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Understanding Formal Languages and Finite State Machines in CSE 467/567

This resource provides a comprehensive overview of formal languages, regular expressions, and finite state machines, aligned with the curriculum of CSE 467/567. It covers the definition of languages as sets of strings, Chomsky's hierarchy, and the recursive definition of regular languages. Key concepts include set operations, finite automata, and the distinction between deterministic and nondeterministic finite automata (DFA and NFA). The material also emphasizes practical examples and techniques for constructing these automata, making it ideal for students pursuing an in-depth understanding of computational theory.

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Understanding Formal Languages and Finite State Machines in CSE 467/567

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  1. Formal languages • regular expressions • regular languages • finite state machines CSE 467/567

  2. Formal languages A string is a (perhaps empty) sequence of symbols. e denotes the empty string. A language is a (perhaps empty) set of strings. Æ denotes the empty set. There are many different classes of languages. Main ones in Chomsky hierarchy are regular, context free, context sensitive and unrestricted. CSE 467/567

  3. Sets and set operations Examples: Æ is the empty set, a set with no members {a, b, c} is a set with three members Operations: Suppose A={a, b, c} and B={1, 2}, then A·B={a1, a2, b1, b2, c1, c2} A´B={(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)} AÈB={a, b, c, 1, 2} A*={e, a, b, c, aa, ab, ac, ba, bb, bc, aaa, aab, aac, …} CSE 467/567

  4. Regular languages The class of regular languages over an alphabet å can be defined recursively: base case 1: Æ is a regular language base case 2: {e} is a regular language base case 3: for each symbol a in å, {a} is a regular language recursive cases: If S and T are regular languages, then so are: {st| s is in S and t is in T}, the concatenation of S and T {x| x is in S or x is in T}, the disjunction of S and T S*, the Kleene closure of S Nothing else is a regular language. CSE 467/567

  5. Finite state automata Formally a 5-tuple (Q, å,q0,F,d) where Q is a finite set of states å is a finite input alphabet of symbols q0 Î Q is the initial state F Í Q is a set of final states d: Q´åQ CSE 467/567

  6. Examples FSA (5-tuple and diagram) accepting each of the following (assume å={a,b,c,…z}): {a} {fred, wilma} {ball, bell, bill, boll, bull} {e, a} {e, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, …} CSE 467/567

  7. RE’s outside of Perl? Can always directly implement the DFA. • switch on state, switch on symbol • table-driven method (put d in an array) Example! CSE 467/567

  8. Construction of FA Base cases • Æ is a regular language • {e} is a regular language • for each symbol a in å, {a} is a regular language Recursive cases If S and T are regular languages, then so are: • {st| s is in S and t is in T}, the concatenation of S and T • {x| x is in S or x is in T}, the disjunction of S and T • S*, the Kleene closure of S CSE 467/567

  9. NFA vs. DFA Accept same set of languages. Simulation of NFA through search: depth-first (stack regime for next node) breadth-first (queue regime) Can mechanically convert NFA to DFA, with possible exponential increase in number of states. CSE 467/567

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