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The Smallest Grammar Problem as Constituents Choice and Minimal Grammar Parsing
Grupo de Procesamiento de Lenguaje Natural Universidad Nacional de Córdoba/Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina
Symbiose Project, IRISA/INRIA Rennes-Bretagne Atlantique, France
* Authors to whom correspondence should be addressed.
Received: 12 October 2011; Accepted: 14 October 2011 / Published: 26 October 2011
Abstract: The smallest grammar problem—namely, finding a smallest context-free grammar that generates exactly one sequence—is of practical and theoretical importance in fields such as Kolmogorov complexity, data compression and pattern discovery. We propose a new perspective on this problem by splitting it into two tasks: (1) choosing which words will be the constituents of the grammar and (2) searching for the smallest grammar given this set of constituents. We show how to solve the second task in polynomial time parsing longer constituent with smaller ones. We propose new algorithms based on classical practical algorithms that use this optimization to find small grammars. Our algorithms consistently find smaller grammars on a classical benchmark reducing the size in 10% in some cases. Moreover, our formulation allows us to define interesting bounds on the number of small grammars and to empirically compare different grammars of small size.
Keywords: smallest grammar problem; hierarchical structure inference; optimal parsing; data discovery
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MDPI and ACS Style
Carrascosa, R.; Coste, F.; Gallé, M.; Infante-Lopez, G. The Smallest Grammar Problem as Constituents Choice and Minimal Grammar Parsing. Algorithms 2011, 4, 262-284.
Carrascosa R, Coste F, Gallé M, Infante-Lopez G. The Smallest Grammar Problem as Constituents Choice and Minimal Grammar Parsing. Algorithms. 2011; 4(4):262-284.
Carrascosa, Rafael; Coste, François; Gallé, Matthias; Infante-Lopez, Gabriel. 2011. "The Smallest Grammar Problem as Constituents Choice and Minimal Grammar Parsing." Algorithms 4, no. 4: 262-284.