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Regular Definitions
• Shorthand
– One or more instances: r+ denotes rr*
– Zero or one Instance: r? denotes r|ε
– Character classes: [a-z] denotes
[a|b|…|z]](https://image.slidesharecdn.com/22specificationoftokens-220323061426/85/2_2Specification-of-Tokens-ppt-15-320.jpg)
2_2Specification of Tokens.ppt
- 2. 2 Strings and Languages • Regular Expressions are an important notation for specifying patterns. • Alphabet – any finite set of symbols e.g. ASCII, binary alphabet, UNICODE, EBCDIC,LATIN-1 • String – A finite sequence of symbols drawn from an alphabet – Banana (ASCII Alphabet) – Length of a string => |s| – Empty String => ε • Other terms relating to strings: prefix; suffix; substring; proper prefix, suffix, or substring (non-empty, not entire string); subsequence • Language – A set of strings over a fixed alphabet
- 3. 3 Languages • A language, L, is simply any set of strings over a fixed alphabet. Alphabet Languages {0,1} {0,10,100,1000,100000…} {0,1,00,11,000,111,…} {a,b,c} {abc,aabbcc,aaabbbccc,…} {A, … ,Z} {FOR,WHILE,GOTO,…} {A,…,Z,a,…,z,0,…9, { All legal PASCAL progs} +,-,…,<,>,…} Special Languages: - EMPTY LANGUAGE - contains string only
- 4. 4 String operations • Given String: banana • Prefix : ban, banana • Suffix : ana, banana • Substring : nan, ban, ana, banana • Subsequence: bnan, nn • Proper Prefix and Suffix
- 5. 5 String Operations • Concatenation – xy; s = s = s; - identity for concatenation – s0 = if i > 0 si = si-1s
- 6. 6 Operations on Languages OPERATION DEFINITION union of L and M written L M concatenation of L and M written LM Kleene closure of L written L* positive closure of L written L+ L M = {s | s is in L or s is in M} LM = {st | s is in L and t is in M} L+= 0 i i L L* denotes “zero or more concatenations of “ L L*= 1 i i L L+ denotes “one or more concatenations of “ L Exponentiation Lo={ε}, L1=L,L2=LL
- 7. 7 Operations on Languages • LUD is the set of letters and digits • LD is the set of strings consisting of a letter followed by a digit • L4 is the set of all four strings • L* is the set of strings including ε • D+ is the set of strings of one or more digits.
- 8. 8 Say What? L = {A, B, C, D } D = {1, 2, 3} • L D {A, B, C, D, 1, 2, 3 } • LD {A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, D3 } • L2 { AA, AB, AC, AD, BA, BB, BC, BD, CA, … DD} • L* { All possible strings of L plus } • L+ L* - • L (L D ) Valid :{ A1,AA2,B345,CD45} Invlaid:{321,4A2} • L (L D )* Valid:{ A,A1,A23,D3,DA5..} Invalid:{31}
- 9. 9 Regular Expressions • A Regular Expression is a Set of Rules / Techniques for Constructing Sequences of Symbols (Strings) from an Alphabet. • Let Be an Alphabet, r a Regular Expression Then L(r) is the Language That is characterized by the Rules of r
- 10. 10 Regular Expressions • Defined over an alphabet Σ • ε represents {ε}, the set containing the empty string • If a is a symbol in Σ, then a is a regular expression denoting {a}, the set containing the string a • If r and s are regular expressions denoting the languages L(r) and L(s), then: – (r)|(s) is a regular expression denoting L(r)U L(s) – (r)(s) is a regular expression denoting L(r)L(s) – (r)* is a regular expression denoting (L(r))* – (r) is a regular expression denoting L(r) • Precedence: * (left associative), then concatenation (left associative), then | (left associative)
- 11. 11 Regular Expressions Alphabet = {a, b} 1. a|b denotes {a, b} 2. (a|b)(a|b) denotes {ab, aa, ba, bb} 3. a* denotes {, a, aa, …} 4. (a|b)* - Strings of a’s and b’s including the 5. a|a*b – a followed by zero/more a’s followed by b
- 12. 12 Algebraic Properties of Regular Expressions AXIOM DESCRIPTION r | s = s | r r | (s | t) = (r | s) | t (r s) t = r (s t) r = r r = r r* = ( r | )* r ( s | t ) = r s | r t ( s | t ) r = s r | t r r** = r* | is commutative | is associative concatenation is associative concatenation distributes over | relation between * and Is the identity element for concatenation * is idempotent
- 13. 13 Regular Definitions • Names maybe given to regular expressions; these names can be used like symbols • Let is an alphabet of basic symbols. The regular definition is a sequence of definitions of the form d1 r1 d2 r2 . . . dn rn Where, each di is a distinct name, and each ri is a regular expression over the symbols in {d1, d2, …, di-1 }
- 14. 14 Regular Definitions • Example 1: – letter A|B|…|Z|a|b|…|z – digit 0|1|…|9 – id letter (letter | digit)* • Example 2 – digit 0 | 1 | 2 | … | 9 – digits digit digit* – optional_fraction . digits | – optional_exponent ( E ( + | -| ) digits) | – num digits optional_fraction optional_exponent
- 15. 15 Regular Definitions • Shorthand – One or more instances: r+ denotes rr* – Zero or one Instance: r? denotes r|ε – Character classes: [a-z] denotes [a|b|…|z]
- 16. 16 Example • digit 0 | 1 | 2 | … | 9 • digits digit+ • optional_fraction (. digits ) ? • optional_exponent ( E ( + | -) ? digits) ? • num digits optional_fraction optional_exponent
- 17. 17 Limitations of Regular Expression • Some languages cannot be described by any regular expression • Cannot describe balanced or nested constructs – Example, all valid strings of balanced parentheses – This can be done with CFG • Cannot describe repeated strings – Example: {wcw|w is a string of a’s and b’s} – This can be done with CFG • Can be used to denote only a fixed or unspecified number of repetitions.