Abstract
In the smallest grammar problem, we are given a word w and we want to compute a preferably small context-free grammar G for the singleton language {w} (where the size of a grammar is the sum of the sizes of its rules, and the size of a rule is measured by the length of its right side). It is known that, for unbounded alphabets, the decision variant of this problem is NP-hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = NP. We settle the long-standing open problem whether these hardness results also hold for the more realistic case of a constant-size alphabet. More precisely, it is shown that the smallest grammar problem remains NP-complete (and its optimisation version is APX-hard), even if the alphabet is fixed and has size of at least 17. The corresponding reduction is robust in the sense that it also works for an alternative size-measure of grammars that is commonly used in the literature (i. e., a size measure also taking the number of rules into account), and it also allows to conclude that even computing the number of rules required by a smallest grammar is a hard problem. On the other hand, if the number of nonterminals (or, equivalently, the number of rules) is bounded by a constant, then the smallest grammar problem can be solved in polynomial time, which is shown by encoding it as a problem on graphs with interval structure. However, treating the number of rules as a parameter (in terms of parameterised complexity) yields W[1]-hardness. Furthermore, we present an mathcal {O}(3^{mid {w}mid }) exact exponential-time algorithm, based on dynamic programming. These three main questions are also investigated for 1-level grammars, i. e., grammars for which only the start rule contains nonterminals on the right side; thus, investigating the impact of the “hierarchical depth” of grammars on the complexity of the smallest grammar problem. In this regard, we obtain for 1-level grammars similar, but slightly stronger results.
Highlights
Context-free grammars are among the most classical concepts in theoretical computer science
From a formal languages point of view, describing a single word by a context-free grammar seems excessive, there are at least two evident motivations: – Compression Perspective:2 The grammar G is a compressed representation of the word w. – Inference Perspective: The grammar G identifies the hierarchical structure of the word w
The inference perspective of computing grammars for single words has been applied in two more PhD-theses, namely by de Marcken [3] in order to investigate whether analysing the structure of small grammars for large English texts could help understanding the structure of the language itself, and by Galle [4] in order to infer hierarchical structures in DNA
Summary
Context-free grammars are among the most classical concepts in theoretical computer science. We are concerned with grammars G that describe singleton languages {w} (or, by slightly abusing notation, grammars describing single words).
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