bct 2083 discrete structure and applications

BCT 2083 DISCRETE STRUCTURE AND APPLICATIONSBCT 2083 DISCRETE STRUCTURE AND APPLICATIONS

CHAPTER 5CHAPTER 5MODELLING COMPUTATIONMODELLING COMPUTATION

SITI ZANARIAH SATARISITI ZANARIAH SATARI

FIST/FSKKP UMP I0910FIST/FSKKP UMP I0910

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CONTENTCONTENT

5.1 Languages and Grammars5.2 Finite-State Machines with Output5.3 Finite-State Machines with No

Output5.4 Language Recognition (not cover)

5.5 Turing Machines (not cover)

• Models of computation help us to answer the following questions:1. Can a task be carried out using computer?2. If YES, How can the task be carried out?

• Three types of structures used in models computation:1. Grammars

– Used to generate words of a language and determine whether a word is in a language.

2. Finite-state Machines– Used in modeling a problem.

3. Turing Machines– Used to classify problems as tractable/intractable and

solvable/unsolvable.

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IntroductionIntroduction

5.15.1 LANGUAGE AND GRAMMARSLANGUAGE AND GRAMMARS

• Understand language and grammars in models of computation

• Construct derivation tree

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5.1 LANG

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►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?

Natural Language & Formal LanguageNatural Language & Formal Language

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► 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.

Basic TerminologiesBasic Terminologies

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► A Phrase-Structure Grammars G = (V, T, S, P) consists of: 1. a vocabulary V, 2. a subset T of V consisting of terminal elements 3. a start symbol S from V4. 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.

Phrase-Structure GrammarsPhrase-Structure Grammars

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► 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.

Language Language LL((GG) of a Grammar ) of a Grammar GG

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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?

2. Find the language L(G) over {a, b, c} generated by the grammar G with the productions P = {S →aSb, aS →Aa, Aab →c}.

3.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}.

EXERCISE 5.1EXERCISE 5.1

5.1 LANG

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Types of GrammarsTypes of GrammarsTYPE RESTRICTIONS ON PRODUCTION w ⇒ w’

0 No restrictions

1Context- sensitive

Every production is of the form α → β where |α|≤ β or of the form α → λ

2Context- free

Every production is of the form A → β where the left side is a nonterminal

3regular

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.



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► 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.

Derivation Tree of Context-Free GrammarDerivation Tree of Context-Free Grammar

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► 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) ⇒ acbabcThus, the derivation tree is given as follows:

Example: Derivation TreeExample: Derivation Tree

S

a A B

B b a

c

b B

c

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4. 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.

5. 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).


5.1 LANG

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EXERCISE 5.1 : EXTRAEXERCISE 5.1 : EXTRA

PAGE : 793, 794, 795 and 796

Rosen K.H., Discrete Mathematics & Its Applications, (Seventh Edition), McGraw-Hill, 2007.

5.25.2 FINITE-STATE MACHINE WITH OUTPUTFINITE-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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• 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.

Finite State MachinesFinite State Machines

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• A Finite State Machine, M = (S, I, O, f, g, s0) consists of: 1. A finite set S of states2. A finite input alphabet I3. A finite output alphabet O4. A transition function f that assigns to each state and input pair

a new state5. An output functions g that assigns to each state and input pair

an output6. 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)

Finite State Machines with OutputFinite State Machines with Output

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• 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 Table and State DiagramState Table and State Diagram

Statef g

Input0 1

Input0 1

s0 s1 s0 1 0

s1 s3 s0 1 1

s2 s1 s2 0 1

s3 s2 s1 0 0

S = {s0, s1, s2, s3}, I = {0, 1}, O = {0, 1}

s1

s3s0

s2

1,1

1,00,1

1,1

1,0

0,1

0,00,0

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1. Draw the state diagrams for the finite state machines with the following tables.

Statef g

Input0 1

Input0 1

s0 s1 s0 1 1

s1 s2 s0 0 0

s2 s0 s3 1 0

s3 s1 s2 0 1


Statef g

Input0 1

Input0 1

s0 s1 s0 0 0

s1 s0 s2 0 1

s2 s1 s1 1 0

TABLE 1 TABLE 2

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2. Construct the state tables for the finite state machines with the following state diagrams.


FIGURE 1 FIGURE 2

s1 s2s0

1,0

0,0 1,0

1,0

0,0

0,1

s1

s3

s0

s2

1,11,0

0,1

1,1

1,00,1

0,0

0,0

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• If the input string is 0111 then the output string is 1100.

• If the input string is 11011011 then the output string is 00110110.

Input and Output StringInput and Output String

Statef g

Input0 1

Input0 1

s0 s1 s0 1 0

s1 s3 s0 1 1

s2 s1 s2 0 1

s3 s2 s1 0 0

S = {s0, s1, s2, s3}, I = {0, 1}, O = {0, 1}

s1

s3s0

s2

1,1

1,00,1

1,1

1,0

0,1

0,00,0

• Given the following finite state machine;

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3. Given the finite state machines with the following state table and state diagrams. Determine the output for each of the following input strings.


TABLE 1FIGURE 1

s1

s3

s0

s2

1,11,0

0,1

1,1

1,00,1

0,0

0,0

Statef g

Inputa b

Inputa b

s0 s0 s2 3 2

s1 s0 s2 2 1

s2 s1 s2 1 3

a) 1010 b) 11011011

c) 1010001001

w = abaabbababaa

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EXERCISE 5.2 : EXTRAEXERCISE 5.2 : EXTRA

PAGE : 802 and 803

Rosen K.H., Discrete Mathematics & Its Applications, (Seventh Edition), McGraw-Hill, 2007.

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THAT’S ALL ; THANK YOU

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

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bct 2083 discrete structure and applications

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