Introduction to Language Theory

1
Introduction to Language Theory
Programming Language Concepts Lecture 3

• Prepared by
• Manuel E. Bermúdez, Ph.D.
• Associate Professor
• University of Florida

2
Introduction to Language Theory

• Definition An alphabet (or vocabulary) S is a
  finite set of symbols.
• Example Alphabet of Pascal
• - / lt (operators)
• begin end if var (keywords)
• ltidentifiergt (identifiers)
• ltstringgt (strings)
• ltintegergt (integers)
• , ( ) (punctuators)
• Note All identifiers are represented by one
  symbol, because S must be finite.

3
Introduction to Language Theory

• Definition A sequence t t1t2tn of symbols
  from an alphabet S is a string.
• Definition The length of a string t t1t2tn
  (denoted t) is n. If n 0, the string is e,
  the empty string.
• Definition Given strings s s1s2sn and
• t t1t2tm, the concatenation of s and t,
  denoted st, is the string s1s2snt1t2tm.

4
Introduction to Language Theory

• Note eu u ue, uev uv, for any strings u,v
  (including e)
• Definition S is the set of all strings of
  symbols from S.
• Note S is called the reflexive, transitive
  closure of S.
• S is described by the graph (S, ), where
  denotes concatenation, and there is a designated
  start node, e.

5
Introduction to Language Theory

• Example S a, b.
• (S, )
• S is countably infinite, so cant compute all of
  S, and can only compute finite subsets of S,
  but can compute whether a given string is in S.

aa
a
a
aba
b
a
ab
a
b
abb

• e

b
ba
a
b
b
bb
6
Introduction to Language Theory

• Example S Pascal vocabulary.
• S all possible alleged Pascal programs,
  i.e. all possible inputs to Pascal compiler.
• Need to specify L ? S, the correct Pascal
  programs.
• Definition A language L over an alphabet S is a
  subset of S.

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Introduction to Language Theory

• Example S a, b.
• L1 ø is a language
• L2 e is a language
• L3 a is a language
• L4 a, ba, bbab is a language
• L5 anbn / n gt 0 is a language
• where an aaa, n times
• L6 a, aa, aaa, is a language
• Note L5 is an infinite language, but described
  finitely.

8
Introduction to Language Theory

• THIS IS THE MAIN GOAL OF LANGUAGE SPECIFICATION
• To describe (infinite) programming languages
  finitely, and to provide corresponding finite
  inclusion-test algorithms.

9
Language Constructors

• Definition The catenation (or product) of two
  languages L1 and L2, denoted L1L2, is the set
• uv u?L1, v?L2.
• Example L1 e, a, bb, L2 ac, c
• L1L2 ac, c, aac, ac, bbac, bbc
• ac, c, aac, bbac, bbc

10
Language Constructors

• Definition Ln LLL (n times),
• and L0 e.
• Example L a, bb
• L3 aaa, aabb, abba,
  abbbb, bbaa, bbabb, bbbba, bbbbbb

11
Language Constructors

• Definition The union of two languages L1 and L2
  is the set L1 L2 u u?L1 v v?L2
• Definition The Kleene star (L) of a language is
  the set L U Ln, n gt0.
• Example L a, bb
• L any string composed of as and
• bbs
• Definition The Transitive Closure (L) of a
  language L is the set L U Ln, n gt 1.

n
n
12
Language Constructors

• Note
• In general, L L U e, but L ? L - e.
• For example, consider L e. Then
• e L ? L e e e ø.

13
Grammars

• Goal Providing a means for describing languages
  finitely.
• Method Provide a subgraph (S, ?) of (S, ),
  and a start node S, such that the set of
  reachable nodes (from S) are the strings in the
  language.

14
Grammars

• Example S a, b
• L anbn / n gt 0

a
aaa

aaba
a
aa
a
aab
b
b
b
a
ab
a
aabb

• e

a
ba
bbaa
a
b
a
b
bba
b
bb
bbab
b
bbb
b
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Grammars

• gt (derives) is a relation defined by a finite
  set of rewrite rules known as productions.
• Definition Given a vocabulary V, a production is
  a pair (u, v) ? V x V, denoted u ? v. u is
  called the left-part v is called the right-part.
  

16
Grammars

• Example Pseudo-English.
• V Sentence, NP, VP, Adj, N, V, boy, girl,
  the, tall, jealous, hit, bit
• Sentence ? NP VP (one production)
• NP ? N
• NP ? Adj NP
• N ? boy
• N ? girl
• Adj ? the
• Adj ? tall
• Adj ? jealous
• VP ? V NP
• V ? hit
• V ? bit
• Note English is much too complicated to be
  described this way.

