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BCT 2083 DISCRETE STRUCTURE AND APPLICATIONS. CHAPTER 5 MODELLING COMPUTATION. SITI ZANARIAH SATARI FIST/FSKKP UMP I0910. 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)
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BCT 2083 DISCRETE STRUCTURE AND APPLICATIONS CHAPTER 5 MODELLING COMPUTATION SITI ZANARIAH SATARI FIST/FSKKP UMP I0910
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) CHAPTER 5 MODELLING COMPUTATION
Introduction • 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 MODELLING COMPUTATION
CHAPTER 5 MODELLING COMPUTATION • Understand language and grammars in models of computation • Construct derivation tree 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? 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, λis 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. 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. 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) = { wЄT*: 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. 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}. 5.1 LANGUAGE AND GRAMMARS
Types of Grammars 5.1 LANGUAGE AND 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.
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. 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: 5.1 LANGUAGE AND GRAMMARS S a A B B b b a B c c
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). 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. 5.1 LANGUAGE AND GRAMMARS
5.2 FINITE-STATE MACHINE WITH OUTPUT CHAPTER 5 MODELLING COMPUTATION • Understand finite state machines with output. • Draw state diagrams for finite state machines with output. • Construct state table for finite state machines with output.
Finite State Machines • 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. 5.2 FINITE-STATE MACHINES WITH OUTPUT
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) 5.2 FINITE-STATE MACHINES WITH OUTPUT
State Table and State Diagram • State TABLE represents the values of the transaction functionf and the output functiong 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 5.2 FINITE-STATE MACHINES WITH OUTPUT 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
EXERCISE 5.2 • Draw the state diagrams for the finite state machines with the following tables. 5.2 FINITE-STATE MACHINES WITH OUTPUT TABLE 1 TABLE 2
EXERCISE 5.2 • Construct the state tables for the finite state machines with the following state diagrams. 5.2 FINITE-STATE MACHINES WITH OUTPUT 0,0 0,0 1,0 s0 0,0 s1 0,1 s0 s1 s2 1,1 0,1 1,0 1,0 0,1 1,0 0,0 s2 s3 1,0 1,1 FIGURE 1 FIGURE 2
Input and Output String • If the input string is 0111 then the output string is 1100. • If the input string is 11011011 then the output string is 00110110. 5.2 FINITE-STATE MACHINES WITH OUTPUT • Given the following finite state machine; 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
EXERCISE 5.2 • Given the finite state machines with the following state table and state diagrams. Determine the output for each of the following input strings. 5.2 FINITE-STATE MACHINES WITH OUTPUT 0,0 s0 s1 0,1 1,1 0,1 1,0 1,0 0,0 s2 s3 1,1 TABLE 1 FIGURE 1 w = abaabbababaa • 1010 b) 11011011 • c) 1010001001
EXERCISE 5.2 : EXTRA PAGE : 802 and 803 Rosen K.H., Discrete Mathematics & Its Applications, (Seventh Edition), McGraw-Hill, 2007. 5.2 FINITE-STATE MACHINES WITH OUTPUT
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