coding alphabet from q to Q, 2 ≤ q < Q, results in an improvement of the complexity by a factor og log q. As a consequence, a string avoiding consistently a fixed letter is not random. In binary, this corresponds to a trivial situation. In the nonbinary case the distinction is relevant: more than 3.2n ternary strings of length n are not random (many of these strings are binary random). This phenomenon is even sharper for infinite sequences.

COUNTING THE NUMBER OF MINIMAL DFCA OBTAINED BY MERGING STATES

Finite Deterministic Cover Automata (DFCA) can be obtained from Deterministic Finite Automata (DFA) using the similarity relation and a method of merging similar states. The DFCA minimization procedure can yield different results depending on the order of merging the similar states, because the minimal DFCA for a finite language is in general not unique. We count the number of minimal DFCA that can be obtained from a given minimal DFA with n states by merging the similar states in the given DFA. We compute an upper-bound for this number and prove that in the worst case, it is n-1 for an unary alphabet, and

A note on blum static complexity measures

\frac{\lceil\frac{4n-9+\sqrt{8n+1}}{8}\rceil !}{(2\lceil \frac{4n-9+\sqrt{8n+1}}{8}\rceil -n+1)!}

An incremental algorithm for constructing minimal deterministic finite cover automata

for a non-unary alphabet. We prove that this upper-bound is reached, i.e., for any given positive integer n one can construct a minimal DFA with n states, which has the number of minimal DFCA obtained by merging similar states equal to this maximum expression.

The number of similarity relations and the number of minimal deterministic finite cover automata

Dual complexity measures have been developed by Burgin, under the influence of the axiomatic system proposed by Blum in [3]. The concept of dual complexity measure is a generalization of Kolmogorov/Chaitin complexity, also known as algorithmic or static complexity. In this paper we continue this effort by extending some of the well known results for plain and prefix-free complexities to the general case of Blum universal static complexity. We also extend some results obtained by Calude in [9] to a larger class of computable measures, proving that transducer complexity is a dual (Blum static) complexity measure.

Note on the topological structure of random strings

We present a fast incremental algorithm for constructing minimal DFCA for a given language. Since it was shown that the DFCA for a language L can have less states than the DFA for L, this technique seems to be the best choice for incrementally building the automaton for a large language, especially when the number of states in the DFCA is significantly less than the number of states in the corresponding minimal DFA. We have implemented the proposed algorithm and have tested it against the best known DFCA minimization technique.