Fundamentals of Programming Languages I Introduction and Logics

Transcription

1 Fundamentals of Programming Languages I Introduction and Logics Guoqiang Li School of Software, Shanghai Jiao Tong University

2 Instructor and Teaching Assistants Guoqiang LI

3 Instructor and Teaching Assistants Guoqiang LI Homepage: liguoqiang Course page: liguoqiang/teaching/prog17/index.htm Office: Rm. 1212, Building of Software Phone:

4 Instructor and Teaching Assistants Guoqiang LI Homepage: liguoqiang Course page: liguoqiang/teaching/prog17/index.htm Office: Rm. 1212, Building of Software Phone: TA: Yuwei WANG: wangyuwei95 (AT) qq (DOT) com

5 Instructor and Teaching Assistants Guoqiang LI Homepage: liguoqiang Course page: liguoqiang/teaching/prog17/index.htm Office: Rm. 1212, Building of Software Phone: TA: Yuwei WANG: wangyuwei95 (AT) qq (DOT) com Office hour: Tue. Software Building 3203

6 What does the lecture aim for?

7 Similar Lectures I Fundamentals of Programming Languages by University of Colorado Boulder bec/courses/csci5535-f13/

8 Similar Lectures I Fundamentals of Programming Languages by University of Colorado Boulder bec/courses/csci5535-f13/ 2010 Spring Programming semantics 2013 Fall Programming analysis and verification

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13 Similar Lectures IV Theory of Programming Languages by CMU aldrich/courses/15-819o-13sp Introduction to Programming Languages Theory by Standford Theory of Programming Languages by SJTU xiaojuan/tapl2016/index.html

14 Similar Lectures IV Theory of Programming Languages by CMU aldrich/courses/15-819o-13sp Introduction to Programming Languages Theory by Standford Theory of Programming Languages by SJTU xiaojuan/tapl2016/index.html Types and Functional Programming Languages

15 Fundamental Requirements Program Verification and Analysis

16 Fundamental Requirements Program Verification and Analysis Propositional logic, predicate logic etc. Automata theory, DFA, NFA, PDS, PN etc. Algorithm.

17 Fundamental Requirements Program Verification and Analysis Propositional logic, predicate logic etc. Automata theory, DFA, NFA, PDS, PN etc. Algorithm. Program Semantics

18 Fundamental Requirements Program Verification and Analysis Propositional logic, predicate logic etc. Automata theory, DFA, NFA, PDS, PN etc. Algorithm. Program Semantics Set theory. Algebra theory, group, ring, domain etc. category theory, maybe...

19 Fundamental Requirements Program Verification and Analysis Propositional logic, predicate logic etc. Automata theory, DFA, NFA, PDS, PN etc. Algorithm. Program Semantics Set theory. Algebra theory, group, ring, domain etc. category theory, maybe... Types and Programming Languages

20 Fundamental Requirements Program Verification and Analysis Propositional logic, predicate logic etc. Automata theory, DFA, NFA, PDS, PN etc. Algorithm. Program Semantics Set theory. Algebra theory, group, ring, domain etc. category theory, maybe... Types and Programming Languages Logic Computability theory Lambda calculus theory...

21 Fundamental of Fundamental Several theories in theoretical computer science are given, which is a minimal requirement and self-contained in this lecture.

22 Fundamental of Fundamental Several theories in theoretical computer science are given, which is a minimal requirement and self-contained in this lecture. All of three directions are taught, which only include very fundamental part,

23 Fundamental of Fundamental Several theories in theoretical computer science are given, which is a minimal requirement and self-contained in this lecture. All of three directions are taught, which only include very fundamental part, if time permitted.

24 Fundamental of Fundamental Several theories in theoretical computer science are given, which is a minimal requirement and self-contained in this lecture. All of three directions are taught, which only include very fundamental part, if time permitted. As simple as possible,

25 Fundamental of Fundamental Several theories in theoretical computer science are given, which is a minimal requirement and self-contained in this lecture. All of three directions are taught, which only include very fundamental part, if time permitted. As simple as possible, although it is very theoretical.

