1. Introduction to Language Theory
Programming Language Translators
Prepared by
Manuel E. Bermúdez, Ph.D.
Associate Professor
University of Florida

2. Introduction to Language Theory
Definition: An alphabet (or vocabulary) Σ is a finite set of symbols.
Example: Alphabet of Pascal:
+ - * / < … (operators)
begin end if var (keywords)
<identifier> (identifiers)
<string> (strings)
<integer> (integers)
; : , ( ) [ ] (punctuators)
Note: All identifiers are represented by one symbol, because Σ must be finite.

3. Introduction to Language Theory
Definition: A sequence t = t1t2…tn of symbols from an alphabet Σ is a string.
Definition: The length of a string t = t1t2…tn (denoted |t|) is n. If n = 0, the string is ε, the empty string.
Definition: Given strings s = s1s2…sn and
t = t1t2…tm, the concatenation of s and t, denoted st, is the string s1s2…snt1t2…tm.

4. Introduction to Language Theory
Note: εu = u = uε, uεv = uv, for any strings u,v (including ε)
Definition: Σ* is the set of all strings of symbols from Σ.
Note: Σ* is called the reflexive, transitive closure of Σ.
Σ* is described by the graph (Σ*, ·), where “·” denotes concatenation, and there is a designated “start” node, ε.

5. Introduction to Language Theory
Example: Σ = {a, b}.
(Σ*, ·)
Σ* is countably infinite, so can’t compute all of Σ*, and can only compute finite subsets of Σ*, but can compute whether a given string is in Σ*. aa a a
aba a b a ab b
abb ε b a ba b b bb

6. Introduction to Language Theory
Example: Σ = Pascal vocabulary.
Σ* = all possible alleged Pascal programs, i.e. all possible inputs to Pascal compiler.
Need to specify L  Σ*, the correct Pascal programs.
Definition: A language L over an alphabet Σ is a subset of Σ*.

7. Introduction to Language Theory
Example: Σ = {a, b}.
L1 = ø is a language
L2 = {ε} is a language
L3 = {a} is a language
L4 = {a, ba, bbab} is a language
L5 = {anbn / n >= 0} is a language
where an = aa…a, 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 = {ε, a, bb}, L2 = {ac, c}
L1L2 = {ac, c, aac, ac, bbac, bbc}
= {ac, c, aac, bbac, bbc}

10. Language Constructors
Definition: Ln = LL…L (n times),
and L0 = {ε}.
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 >0.
Example: L = {a, bb}
L* = {any string composed of a’s and
bb’s}
Definition: The Transitive Closure (L+) of a language L is the set L+ = U Ln, n > 1. ∩ ∩

12. Language Constructors
Note:
In general, L* = L+ U {ε}, but L+ ≠ L* - {ε}.
For example, consider L = {ε}. Then
{ε} = L+ ≠ L* – {ε} = {ε} – {ε} = ø.

13. Grammars Goal: Providing a means for describing languages finitely.
Method: Provide a subgraph (Σ*, →*) of (Σ*, ·), and a start node S, such that the set of reachable nodes (from S) are the strings in the language.

14. Grammars Example: Σ = {a, b} L = {anbn / n > 0} ε a aaa a aaba a aa
aabb ε a ba
bbaa a b a b
bba b bb b
bbab b
bbb

15. Grammars
“=>” (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 => is defined such that
, β, u, v  V* , uβ => vβ 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

18. Grammars => Adj NP VP => the NP VP => the Adj NP VP
Sentence => NP VP
=> Adj NP VP
=> the NP VP
=> the Adj NP VP
=> the jealous NP VP
=> the jealous N VP
=> the jealous girl VP
=> the jealous girl V NP
=> the jealous girl hit NP
=> the jealous girl hit Adj NP
=> the jealous girl hit the NP
=> the jealous girl hit the N
=> the jealous girl hit the boy

19. Grammars Definition: A grammar is a 4-tuple G = (Φ, Σ, P, S) where
Φ is a finite set of nonterminals,
Σ is a finite set of terminals,
V = Φ U Σ is the grammar’s vocabulary,
S  Φ is called the start or goal symbol,
and P  V* x V* is a finite set of productions.
Example: Grammar for {anbn / n > 0}.
G = (Φ, Σ, P, S), where
Φ = {S},
Σ = {a, b},
and P = {S → aSb, S → ε}

20. Grammars Derivations:
S => aSb => aaSbb => aaaSbbb => aaaaSbbbb → …
ε ab aabb aaabbb aaaabbbb
Note: Normally, grammars are given by simply listing the productions.
=>
=>
=>
=>
=>

21. 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 → ε
Left part omitted if same as for preceeding production, e.g.
→ ε

22. Grammars Example: Grammar for identifiers. Identifier → Letter
→ Identifier Digit
Letter → ‘a’ → ‘A’
→ ‘b’ → ‘B’ .
→ ‘z’ → ‘Z’
Digit → ‘0’
→ ‘1’
→ ‘9’

23. Grammars
Definition: The language generated by a grammar G, is the set L(G) = {  Σ* | S =>*  }
Definition: A sentential form generated by a grammar G is any string α such that S =>*  .
Definition: A sentence generated by a grammar G is any sentential form  such that   Σ*.

24. Grammars Example: sentential forms
S => aSb => aaSbb => aaaSbbb => aaaaSbbbb > …
ε ab aabb aaabbb aaaabbbb
Lemma: L(G) = { | is a sentence}
Proof: Trivial.
=>
=>
=>
=>
=>
sentences

25. Grammars Example: A → aABC → aBC aB → ab bB → bb bC → bc CB → BC
cC → cc

26. Grammars Derivations: A => aABC => aaABCBC => … =>
aBC aaBCBC aaaBCBCBC
abC aabCBC aaaBBCBCC
abc aabBCC aaaBBBCCC
aabbCC aaabBBCCC
(2)
aabbcC aaabbbCCC
aabbcc aaabbbcCC
aaabbbccc
L (G) = {anbncn | n > 1}
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>

27. 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 Restrictions
On grammar
Accepting
Machine
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
Φ.
Push-Down Automaton
(parser) 3
Regular
Φ, U
ΦU{}
Finite- State Automaton

29. 24/04/2017
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 > 0}
Be selective. You do not need to cover both research and education. It does not need to be a long list. You can put down just one opportunity that you are really excited about. Just identify what you think are the biggest opportunities for your department faculty. Strike a balance between “thinking big” and being realistic. One way to think would be to say that if you were the Dean, you would invest in these opportunities. Remember the goal is to have national level prominence and visibility where our peer group will recognize our activities and accomplishments. For example, the NSF ERC on Particle Science and Technology
As you go to the next slide, please bear in mind that there may well be very strong connections between this slide and the next on multi-disciplinary collaborations.
{anbn | n>0}
{anbncn | n>0}
English?