A hierarchy of deterministic context-free ω-languages

Twenty years ago, Klaus. W. Wagner came up with a hierarchy of ω-regular sets that actually bears his name. It turned out to be exactly the Wadge hierarchy of the sets of ω-words recognized by deterministic finite automata. We describe the Wadge hierarchy of context-free ω-languages, which stands as an extension of Wagner's work from automata to pushdown automata.     

Extensions of sokolowski's theorem to prove languages are not context free or not regular

An extension of a theorem of Sokolowski's is given which is more generally applicable and illustrated by showing several languages are not context free. This is then modified to prove a similar kind of theorem for regular languages.     

Recursive functions of context free languages (I)

It is intended to establish the recursive function theory on context free languages (CFLs). In this paper, the function class CFRF and its proper subclass CFPRF were defined on CFLs; it is quite straightforward to use them for describing non-numerical algorithms. In fact, they are respectively the partial recursive functions and primitive recursive functions of context free languages. The structure induction method for proving CFPRF function properties was presented. A method for CFL sentence enumeration was given, the minimization operator was defined. Based on CFL sentence enumeration, the minimization operator evaluation method was given. Finally, the design and implementation principles of executable specification languages with the CFRF as theoretical basis were discussed. This paper is based on the Technical Report ISCAS-LCS-2k-03 (SAQ Report No. 30): Recursive Functions Defined on Context Free Languages (I), August 2000, with minor revisions.     

Recursive functions of context free languages (I)--The definitions of CFPRF and CFRF

It is intended to establish the recursive function theory on context free languages (CFLs). In this paper, the function class CFRF and its proper subclass CFPRF were defined on CFLs; it is quite straightforward to use them for describing non_numerical algorithms. In fact, they are respectively the partial recursive functions and primitive recursive functions of context free languages. The structure induction method for proving CFPRF function properties was presented. A method for CFL sentence enumeration was given, the minimization operator was defined. Based on CFL sentence enumeration, the minimization operator evaluation method was given. Finally, the design and implementation principles of executable specification languages with the CFRF as theoretical basis were discussed.     

Incremental learning of context free grammars based on bottom-up parsing and search

This paper describes approaches for machine learning of context free grammars (CFGs) from positive and negative sample strings, which are implemented in Synapse system. The grammatical inference consists of a rule generation by “inductive CYK algorithm,” mechanisms for incremental learning, and search. Inductive CYK algorithm generates minimum production rules required for parsing positive samples, when the bottom-up parsing by CYK algorithm does not succeed. The incremental learning is used not only for synthesizing grammars by giving the system positive strings in the order of their length but also for learning grammars from other similar grammars. Synapse can synthesize fundamental ambiguous and unambiguous CFGs including nontrivial grammars such as the set of strings not of the form ww with w∈{a,b}⁺.     

Recursive functions of context free languages (Ⅱ)——Validity of CFPRF and CFRF definitions

In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains . It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.