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For example, the question of whether a given regular language is positive (does not include the empty string) is algorithmically decidable.
"Positiveness Problem".
Note that each instance of the Positiveness Problem is a regular language. (Each instance itself is, not the set of solved instances.) Clearly, we cannot take the set of strings in the language to be our instance, (since, in general, this is likely to be in?nite in size. But we have at least two means of specifying any regular language using ?nite objects: we can give a Finite State Automaton that recognizes the language as a ?ve-tuple, each component of which is ?nite, (or, equivalently, the transition graph in some other form) or we can give a regular expression. Since we have algorithms for converting back and forth between these two forms, we can choose whichever is convenient for us. In this case, lets assume we are given the ?ve-tuple. Since we have an algorithm for converting NFAs to DFAs as well, we can also assume, without loss of generality, that the automaton is a DFA.
A solution to the Positiveness Problem is just "True" or "False". It is a decision problem a problem of deciding whether the given instance exhibits a particular property. (We are familiar with this sort of problem. They are just our "checking problems"-all our automata are models of algorithms for decision problems.) So the Positiveness Problem, then, is just the problem of identifying the set of Finite State Automata that do not accept the empty string. Note that we are not asking if this set is regular, although we could. (What do you think the answer would be?) We are asking if there is any algorithm at all for solving it.
We'll close our consideration of regular languages by looking at whether (certain) problems about regular languages are algorithmically decidable.
The fact that regular languages are closed under Boolean operations simpli?es the process of establishing regularity of languages; in essence we can augment the regular operations
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Kleene called this the Synthesis theorem because his (and your) proof gives an effective procedure for synthesizing an automaton that recognizes the language denoted by any given r
Theorem The class of recognizable languages is closed under Boolean operations. The construction of the proof of Lemma 3 gives us a DFA that keeps track of whether or not a give
A problem is said to be unsolvable if no algorithm can solve it. The problem is said to be undecidable if it is a decision problem and no algorithm can decide it. It should be note
Normal forms are important because they give us a 'standard' way of rewriting and allow us to compare two apparently different grammars G1 and G2. The two grammars can be shown to
#Your company has 25 licenses for a computer program, but you discover that it has been copied onto 80 computers. You informed your supervisor, but he/she is not willing to take an
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