The Smallest Grammar Problem

Inductive Synthesis of Cover-Grammars with the Help of Ant Colony Optimization

“…A similar concept has been studied in the context of automata [3]. Although any n-state deterministic finite automaton for some finite language L can be converted into a corresponding minimal cover-automaton, using only O(n log n) time [17], the construction of a minimal covergrammar seems to be intractable, specially in view of the following facts: (1) there is no polynomial-time algorithm for obtaining the smallest context-free grammar that generates exactly one given word (unless P = NP) [4]; (2) context-free grammar equivalence and even equivalence between a context-free grammar and a regular expression are undecidable [15]; (3) for any alphabet of a size of at least 2, the class of context-free grammars is not polynomially characterizable [12]; (4) the grammar can be exponentially smaller than any word in the language (an example is given in a book [14]). To our best knowledge there are no published algorithms for a cover-grammar problem defined as above.…”

Dictionary-Based Data Compression

“…Charikar et al [2] considered LZ78 as an approximation algorithm for the NP-hard problem of finding the smallest context-free grammar that generates only the string T.T h eLZ78 parsing of T can be viewed as a contextfree grammar in which for each dictionary word T i = T jt here is a production X i ! X j˛.…”

Approximability of Minimum AND-Circuits

“…, M k } be a Min-3-AC instance with opt Min-3-AC (M) = k + . Then Greedy outputs a circuit C Greedy for M of size at most min 4 3 · k + , 1 + 1 e 2 k + 2 .…”

“…Note that the greedy algorithms proposed in Section 6.1 are closely related to the so-called global algorithms Re-Pair, Greedy, and Longest Match for the smallest grammar problem [4], which deals with the compression of a given string by a context-free grammar that generates exactly that string. Global algorithms are of particular interest for this problem since they are believed to have low approximation ratios.…”

“…The monomials may for example arise in the DNF-representation of a Boolean function or in some decomposed or factored form. Thus, the Minimum-AND-Circuit problem is of fundamental interest for automated circuit design, see Charikar et al [4,Sect. VII.B] and references therein.…”