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give the myhill graph of your automatonyou may use a single node to represent the entire set of symbols of the english alphabet another to represent
another way of representing a strictly 2-local automaton is with a myhill graph these are directed graphs in which the vertices are labeled with
sketch an algorithm for the universal recognition problem for sl2 this takes an automaton and a string and returns true if the string is accepted by
suppose a sigma t is an sl2 automaton sketch an algorithm for recognizing la by in essence implementing the automaton your algorithm should work
the computation of an sl2 automaton a sigma t on a string w is the maximal sequence of ids in which each sequential pair of ids is related by -a
we will assume that the string has been augmented by marking the beginning and the end with the symbols lsquo and lsquo respectively and that these
the fundamental idea of strictly local languages is that they are specied solely in terms of the blocks of consecutive symbols that occur in a word
we can then specify any language in the class of languages by specifying a particular automaton in the class of automata we do that by specifying
as we are primarily concerned with questions of what is and what is not computable relative to some particular model of computation we will usually
the upper string r isin q is the sequence of states visited by the automaton as it scans the lower string w isin sigma we will refer to this string
the sl2 languages are specied with a set of 2-factors in sigma2 plus some factors in sigma and some factors in sigma distinguishing symbols that may
we developed the idea of fsa by generalizing ltk transition graphs not surprisingly then every ltk transition graph is also the transition graph of a
the initial id of the automaton given in figure 3 running on input lsquoaabbba isa aabbbathe id after the rst three transitions of the computation
denition instantaneous description of an fsaan instantaneous description id of a fsa a qsigma t q0 f is a pair qw isin qtimessigma where q the
denition deterministic finite state automaton for any state set q and alphabet sigma both nite a nite state automaton fsa over qandsigma is a
since the signicance of the states represented by the nodes of these transition graphs is arbitrary we will allow ourselves to use any nite set such
another striking aspect of ltk transition graphs is that they are generally extremely inecient all we really care about is whether a path through the
lemma 1 a string w isin sigma is accepted by an ltk automaton iff w is the concatenation of the symbols labeling the edges of a path through the ltk
we represented slk automata as myhill graphs directed graphs in which the nodes were labeled with k-1-factors of alphabet symbols along with a node
so we have that every language that can be constructed from sl languages using boolean operations and concatenation that is every language in lto is
we saw earlier that lt is not closed under concatenation if we think in terms of the lt graphs recognizing the concatenation of lt languages would
theorem the class of recognizable languages is closed under boolean operationsthe construction of the proof of lemma 3 gives us a dfa that keeps
we got the class lt by taking the class sl and closing it under boolean operations we have observed that lt sube recog so certainly any boolean
in general non-determinism by introducing a degree of parallelism may increase the accepting power of a model of computation but if we subject nfas
automaton nfa with epsilon-transitions is a 5-tuple qsigma delta q0 fi where q sigma q0 and f are as in a dfa and t sube q times q times sigma cup