1.0 Languages, Expressions, Automata
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1 .0 Languages, Expressions, Automata Alphaet: Language: a finite set, typically a set of symols. a particular suset of the strings that can e made from the alphaet. ex: an alphaet of digits = {-,0,,2,3,4,5,6,7,8,9} a language of integers = {0,,2,...,0,02,03,...,-,-2,etc.} Note that strings such as 2-20 would not e included in this language. Regular Expression: A pattern that generates (only) the strings of a desired language. It is made up of letters of the language s alphaet, as well as of the following special characters: ( ) used for grouping repetition concatenation (usually omitted) + denotes a choice ( or ). ë a special symol denoting the null string Precedence from highest to lowest: ( ) + formal (recursive) definition: If A is an alphaet, and a A, then a is a regular expression. ë is a regular expression. If r and s are regular expressions, then the following are also regular expressions: r*, r s = rs, r + s, and ( r ) examples: (assume that A = {a, } ) a a (or just aa ) matched only y the string aa a + a matched y exactly two strings: a and a * matched y { ë,,,,...} (a + a*)*a ( + ë) matched y aaa, and many others Some convenient extensions to regular expression notation: 2 4 aa = a, =, etc. + a = aa* = { any string of a's of positive length, i.e. excludes ë } ex: (a) = aa a, so don't try to use "algera. 2 ex: (a+) = (a+)(a+) = aa or a or a or. ex: (a+)* any string made up of a s and s..0 Languages, Expressions, Automata
2 Examples of regular expressions over {a, } : all strings that egin with a and end with a (a + )* all non empty strings of even length (aa + a + a + ) + all strings with at least one a (a + )* a (a + )* all strings with at least two a's (a + )* a (a + )* a (a + )* all strings of one or more 's with an optional single leading a (a + ë) + Tips: the language { a, a, aaa, } a + a + aaa + or a (ë + aa) + (a + ) or (a + ) + ( + aa) a or? Check the simplest cases Check for sins of omission (forgot some strings) Check for sins of commission (included some unwanted strings) More examples Find a regular expression for the following sets of strings on { a, }: All strings with at least two s. (a + )* (a + )* (a + )* All strings with exactly two s. a* a* a* All strings with at least one a and at least one. (a + )* (a + a) (a + )* All strings which end in a doule letter (two a s or two s). (a + )* (aa + ) All strings of even length (include length). (aa + + a + a)*.0 Languages, Expressions, Automata 2
3 Finite Automata: a particular, simplified model of a computing machine, that is a language recognizer : yes aaa text recognizer no A finite automaton (FSA) has five pieces:. S = a finite numer of states, 2. A = the alphaet, 3. S i = the state, 4. Y = one or more final or accept states, and 5. F = a transition function (mapping) etween states, F: S x A S. The transition function F is usually presented in one of two ways: as a tale (called a transition tale), or as a graph (called a transition diagram). Transition Tale (example): A= { a, }, S = { s, s, s }, S = s, Y = { s, s } 0 2 i 0 2 current input F a current state gives the next state s0 s0 s2 s s s0 s2 s0 s0.0 Languages, Expressions, Automata 3
4 Transition Diagram (example): Note that this FSA is: Complete (no undefined transitions) Deterministic (no choices) Skeleton Method - a useful solution technique in limited cases: The skeleton is a sequence of states assuming legal input. Construct the skeleton, presume that no additional states will e needed. The FSA must e complete and deterministic: for A= { a, }, every state has exactly two arcs leaving it, one laeled a and one laeled. example (skeleton): All strings containing aaa.0 Languages, Expressions, Automata 4
5 Examples Assume A= { a, }. Construct the following automata which:. Accepts strings of the form (a+)* a, 2. Accepts ë only. a, a, s 3. Accepts strings which egin with a a s a, s0 s a found at egin. s 2 a, s ing 2 4. Accepts strings containing aa (skeleton method) a a a, s s s2 0.0 Languages, Expressions, Automata 5
6 5. All words containing at least two a's a a a, s s s All words containing exactly two a's a a s s2 a s0 no a found s one a found s2 two a found s too many a a, 3 s 3 Equivalence of Regular Expressions and Finite-State Automata. For every regular expression R, defining a language L, there is a FSA M recognizing exactly L. 2. For every FSA M, recognizing a language L, there is a regular expression R matching all the strings of L and no others. (we will prove this later) Question: is there a FSA that can recognize { ë, a, aa, aaa,... }?? Answer: No, ecause we need to rememer how many a s have een seen to verify that there are as many s. Since an FSA can only have a finite numer of states there cannot e enough states to count the a s. We need a more powerful kind of recognizer... that is, a grammar..0 Languages, Expressions, Automata 6
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. Finite automata: language recognizers n F can e descried y a laeled directed graph, where the nodes, called states, are laeled with a (unimportant) name edges, called transitions, are laeled with symols
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