Introduction
Wen-Guey Tzeng
Computer Science Department
National Chiao Tung University
Formal languages and finite automata
• We have been building computing machines for solving
computing problems
• Questions
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What is a computing machine?
How do we study it?
What can it do?
What is a problem to be solved by a computing machine?
• Finite automata: abstract models of computing
machines
• Formal languages: formal representation of computing
problems
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Classification
• Finite automata: by types of memories
– Finite automata
– Pushdown automata
– Turing machines
• Formal languages: by types of grammars
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Regular languages
Context-free languages
Recursive languages (solvable)
Recursively enumerable languages (semi-solvable)
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Relations between languages and
automata
• Finite automata and regular languages
• Pushdown down automata and context-free
languages
• Turing machines and recursive languages
• Complexity classes: by runtime or space used by
Turing machines
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P: polynomial time computable
NP: non-deterministic polynomial time computable
EXP: solvable in exponential time
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Computing
machines
Formalization
Abstraction
Yes/No
problems
Computing
problems
Formal
languages
Classifications:
grammars,
machines, etc.
Relations
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Finite
automata
Classifications:
memory,
runtime,
space, etc.
Regular vs. finite automata
Context-free vs. pushdown
Recursive vs. Turing machines
Complexity classes
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Applications
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Programming languages
Compilers
Natural language processing
Algorithm design
Software engineering
Pattern recognitions
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• How to compile a Java program, like the above?
• We need a grammar for the Java programming
language
• Context-free grammar
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Mathematical notation
• Set: a collection of elements
– |S|
– SxT={(x, y) | xS, yT}
• Function
– f: S1S2
• Binary relation
– R  S1xS2
– xS1 and yS2 has relation R if (x, y)R
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• Graph: G=(V, E)
– V: node set
– EVxV: edge set
• Tree
– Level
– Height
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Proof Techniques
• Proof by induction
– For some k, we know (prove) that P1, P2, …, Pk are
true
– The problem is that for nk, the truths of P1, P2,…,
Pn imply the truth of Pn+1
• Example
– Theorem: a binary tree of height n has at most 2n
leaves
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• Proof by contradiction
– To show that P is true.
– We assume that if P is false, then …
• Example:
– Theorem: sqrt(2) is not rational.
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Three Basic Concepts
• Languages (yes-no problems)
– Symbol set (alphabet) ={a, b, c, …}
– String: w=a1a2…an, n0, ai
– |w|: length of string
– String concatenation: w1w2
– String reversing: wR=anan-1…a1
– Prefix(w)={, a1, a1a2, …, a1a2…an}
– Suffix(w)={, an, an-1an, …, a1a2…an}
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– Empty string
•
• ||=0
• w= w=w
– *: the set of all strings over 
• * over {a,b} = {, a, b, aa, ab, ba, bb, …}
– + = * - {}: set of all non-empty strings
– A language L over  is a subset of *: a set of
strings
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– LR = {wR: wL }
– L c = * - L
– L1L2 = {xy: xL1, yL2}
– Ln = {x1x2…xn: xiL}
• L0 = {}
• L* = L0  L1  …
• L+ = L1  L2  …
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• Grammar: a set of rules for generating strings
– G=(V, T, P, S)
– V: set of variables
– T: set of terminals (symbols)
– P: set of production rules
– SV: the start variable
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Example
G=({S,A,B}, {a,b}, {SaA, AbS, S}, S)
• String generation/derivation
• The language generated by G
– L(G) = {w | S * w}
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• Problem: find grammars for languages over ={a, b}
– L={anbn | n0}
– L={w | na(w)=nb(w)}
– L={anbm | nm0}
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• Automaton: an abstract model of a digital
computer
– Input
– Control unit: represented by “states”
– Storage
– Output
– Operating in discrete time frame
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