On the arithmetic power of context-free languages

• Authors: Albert Goldfain
• Venue: CCSC '01 Proceedings of the sixth annual CCSC northeastern conference on The journal of computing in small colleges
• Year: 2001

Our research focuses on characterizing the computational limits of context-free languages based on a non-classical definition of “computation”. In our study we identify a computation as a process of recognizing a string of a's and b's in which the number b's is a function of the number of a's. We feel that this ability to “compute” arithmetic functions in this sense is a fundamental property that should be examined for all models of computation. Our poster presents a proof of what we have called the Arithmetic Power Theorem for Context-Free Languages that we state as follows:Arithmetic Power Theorem for Context-Free Languages: A language L of the form L=ai bf(i) is context-free only if f(i)= &THgr;(i) or f(i) = &THgr;(1).Our approach in proving this theorem was directed towards a reformulation of our question based on subsets of L that we called pumping sets. The notion of a pumping set is a generalization, and in many ways is an extension, of the well-known Pumping Lemma that exposes the ability to “pump” a given string in a context-free language into another string in the same language via replication. Pumping sets are not found in the standard textbooks on computational theory [1][2][3] and to our knowledge are absent from the existing research on context-free grammars. However, the study of sets of pumped strings (known as paired loops) has yielded interesting results in the area of slender languages [4].We define a pumping set as follows: A pumping set of a derivation is the set of all strings obtained by replicating a path between two occurrences of some non-terminal in the derivation tree. This is illustrated in the following example:We call the derivation in the definition of pumping set a generator of a pumping set. It was necessary to establish several properties which would allow us to use pumping sets to characterize the ability of a production rule in a grammar (or a sequence of such rules) to affect the growth of b's in the strings of our language L.