26 Lecture Agenda Introduction and logic basics (1 lecture) Formal basics (3 lectures) Programming verification (2 or 3 lectures) Exercise I. (1 lecture) Programming semantics (2 lectures) Basic functional programming (3 lectures) Exercise II. (1 lecture) Conclusion and wrap up (1 lecture)

27 Lecture Agenda Introduction and logic basics (1 lecture) Formal basics (3 lectures) Model checking Finite and Büchi automata LTL model checking Programming verification (2 or 3 lectures) Exercise I. (1 lecture) Programming semantics (2 lectures) Basic functional programming (3 lectures) Exercise II. (1 lecture) Conclusion and wrap up (1 lecture)

28 Lecture Agenda Introduction and logic basics (1 lecture) Formal basics (3 lectures) Programming verification (2 or 3 lectures) Abstract interpretation Pushdown automata and interprocedural programs Petri Net and concurrent programs Exercise I. (1 lecture) Programming semantics (2 lectures) Basic functional programming (3 lectures) Exercise II. (1 lecture) Conclusion and wrap up (1 lecture)

29 Lecture Agenda Introduction and logic basics (1 lecture) Formal basics (3 lectures) Programming verification (2 or 3 lectures) Exercise I. (1 lecture) Programming semantics (2 lectures) Denotational semantics Operational semantics Axiomatic semantics Basic functional programming (3 lectures) Exercise II. (1 lecture) Conclusion and wrap up (1 lecture)

30 Lecture Agenda Introduction and logic basics (1 lecture) Formal basics (3 lectures) Programming verification (2 or 3 lectures) Exercise I. (1 lecture) Programming semantics (2 lectures) Basic functional programming (3 lectures) Lambda calculus Simple types Functional programming Exercise II. (1 lecture) Conclusion and wrap up (1 lecture)

31 References No particular textbook that can cover all the parts. Here are three Reference books: Edmund M. Clarke Jr., Orna Grumberg, Doron A. Peled. Model Checking. MIT Press, 1999 Glynn Winskel. Formal Semantics of Programming Languages: An Introduction. MIT Press, 1993 Benjamin C. Pierce. Types and Programming Languages. MIT Press, 2002

32 References No particular textbook that can cover all the parts. Here are three Reference books: Edmund M. Clarke Jr., Orna Grumberg, Doron A. Peled. Model Checking. MIT Press, 1999 Glynn Winskel. Formal Semantics of Programming Languages: An Introduction. MIT Press, 1993 Benjamin C. Pierce. Types and Programming Languages. MIT Press, Several famous papers + Lecture notes shared in the course webpage.

33 Scoring Policy 10% Attendance. 20% Homework. 70% Final exam.

34 Scoring Policy 10% Attendance. 20% Homework. Four assignments. 70% Final exam.

35 Scoring Policy 10% Attendance. 20% Homework. Four assignments. Each one is 5pts. 70% Final exam.

36 Scoring Policy 10% Attendance. 20% Homework. Four assignments. Each one is 5pts. Work out individually. 70% Final exam.

37 Scoring Policy 10% Attendance. 20% Homework. Four assignments. Each one is 5pts. Work out individually. Each assignment will be evaluated by A, B, C, D, F (Excellent(5), Good(5), Fair(4), Delay(3), Fail(0)) 70% Final exam.

38 Scoring Policy 10% Attendance. 20% Homework. Four assignments. Each one is 5pts. Work out individually. Each assignment will be evaluated by A, B, C, D, F (Excellent(5), Good(5), Fair(4), Delay(3), Fail(0)) 70% Final exam. Maybe replaced by report, if the condition is satisfied!

