M Sipser Introduction to the Theory of Computation

ﻣﺮﺍﺟﻊ ﺩﺭﺱ : • ﻣﺮﺟﻊ ﺍﺻﻠی M. Sipser, ”Introduction to the Theory of Computation, ” 2 nd Ed. , Thompson Learning Inc. , 2006. : • ﻣﺮﺍﺟﻊ کﻤکی P. Linz, “An Introduction to Formal Languages and Automata, ” 3 rd Ed. , Jones and Barlett Publishers, Inc. , 2001. J. E. Hopcroft, R. Motwani and J. D. Ullman, “Introduction to Automata Theory, Languages, and Computation, ” 2 nd Ed. , Addison-Wesley, 2001. P. J. Denning, J. B. Dennnis, and J. E. Qualitz, “Machines, Languages, and Computation, ” Prentice-Hall, Inc. , 1978. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Logic

Intuitive (Naïve) Set Theory There are three basic concepts in set theory: • Membership • Extension • Abstraction 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Membership is a relation that holds between a set and an object to mean “the object x is a member of the set A”, or “x belongs to A”. The negation of this assertion is written as an abbreviation for the proposition One way to specify a set is to list its elements. For example, the set A = {a, b, c} consists of three elements. For this set A, it is true that but 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Extension The concept of extension is that two sets are identical if and if only if they contain the same elements. Thus we write A=B to mean 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Abstraction Each property defines a set, and each set defines a property • If p(x) is a property then we can define a set A • If A is a set then we can define a predicate p(x) 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Intuitive versus axiomatic set theory • The theory of set built on the intuitive concept of membership, extension, and abstraction is known as intuitive (naïve) set theory. • As an axiomatic theory of sets, it is not entirely satisfactory, because the principle of abstraction leads to contradictions when applied to certain simple predicates. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Russell’s Paradox Let p(X) be a predicate defined as P(X) = (X X) Define set R as R={X|p(X)} • Is p(R) true? • Is p(R) false? 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Gottlob Frege’s Comments • The logician Gottlob Frege was the first to develop mathematics on the foundation of set theory. He learned of Russell Paradox while his work was in press, and wrote, “A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr. Bertrand Russell as the work was nearly through the press. ” 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Power Set The set of all subsets of a given set A is known as the power set of A, and is denoted by P(A): P(A) = {B | B A} 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Set Operation-Union • The union of two sets A and B is A B = {x | (x A) (x B)} and consists of those elements in at least one of A and B. • If A 1, …, An constitute a family of sets, their union is = (A 1 … An) = {x| x Ai for some i, 1≤ i ≤ n} 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Set Operation-Intersection • The union of two sets A and B is A B = {x | (x A) (x B)} and consists of those elements in at least one of A and B. • If A 1, …, An constitute a family of sets, their intersection is = (A 1 … An) = {x| x Ai for all i, 1≤ i ≤ n} 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Set Operation-Complement • The complement of a set A is a set Ac defined as: Ac = {x | x A} • The complement of a set B with respect to A, also denoted as A-B, is defined as: Ac = {x A | x B} 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Ordered Pairs and n-tuples • An ordered pair of elements is written (x, y) where x is known as the first element, and y is known as the second element. • An n-tuple is an ordered sequence of elements (x 1, x 2, …, xn) And is a generalization of an ordered pair. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Ordered Sets and Set Products • By Cartesian product of two sets A and B, we mean the set A×B = {(x, y)|x A , y B} • Similarly, A 1×A 2×…An = {x 1 A 1, x 2 A 2, …, xn An} 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Relations A relation ρ between sets A and B is a subset of A×B: ρ A×B • The domain of ρ is defined as Dρ = {x A | for some y B, (x, y) ρ} • The range of ρ is defined as Rρ = {y B | for some x A, (x, y) ρ} • If ρ A×A, then ρ is called a ”relation on A”. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Types of Relations on Sets • A relation ρ is reflexive if (x, x) ρ, for each x A • A relation ρ is symmetric if, for all x, y A (x, y) ρ implies (y, x) ρ • A relation ρ is antisymmetric if, for all x, y A (x, y) ρ and (y, x) ρ implies x=y • A relation ρ is transitive if, for all x, y, z A (x, y) ρ and (y, z) ρ implies (x, z) ρ 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Partial Order and Equivalence Relations • A relation ρ on a set A is called a partial ordering of A if ρ is reflexive, anti-symmetric, and transitive. • A relation ρ on a set A is called an equivalence relation if ρ is reflexive, symmetric, and transitive. