Table-Automata / Finite-coFinite Languages Paul Cull Oregon State University pc©cs .orst .edu The equivalence between a class of machines and a corresponding class o f languages can be obscured by technical details . Students may lose sight of th e problem while trying to master the techniques needed for the correspondence . For a recent class I came up with the following simple example which ma y be useful to others . Let T be the class of languages which consists of the Finite languages an d the coFinite languages . A language L over the alphabet E is coFinite when â L is Finite . (coFinite is a new idea to many students so a few simpl e examples are necessary . Also . ask the students to classify some example set s as finite, cofinite . or infinite not cofinite . ) One can then ask students to decide whether or not .F is closed unde r the usual set operations : union . complement . and intersection . (Showing intersection closure is a nice example of using DeMorgan's law . ) Next . one can introduce the string operations of concatenation and reversal. Extending reversal to a set of strings is clear . Concatenating two sets o f strings is less clear and needs some examples . After concatenation has bee n explained, star can be presented and explained . Is .F closed under reversal . concatenation, and star? (For concatenation, the answer depends on the siz e of the alphabet . ) .F can be tied to a class of automata, the table automata . A table automaton consists of a table which is a finite list of strings and for each string . x . in the table. there is a value v(x) . where v(x) E {Yes . NO . and a convention so that if x is not in the table . then e(x) has the conventional value e . Of course . e E {Yes . NO . So there are two types of table automata-, the YES type which says YE S for all strings not in the table, and the NO type which says NO for all string s not in the table . The obvious exercises ar e 1. Define L(M) the language recognized by a table-automaton M . 2. Show that a language is in T if and only if the language is recognize d by a table-automaton .
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Table-automata/ finite co-finite languages
Cull, Paul
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, Volume 30 (1)
Association for Computing Machinery – Mar 1, 1999
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