39 Any Questions?

40 Logic Basics

41 Brief Historical Notes on Logic

42 Historical View Philosophical Logic 500 BC to 19th Century Symbolic Logic Mid to late 19th Century Mathematical Logic Late 19th to mid 20th Century Logic in Computer Science

43 Philosophical Logic

44 500 B.C - 19th Century Philosophical Logic

45 Philosophical Logic 500 B.C - 19th Century Logic dealt with arguments in the natural language used by humans.

46 Philosophical Logic 500 B.C - 19th Century Logic dealt with arguments in the natural language used by humans. Example: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

47 Philosophical Logic

48 Philosophical Logic Natural languages are very ambiguous.

49 Philosophical Logic Natural languages are very ambiguous. Eric does not believe that Mary can pass any test. does not believe that she can pass some test, or does not believe that she can pass all tests I only borrowed your car. And not borrowed and used, or And not car and coat Tom hates Jim and he likes Mary. Tom likes Mary, or Jim likes Mary

50 Philosophical Logic Natural languages are very ambiguous. Eric does not believe that Mary can pass any test. does not believe that she can pass some test, or does not believe that she can pass all tests I only borrowed your car. And not borrowed and used, or And not car and coat Tom hates Jim and he likes Mary. Tom likes Mary, or Jim likes Mary It led to many paradoxes.

51 Philosophical Logic Natural languages are very ambiguous. Eric does not believe that Mary can pass any test. does not believe that she can pass some test, or does not believe that she can pass all tests I only borrowed your car. And not borrowed and used, or And not car and coat Tom hates Jim and he likes Mary. Tom likes Mary, or Jim likes Mary It led to many paradoxes. This sentence is a lie. (The Liar s Paradox)

52 Sophism Sophism generally refers to a particularly confusing, illogical and/or insincere argument used by someone to make a point, or, perhaps, not to make a point. Sophistry refers to [ ] rhetoric that is designed to appeal to the listener on grounds other than the strict logical cogency of the statements being made.

53 The Sophist s Paradox A Sophist is sued for his tuition by the school that educated him. He argues that he must win, since, if he loses, the school did not educate him well enough, and does not deserve the money.

54 The Sophist s Paradox A Sophist is sued for his tuition by the school that educated him. He argues that he must win, since, if he loses, the school did not educate him well enough, and does not deserve the money. The school argues that he must lose, since, if he wins, he was educated well enough, and therefore should pay for it.

55 Logic in Computer Science Logic has a profound impact on computer science. Some examples: Propositional logic - the foundation of computers and circuitry Databases - query languages Programming languages (e.g. prolog) Design Validation and verification AI (e.g. inference systems)

56 Logic in Computer Science Propositional Logic First Order Logic Higher Order Logic Temporal Logic

57 Propositional Logic: Syntax

58 Propositional Logic

59 Propositional Logic A proposition: a sentence that can be either true or false.

60 Propositional Logic A proposition: a sentence that can be either true or false. Propositions: x is greater than y Noam wrote this letter

61 Propositional Logic: Syntax

62 Propositional Logic: Syntax The symbols of the language: Propositional symbols (Prop): A, B, C,... Connectives: and or not implies equivalent to xor (different than), False, True Parenthesis: (, ).

63 Propositional Logic: Syntax The symbols of the language: Propositional symbols (Prop): A, B, C,... Connectives: and or not implies equivalent to xor (different than), False, True Parenthesis: (, ). Q1: How many different binary symbols can we define?

64 Propositional Logic: Syntax The symbols of the language: Propositional symbols (Prop): A, B, C,... Connectives: and or not implies equivalent to xor (different than), False, True Parenthesis: (, ). Q1: How many different binary symbols can we define? Q2: What is the minimal number of such symbols?

65 Formulas Grammar of well-formed propositional formulas Formula := prop (Formula) (Formula Formula) where prop Prop and is one of the binary relations.

66 Formulas Examples of well-formed formulas: ( A) ( ( A)) (A (B C)) (A (B C)) Correct expressions of Propositional Logic are full of unnecessary parenthesis.

67 Formulas: Abbreviations We write A B C...

68 Formulas: Abbreviations We write in place of A B C... (A (B (C...)))

69 Formulas: Abbreviations We write in place of A B C... (A (B (C...))) Thus, we write A B C, A B C,...

70 Formulas: Abbreviations We write in place of A B C... (A (B (C...))) Thus, we write A B C, A B C,... in place of (A (B C)), (A (B C)),...

