Formal Languages
Formal Languages Wednesday, September 29, 2010 Reading: Sipser pp 13-14, 44-45 Stoughton 2.1, 2.2, end of 2.3, beginning of 3.1 CS235 Languages and Automata Department of Computer Science Wellesley College Alphabets, Strings, and Languages An alphabet is a set of symbols. E.g.: 1 = {0,1}; 2 = {-,0,+} 3 = {a,b, …, y, z}; 4 = {, , a , aa } A string over is a seqqfyfuence of symbols from . The empty string is ofte written (Sipser) or ; Stoughton uses %. * denotes all strings over . E.g.: * • 1 contains , 0, 1, 00, 01, 10, 11, 000, … * • 2 contains , -, 0, +, --, -0, -+, 0-, 00, 0+, +-, +0, ++, ---, … * • 3 contains , a, b, …, aa, ab, …, bar, baz, foo, wellesley, … * • 4 contains , , , a , aa, …, a , …, a aa , aa a ,… A lnlanguage over (Stoug hto nns’s -lnlanguage ) is any sbstsubset of *. I.e., it’s a (possibly countably infinite) set of strings over . E.g.: •L1 over 1 is all sequences of 1s and all sequences of 10s. •L2 over 2 is all strings with equal numbers of -, 0, and +. •L3 over 3 is all lowercase words in the OED. •L4 over 4 is {, , a aa }. Formal Languages 11-2 1 Languages over Finite Alphabets are Countable A language = a set of strings over an alphabet = a subset of *. Suppose is finite. Then * (and any subset thereof) is countable. Why? Can enumerate all strings in lexicographic (dictionary) order! • 1 = {0,1} 1 * = {, 0, 1 00, 01, 10, 11 000, 001, 010, 011, 100, 101, 110, 111, …} • for 3 = {a,b, …, y, z}, can enumerate all elements of 3* in lexicographic order -- we’ll eventually get to any given element.
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