Note that strings such as 2-20 would not be included in this language Regular Expression: A pattern that generates (only) the strings of a desired language
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Note that strings such as 2-20 would not be included in this language Regular Expression: A pattern that generates (only) the strings of a desired language
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1.0 Languages, Expressions, Automata11.0 Languages, Expressions, AutomataAlphabet: a finite set, typically a set of symbols.Language: a particular subset of the strings that can be made from the alphabet.
ex: an alphabet of digits = {-,0,1,2,3,4,5,6,7,8,9} a language of integers = {0,1,2,...,101,102,103,...,-1,-2,etc.} Note that strings such as 2-20 would not be 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 alphabet, as well as of the following special characters: ( ) used for grouping ( repetitionCconcatenation (usually omitted)
+ denotes a choice ("or").ëa special symbol denoting the null string
Precedence from highest to lowest: ( ) ( C +
formal (recursive) definition: If A is an alphabet, and a 0 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 C s = rs , r + s , and ( r ) examples:(assume that A = {a, b} ) a C b C a (or just aba ) matched only by the string aba ab + ba matched by exactly two strings: ab and ba b*matched by { ë , b, bb, bbb, ....} b(a + ba*)*a (b + ë)matched by bbaaab, and many others Some convenient extensions to regular expression notation: aa = a , bbbb = b , etc. 24a = aCa* = { any string of a's of positive length, i.e. excludes ë }+ ex:(ab) = abab ... a b , so don't try to use "algebra".222 ex:(a+b) = (a+b)(a+b) = aa or ab or ba or bb.2 ex:(a+b)* any string made up of a's and b's.
1.0 Languages, Expressions, Automata 2Examples of regular expressions over {a, b} :
Call strings that begin with a and end with b
a (a + b)* bCall non empty strings of even length
(aa + ab + ba + bb)Call strings with at least one a
(a + b)* a (a + b)*Call strings with at least two a's
(a + b)* a (a + b)* a (a + b)* Call strings of one or more b's with an optional single leading a (a + ë) b+Cthe language { ab, ba, abaa, bbb }
ab + ba + abaa + bbbor ab (ë + aa) + b (a + bb)or (a + bb) b + (b + aba) a or? Tips: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, b }:CAll strings with at least two b's.
(a + b)* b (a + b)* b (a + b)*CAll strings with exactly two b's.
a* b a* b a* CAll strings with at least one a and at least one b. (a + b)* (ab + ba) (a + b)* CAll strings which end in a double letter (two a's or two b's). (a + b)* (aa + bb)CAll strings of even length (includes 0 length).
(aa + bb + ab + ba)*1.0 Languages, Expressions, Automata 3aaba"recognizer"textyes
noFinite Automata: a particular, simplified model of a computing machine, that is a "language recognizer":A finite automaton (FSA) has five pieces:
1. S = a finite number of states,
2. A = the alphabet,
i3. S = the start state,
4. Y = one or more final or "accept" states, and
5. F = a transition function (mapping) between states, F: S x A º S.
The transition function F is usually presented in one of two ways: • as a table (called a transition table), or • as a graph (called a transition diagram).Transition Table (example):
012 i 1 02
A= { a, b }, S = { s , s , s }, S = s , Y = { s , s } current input F a b current state002 sss 110sss 200
sss gives the next state <
1.0 Languages, Expressions, Automata 4Transition 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 be needed. • The FSA must be complete and deterministic: for A= { a,b }, every state has exactly two arcs leaving it, one labeled "a" and one labeled "b". example (skeleton): All strings containing abaa1.0 Languages, Expressions, Automata 5aa, b
start 0 s startb 1 s a found at begin.a, b 2 s starting b0 s1 s1 saaa, b start b b0 s1 s2sstarta, b0 sa, ba, b start1 s0 sExamples Assume A= { a, b }. Construct the following automata which:1. Accepts strings of the form (a+b)*
2. Accepts ë only.
3. Accepts strings which begin with a
4. Accepts strings containing 'aa' (skeleton method)2
s1.0 Languages, Expressions, Automata 6aaa, b
start bb0 s1 s2saa b start bb a 0 s no 'a' found 1 s one 'a' found 2 s two 'a' found 3 s too many 'a'a, b0 s0 s1 s2s3 s5. All words containing at least two a's4. All words containing exactly two a'sEquivalence of Regular Expressions and Finite-State Automata
1. 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, ab, aabb, aaabbb, . . . } ?? Answer: No, because we need to "remember" how many a's have been seen to verify that there are as many b's. Since an FSA can only have a finitenumber of states there cannot be enough states to count the a's.We need a more powerful kind of recognizer... that is, a grammar.
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