BCT 2083 DISCRETE STRUCTURE AND APPLICATIONS

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BCT 2083 DISCRETE STRUCTURE AND APPLICATIONS

• CHAPTER 5
• MODELLING COMPUTATION

SITI ZANARIAH SATARI FIST/FSKKP UMP I0910
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CHAPTER 5 MODELLING COMPUTATION

• CONTENT

• 5.1 Languages and Grammars
• 5.2 Finite-State Machines with Output
• 5.3 Finite-State Machines with No Output
• 5.4 Language Recognition (not cover)
• 5.5 Turing Machines (not cover)

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Introduction
CHAPTER 5 MODELLING COMPUTATION

• Models of computation help us to answer the
  following questions
• Can a task be carried out using computer?
• If YES, How can the task be carried out?
• Three types of structures used in models
  computation
• Grammars
• Used to generate words of a language and
  determine whether a word is in a language.
• Finite-state Machines
• Used in modeling a problem.
• Turing Machines
• Used to classify problems as tractable/intractable
  and solvable/unsolvable.

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5.1 LANGUAGE AND GRAMMARS
CHAPTER 5 MODELLING COMPUTATION

• Understand language and grammars in models of
  computation
• Construct derivation tree

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5.1 LANGUAGE AND GRAMMARS
Natural Language Formal Language

• Natural Language is spoken language.
• It is not possible to specify all the rules of
  syntax (form) in Natural language
• One of the rules is English grammars.
• Formal Language which are generated by grammars,
  provide models for both natural languages and
  programming languages.
• Formal Language specified by a well defined set
  of rules/syntax.
• Grammars help us answer the following questions
• How can we determine whether a combination of
  words is a valid sentence in a formal language?
• How can we generate the valid sentences of a
  formal language?

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5.1 LANGUAGE AND GRAMMARS
Basic Terminologies

• A vocabulary/alphabet, V is a finite nonempty set
  of elements called symbols.
• Example V a, b, c, A, B, C, S
• A word/sentence over V is a string of finite
  length of elements of V.
• Example Aba
• The empty/null string, ? is the string with no
  symbols.
• V is the set of all words over V.
• Example V Aba, BBa, bAA, cab
• A language over V is a subset of V.
• We can give some criteria to a word to be in a
  language.
• Example cab is a subset in V and is a language.
  

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5.1 LANGUAGE AND GRAMMARS
Phrase-Structure Grammars

• A Phrase-Structure Grammars G (V, T, S, P)
  consists of
• a vocabulary V,
• a subset T of V consisting of terminal elements
• a start symbol S from V
• a finite set of productions P
• Elements of N V T are called nonterminal
  symbols.
• Every production in P must contain at least one
  nonterminal on its left side.
• EXAMPLE
• Let G (V, T, S, P), where V a, b, A, B, S,
  T a, b, S is a start symbol and P S ? ABa,
  A ? BB, B ? ab, A ? Bb. Then, G is a
  Phrase-Structure Grammar.

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5.1 LANGUAGE AND GRAMMARS
Language L(G) of a Grammar G

• Suppose w and w are words over the vocabulary
  set V of a grammar G. Then
• We write w ? w if w can be obtained from w by
  using one of the productions
• We write w ? ? w if w can be obtained from w by
  using a finite number of productions
• The language of G consists of all words in
  terminal set T that can be obtained from the
  start symbol S by the above process
• L (G) w ? T S ? ? w
• EXAMPLE
• Let G (V, T, S, P), where V a, b, A, S, T
  a, b, S is a start symbol and P S ? aA, S
  ? b, A ? aa.
• The language of this grammar is given by L (G)
  b, aaa since we can derive aA from using S ?
  aA, and then derive aaa using A ? aa. We can also
  derive b using S ? b.

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5.1 LANGUAGE AND GRAMMARS
EXERCISE 5.1

• Let G (V, T, S, P), where V a, b, A, B, S,
  T a, b, S is a start symbol and P S ?AB, A
  ?Aa, B ?Bb, A ?a, B ?b. What is L(G), the
  language of this grammar?
• Find the language L(G) over a, b, c generated
  by the grammar G with the productions
  P S
  ?aSb, aS ?Aa, AaB ?c.
• Let V a, b, A, B, S, and T a, b. Find the
  language L(G) generated by the grammar G (V, T,
  S, P), with S is a start symbol and a set of
  productions P S ?AA, S ?B, A ?aaA, A ?aa, B
  ?bB, B ?b.