17
Grammars

• Definition
• Given a finite set of productions P ? V x V
  the relation gt is defined such that
• ?, ß, u, v ? V , ?uß gt ?vß iff
• u ? v ? P is a production.
• Example
• Sentence ? NP VP Adj ? the
• NP ? N Adj ? tall
• NP ? Adj NP Adj ? jealous
• N ? boy VP ? V NP
• N ? girl V ? hit
• V ? bit

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Grammars

• Sentence gt NP VP
• gt Adj NP VP
• gt the NP VP
• gt the Adj NP VP
• gt the jealous NP VP
• gt the jealous N VP
• gt the jealous girl VP
• gt the jealous girl V NP
• gt the jealous girl hit NP
• gt the jealous girl hit Adj NP
• gt the jealous girl hit the NP
• gt the jealous girl hit the N
• gt the jealous girl hit the boy

19
Grammars

• Definition A grammar is a 4-tuple G (F, S, P,
  S)
• where
• F is a finite set of nonterminals,
• S is a finite set of terminals,
• V F U S is the grammars vocabulary,
• S ? F is called the start or goal symbol,
• and P ? V x V is a finite set of productions.
• Example Grammar for anbn / n gt 0.
• G (F, S, P, S), where
• F S,
• S a, b,
• and P S ? aSb, S ? e

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Grammars

• Derivations
• S gt aSb gt aaSbb gt aaaSbbb gt aaaaSbbbb ?
• e ab aabb aaabbb
  aaaabbbb
• Note Normally, grammars are given by simply
  listing the productions.

gt
gt
gt
gt
gt
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Grammar Conventions

• TWS
  convention
• Upper case letter (identifier) nonterminal
• Lower case letter (string) terminal
• Lower case greek letter strings in V
• Left part of the first production is assumed to
  be the start symbol, e.g.
• S ? aSb
• S ? e
• Left part omitted if same as for preceeding
  production, e.g.
• S ? aSb
• ? e

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Grammars

• Example Grammar for identifiers.
• Identifier ? Letter
• ? Identifier Letter
• ? Identifier Digit
• Letter ? a ? A
• ? b ? B
• .
• .
• ? z ? Z
• Digit ? 0
• ? 1
• .
• .
• ? 9

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Grammars

• Definition The language generated by a grammar
  G, is the set L(G) ? ? S S gt ?
• Definition A sentential form generated by a
  grammar G is any string a such that S gt ? .
• Definition A sentence generated by a grammar G
  is any sentential form ? such that ? ? S.

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Grammars

• Example
• sentential forms
• S gt aSb gt aaSbb gt aaaSbbb gt aaaaSbbbb gt
• e ab aabb aaabbb
  aaaabbbb
• Lemma L(G) ? is a sentence
• Proof Trivial.

gt
gt
gt
gt
gt
sentences
25
Grammars

• Example A ? aABC
• ? aBC
• aB ? ab
• bB ? bb
• bC ? bc
• CB ? BC
• cC ? cc

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Grammars

• Derivations A gt aABC gt aaABCBC gt
• aBC aaBCBC
  aaaBCBCBC
• abC aabCBC aaaBBCBCC
• abc aabBCC
  aaaBBBCCC
• aabbCC aaabBBCCC
• (2)
• aabbcC aaabbbCCC
• aabbcc aaabbbcCC
• 
  (2)
• 
  aaabbbccc
• L (G) anbncn n gt 1

gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
gt
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The Chomsky Hierarchy

• A hierarchy of grammars, the languages they
  generate, and the machines the accept those
  languages.

28
The Chomsky Hierarchy
Type Language Name Grammar Name Restrictions On grammar Accepting Machine
0 Recursively Enumerable Unrestricted re-writing system None Turing Machine
1 Context-Sensitive Language Context- Sensitive Grammar For all ???, ?? Linear Bounded Automaton
2 Context- Free Language Context- Free Grammar For all ???, ??F. Push-Down Automaton (parser)
3 Regular Language Regular Grammar For all ???, ??F, ???U ?FU? Finite- State Automaton
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Language Hierarchy
0 Recursively Enumerable Languages
1 Context-Sensitive Languages
2 Context-free Languages
We will deal with type 2 (syntax) and type 3
(lexicon) languages.
3 Regular Languages an n gt 0
anbn ngt0
anbncn ngt0
English?