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Total Ordering • A relation ρ on a set A is |of A if ρ is a partial ordering and, for each pair of elements (x, y) in A×A at least one of (x, y) ρ or (y, x) ρ is true. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Inverse Relation • For any relation ρ A×B, the inverse of ρ is defined by ρ-1 = {(y, x) | (x, Y) ρ} • If Dρ and Rρ are the domain and range of ρ, then Dρ-1 = Dρ and Rρ-1 = Rρ 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Equivalence Class • Let ρ A×A be an equivalence relation on A. The equivalence class of an element x is defined as [x] = {y A | (x, y) ρ} • An equivalence relation on a set partitions the set. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Functions A relation f A× B is a function if it has the property (x, y) f and (x, z) f implies y=z • If f A× B is a function, we write f: A → B and say that f maps A into B. We use the common notation Y = f(x) to mean (x, y) f. • As before, the domain of f is the set Df = {x A | for some y B, (x, y) f} and the range of f is the set Rf = {y B | for some x A, (x, y) f} • If Df A, we say the function is a partial function; if Df = A, we say that f is a total function. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Functions (continued) x Df, we say that f is defined at x; otherwise f undefined at x. Rf = B, we say that f maps Df onto B. a function f has the property f(x) = z and f(y) = z implies x = y then f is a one-to-one function. If f: A → B is a one-to-one function, f gives a one-to-one correspondence between elements of its domain and range. • If is • If 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Functions (continued) • Let X be a set and suppose A X. Define function CA: X → {0, 1} such that CA(x) = 1, if x A; CA(x) = 0, otherwise. CA(x) is called the characteristic function of set A with respect to set X. • If f A× B is a function, then the inverse of f is the set f-1 = {(y, x) | (x, y) f} f-1 is a function if and only if f is one-to-one. • Let f: A → B be a function, and suppose that X A. Then the set Y = f(X) = {y B | y = f(x) for some x X} is known as the image of X under f. • Similarly, the inverse image of a set Y included in the range of f is f-1(Y) = {x A | y = f(x) for some y Y} 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Cardinality • Two sets A and B are of equal cardinality, written as |A| = |B| if and only if there is a one-to-one function f: A → B that maps A onto B. • We write |A| ≤ |B| if B includes a subset C such that |A| = |C|. • If |A| ≤ |B| and |A| ≠ |B|, then A has cardinality less than that of B, and we write |A| < |B| 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Cardinality (continued) • Let J = {1, 2, …} and Jn = {1, 2, …, n}. • A sequence on a set X is a function f: J → X. A sequence may be written as f(1), f(2), f(3), … However, we often use the simpler notation x 1, x 2, x 3, …. , xi X • A finite sequence of length n on X is a function f: Jn → X, usually written as x 1, x 2, x 3, …. , xn, xi X • The sequence of length zero is the function f: → X. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Finite and Infinite Sets • A set A is finite if |A| = |jn| for some integer n≥ 0, in which case we say that A has cardinality n. • A set is infinite if it is not finite. • A set X is denumerable if |X| = |j|. A set is countable if it is either finite or denumerable. • A set is uncountable if it is not countable. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Some Properties • Proposition: Every subset of J is countable. Consequently, each subset of any denumerable set is countable. • Proposition: A function f: J → Y has a countable range. Hence any function on a countable domain has a countable range. • Proposition: The set J × J is denumerable. Therefore, A × B is countable for arbitrary countable sets A and B. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Some Properties • Proposition: The set A B is countable whenever A and B are countable sets. • Proposition: Every infinite set X is at least denumerable ; that is |X| ≥ |J|. • Proposition: The set of all infinite sequence on {0, 1} is uncountable. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Some Properties • Proposition: (Schröder-Bernestein Theorem) For any set A and B, if |A| ≥ |B| and |B| ≥ |A|, then |A| = |B|. • Proposition: (Cantor’s Theorem) For any set X, |X| < |P(X)|. 8 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ

Gödel Numbering • Let be an alphabet containing n objects. Let h: → Jn be an arbitrary one-to-one correspondence. Define function f as: f: → N such that f(ε) =0; f(w. v) = n f(w) + h(v), for w and v . f is called a Gödel Numbering of. • Proposition: is denumerable. • Proposition: Any language on an alphabet is countable. 86 -87 ﻧیﻤﺴﺎﻝ ﺩﻡ ﺩﺍﻧﺸگﺎﻩ ﺻﻨﻌﺘی ﻧﻈﺮیﻪ ﺯﺑﺎﻥ ﻫﺎ ﻭ ﻣﺎﺷیﻦ ﻫﺎ