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5.1 LANGUAGE AND GRAMMARS
Types of Grammars
Every Type 3 grammar is a Type 2 grammar. Every
Type 2 grammar is a Type 1 grammar. Every Type 1
grammar is a Type 0 grammar.
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5.1 LANGUAGE AND GRAMMARS
Derivation Tree of Context-Free Grammar

• Represents the language using an ordered rooted
  tree.
• Root represents the starting symbol.
• Internal vertices represent the nonterminal
  symbol that arise in the production.
• Leaves represent the terminal symbols.
• If the production A ? w arise in the derivation,
  where w is a word, the vertex that represents A
  has as children vertices that represent each
  symbol in w, in order from left to right.

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5.1 LANGUAGE AND GRAMMARS
Example Derivation Tree

• Let G be a context-free grammar with the
  productions P S ?aAB, A ?Bba, B ?bB, B
  ?c. The word w acbabc can be derived from S as
  follows
• S ? aAB ?a(Bba)B ? acbaB ? acba(bB) ? acbabc
• Thus, the derivation tree is given as follows

S
a
A
B
B
b
b
a
B
c
c
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5.1 LANGUAGE AND GRAMMARS
EXERCISE 5.1

• The word w cbab belongs to the language
  generated by the grammar G (V, T, S, P), where
  V a, b, c, A, B, C, S, T a, b, c, S is
  the start symbol and P S ?AB, A ?Ca, B ?Ba, B
  ?Cb, B ?b , C ?cb, C ?b. Construct the
  derivation tree for w.
• The production rules of a grammar for simple
  arithmetic expression are
• expression digit ( expression
  )
• ( expression ) - ( expression )
  
• expression
  operator expression
• digit 0 1 2 3 4
  5 6 7 8 9
• operator -  / ?
• Construct a derivation tree for arithmetic
  expression (2 ? 5 ) - 8 and (3 7) / (9 1).

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5.1 LANGUAGE AND GRAMMARS
EXERCISE 5.1 EXTRA

• PAGE 793, 794, 795 and 796
• Rosen K.H., Discrete Mathematics Its
  Applications, (Seventh Edition), McGraw-Hill,
  2007.

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CHAPTER 5 MODELLING COMPUTATION

• 5.2 FINITE-STATE MACHINE WITH OUTPUT

• Understand finite state machines with output.
• Draw state diagrams for finite state machines
  with output.
• Construct state table for finite state machines
  with output.

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5.2 FINITE-STATE MACHINES WITH OUTPUT
Finite State Machines
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5.2 FINITE-STATE MACHINES WITH OUTPUT
EXERCISE 5.2
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5.2 FINITE-STATE MACHINES WITH OUTPUT
EXERCISE 5.2 EXTRA

• PAGE 802 and 803
• Rosen K.H., Discrete Mathematics Its
  Applications, (Seventh Edition), McGraw-Hill,
  2007.

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CHAPTER 5 MODELLING COMPUTATION

• 5.3 FINITE-STATE MACHINES WITH
• NO OUTPUT

• Understand finite state machines with no output.
• Draw state diagrams for finite state machines
  with no output.
• Construct state table for finite state machines
  with no output.

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5.3 FINITE-STATE MACHINES WITH NO OUTPUT
Finite State Machines
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5.3 FINITE-STATE MACHINES WITH NO OUTPUT
EXERCISE 5.3
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5.3 FINITE-STATE MACHINES WITH NO OUTPUT
EXERCISE 5.3 EXTRA

• PAGE 814, 815, 816 and 817
• Rosen K.H., Discrete Mathematics Its
  Applications, (Seventh Edition), McGraw-Hill,
  2007.

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CHAPTER 5 MODELLING COMPUTATION

• Grammars, finite state machine and Turing
  machines are three structures used in models of
  computation

SUMMARY
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