Specification-of-tokens
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Types of strings, How to specify token, regular expression representation, notations of regular expression
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The Theory of Automata is a fundamental branch of theoretical computer science that studies abstract machines and their computational capabilities. It is a critical area of study in computer science, mathematics, and computational theory, providing insights into the fundamental nature of computation and computational processes.
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Specification-of-tokens
- 1. Specification of Tokens • Definitions: • The ALPHABET (often written ∑) is the set of legal input symbols • A STRING over some alphabet ∑ is a finite sequence of symbols from ∑ • The LENGTH of string s is written |s| • The EMPTY STRING is a special 0-length string denoted ε • REGULAR EXPRESSIONS (REs) are the most common notation for pattern specification. • Every pattern specifies a set of strings, so an RE names a set of strings.
- 2. More definitions: strings and substrings • A PREFIX of s is formed by removing 0 or more trailing symbols of s • A SUFFIX of s is formed by removing 0 or more leading symbols of s • A SUBSTRING of s is formed by deleting a prefix and a suffix from s • A PROPER prefix, suffix, or substring is a nonempty string x that is, respectively, a prefix, suffix, or substring of s but with x ≠ s.
- 3. More definitions • A LANGUAGE is a set of strings over a fixed alphabet ∑. • Example languages: – Ø (the empty set) – { ε } – { a, aa, aaa, aaaa } • The CONCATENATION of two strings x and y is written xy • String EXPONENTIATION is written si, where s0 = ε and si = si-1s for i>0.
- 4. Regular expressions • REs let us precisely define a set of strings. • For C identifiers, we might use letter ( letter | digit )* • Parentheses are for grouping, | means “OR”, and * means zero or more instances. • Every RE ‘r’ defines a language L(r).
- 5. Regular expressions • Here are the rules for writing REs over an alphabet ∑ : 1. ε is an RE denoting { ε }, the language containing only the empty string. 2. If ‘a’ is in ∑, then a is a RE denoting { a }. 3. If r and s are REs denoting L(r) and L(s), then 1. (r)|(s) is a RE denoting L(r) ∪ L(s) 2. (r)(s) is a RE denoting L(r) L(s) 3. (r)* is a RE denoting (L(r))* 4. (r) is a RE denoting L(r)
- 6. Additional conventions • To avoid too many parentheses, we assume: 1. * has the highest precedence, and is left associative. 2. Concatenation has the 2nd highest precedence, and is left associative. 3. | has the lowest precedence and is left associative.
- 7. Example REs 1. a | b 2. ( a | b ) ( a | b ) 3. a* 4. (a | b )* 5. a | a*b
- 8. Equivalence of REs Axiom Description r|s = s|r | is commutative r|(s|t) = (r|s)t | is associative (rs)t = r(st) Concatenation is associative r(s|t) = rs|rt (s|t)r = sr|tr Concatenation distributes over | ε r = r r ε = r ε Is the identity element for concatenation r* = (r| ε)* Relation between * and ε r** = r* * is idempotent
- 9. Regular definitions • Example for identifiers in C: letter -> A | B | … | Z | a | b | … | z digit -> 0 | 1 | … | 9 id -> letter ( letter | digit )* • Example for numbers in Pascal: digit -> 0 | 1 | … | 9 digits -> digit digit* optional_fraction -> . digits | ε optional_exponent -> ( E ( + | - | ε ) digits ) | ε num -> digits optional_fraction optional_exponent