1. BCT 2083 DISCRETE STRUCTURE AND APPLICATIONS
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
CHAPTER 5
MODELLING COMPUTATION
BCT 2083 DISCRETE STRUCTURE AND APPLICATIONS
SITI ZANARIAH SATARI
FIST/FSKKP UMP I0910
CHAPTER 5

2. CHAPTER 5 MODELLING COMPUTATION
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
CONTENT
CHAPTER 5 MODELLING COMPUTATION
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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3. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
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.
CHAPTER 5

4. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
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. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Natural Language & Formal Language
5.1 LANGUAGE AND GRAMMARS
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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6. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Basic Terminologies
5.1 LANGUAGE AND GRAMMARS
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, λ 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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7. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Phrase-Structure Grammars
5.1 LANGUAGE AND 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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8. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Language L(G) of a Grammar G
5.1 LANGUAGE AND GRAMMARS
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 Є 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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9. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.1
5.1 LANGUAGE AND GRAMMARS
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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10. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Types of Grammars
5.1 LANGUAGE AND GRAMMARS
TYPE
RESTRICTIONS ON PRODUCTION w ⇒ w’
No restrictions 1
Context- sensitive
Every production is of the form α → β where |α|≤ β or of the form α → λ 2
Context- free
Every production is of the form A → β where the left side is a nonterminal 3
regular
Every production is of the form A → a or A → aB where the left side is a single nonterminal and the right side is a single terminal or a terminal followed by a terminal or, of the form S → λ
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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11. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Derivation Tree of Context-Free Grammar
5.1 LANGUAGE AND GRAMMARS
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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12. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Example: Derivation Tree
5.1 LANGUAGE AND GRAMMARS
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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13. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.1
5.1 LANGUAGE AND GRAMMARS
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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14. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.1 : EXTRA
5.1 LANGUAGE AND GRAMMARS
PAGE : 793, 794, 795 and 796
Rosen K.H., Discrete Mathematics & Its Applications, (Seventh Edition), McGraw-Hill, 2007.
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15. BCT2083 DISCRETE STRUCTURE & APPLICATIONS
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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16. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Finite State Machines
5.2 FINITE-STATE MACHINES WITH OUTPUT
A Finite State Machine are used
to model many kinds of machine (computer components, vending machine)
as the basis for programs for spell checking, grammar checking, indexing or searching large bodies of text, recognizing speech, transforming text and network protocol.
A Finite State Machine include
a finite set of states, with a designated starting state, an input alphabet
a transition function that assigns a next state to every state and input pair.
A Finite State Machine can produce output or no output.
CHAPTER 5

17. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Finite State Machines with Output
5.2 FINITE-STATE MACHINES WITH OUTPUT
A Finite State Machine, M = (S, I, O, f, g, s0) consists of:
A finite set S of states
A finite input alphabet I
A finite output alphabet O
A transition function f that assigns to each state and input pair a new state
An output functions g that assigns to each state and input pair an output
An initial state s0
A Finite State Machine can be represented by
State TABLE
State DIAGRAM (a directed graph with labeled edge, state represented by a circle)
CHAPTER 5

18. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
State Table and State Diagram
5.2 FINITE-STATE MACHINES WITH OUTPUT
State TABLE represents the values of the transaction function f and the output function g for all pairs of states and input.
State DIAGRAM is a directed graph with labeled edge. Each state represented by a circle. Arrows labeled with the input and output pair are shown for each transition
State f g
Input s0
s s0 s1
s s0 s2
s s2 s3
s s1
S = {s0, s1, s2, s3}, I = {0, 1}, O = {0, 1}
0,1
0,1
1,0 s1
1,0 s0 s3
1,1
0,0
0,0 s2
1,1
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19. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.2
5.2 FINITE-STATE MACHINES WITH OUTPUT
Draw the state diagrams for the finite state machines with the following tables.
State f g
Input s0
s s0 s1
s s0 s2
s s3 s3
s s2
State f g
Input s0
s s0 s1
s s2 s2
s s1
TABLE 1
TABLE 2
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20. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.2
5.2 FINITE-STATE MACHINES WITH OUTPUT
Construct the state tables for the finite state machines with the following state diagrams.
0,0
0,0
1,0 s0
0,0 s1
0,1 s0 s1 s2
1,1
0,1
1,0
1,0
1,0
0,1
0,0 s2 s3
1,0
1,1
FIGURE 1
FIGURE 2
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21. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
Input and Output String
5.2 FINITE-STATE MACHINES WITH OUTPUT
Given the following finite state machine;
State f g
Input s0
s s0 s1
s s0 s2
s s2 s3
s s1
S = {s0, s1, s2, s3}, I = {0, 1}, O = {0, 1}
0,1
0,1
1,0 s1
1,0 s0 s3
1,1
0,0
0,0 s2
1,1
If the input string is 0111 then the output string is 1100.
If the input string is then the output string is
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22. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.2
5.2 FINITE-STATE MACHINES WITH OUTPUT
Given the finite state machines with the following state table and state diagrams. Determine the output for each of the following input strings.
0,0
State f g
Input
a b s0
s s2 s1
s s2 s2
s s2 s0 s1
0,1
1,1
0,1
1,0
1,0
0,0 s2 s3
1,1
TABLE 1
FIGURE 1
w = abaabbababaa b) c)
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23. 5.2 FINITE-STATE MACHINES WITH OUTPUT
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
EXERCISE 5.2 : EXTRA
5.2 FINITE-STATE MACHINES WITH OUTPUT
PAGE : 802 and 803
Rosen K.H., Discrete Mathematics & Its Applications, (Seventh Edition), McGraw-Hill, 2007.
CHAPTER 5

24. CHAPTER 5 MODELLING COMPUTATION
BCT2083 DISCRETE STRUCTURE & APPLICATIONS
CHAPTER 5 MODELLING COMPUTATION
Grammars, finite state machine and Turing machines are three structures used in models of computation
SUMMARY
What NEXT?
FINAL EXAMINATION
THAT’S ALL ; THANK YOU
CHAPTER 5