71 Formulas: Abbreviations We omit parenthesis whenever we may restore them through operator precedence:

72 Formulas: Abbreviations We omit parenthesis whenever we may restore them through operator precedence: binds more strictly than,, and, bind more strictly than,.

73 Formulas: Abbreviations We omit parenthesis whenever we may restore them through operator precedence: binds more strictly than,, and, bind more strictly than,. Thus, we write: A for ( ( A)), A B for (( A) B) A B C for ((A B) C)

74 Propositional Logic: Semantics

75 Propositional Logic: Semantics Truth tables define the semantics (=meaning) of the operators Convention: 0 = false, 1 = true A B A B A B A B

76 Propositional Logic: Semantics Truth tables define the semantics (=meaning) of the operators Convention: 0 = false, 1 = true A B A A B A B

77 Back to Q1 Q1: How many binary operators can we define that have different semantic definition?

78 Back to Q1 Q1: How many binary operators can we define that have different semantic definition? A: 16

79 Satisfiability and Validity

80 Assignments Definition: A truth-values assignment, α, is an element of 2 Prop (i.e., α 2 Prop ). In other words, α is a subset of the variables that are assigned true. Equivalently, we can see α as a mapping from variables to truth values: α : Prop {0, 1} Example: α = {A 0, B 1,...}

81 Satisfaction Relation ( =): Intuition An assignment can either satisfy or not satisfy a given formula. α = φ means α satisfies φ or φ holds at α or α is a model of φ We will first see an example. Then we will define these notions formally.

82 Let φ = (A (B C)) Example

83 Example Let φ = (A (B C)) Let α = {A 0, B 0, C 1}

84 Example Let φ = (A (B C)) Let α = {A 0, B 0, C 1} Q: Does α satisfy φ (α = φ?)

85 Example Let φ = (A (B C)) Let α = {A 0, B 0, C 1} Q: Does α satisfy φ (α = φ?) A: (0 (0 1)) = (0 1) = 1

86 Example Let φ = (A (B C)) Let α = {A 0, B 0, C 1} Q: Does α satisfy φ (α = φ?) A: (0 (0 1)) = (0 1) = 1 Hence, α = φ.

87 Example Let φ = (A (B C)) Let α = {A 0, B 0, C 1} Q: Does α satisfy φ (α = φ?) A: (0 (0 1)) = (0 1) = 1 Hence, α = φ. Let us now formalize an evaluation process.

88 Satisfaction Relation ( =): Formalities = is a relation: = (2 Prop Formula)

89 Satisfaction Relation ( =): Formalities = is a relation: = (2 Prop Formula) Examples: ({A}, A B): the assignment α = {A} satisfies A B ({A, B}, A B)

90 Satisfaction Relation ( =): Formalities = is a relation: = (2 Prop Formula) Examples: ({A}, A B): the assignment α = {A} satisfies A B ({A, B}, A B) Alternatively: = ({0, 1} Prop Formula)

91 Satisfaction Relation ( =): Formalities = is a relation: = (2 Prop Formula) Examples: ({A}, A B): the assignment α = {A} satisfies A B ({A, B}, A B) Alternatively: = ({0, 1} Prop Formula) Examples: (01, A B): the assignment α = {A 0, B 1} satisfies A B (11, A B)

92 Satisfaction Relation ( =): Formalities = is defined recursively: α = A if α(a) = true α = ϕ if α = ϕ α = ϕ 1 ϕ 2 if α = ϕ 1 and α = ϕ 2 α = ϕ 1 ϕ 2 if α = ϕ 1 or α = ϕ 2 α = ϕ 1 ϕ 2 if α = ϕ 1 implies α = ϕ 2 α = ϕ 1 ϕ 2 if α = ϕ 1 iff α = ϕ 2

93 From Definition to an Evaluation Algorithm Truth Evaluation Problem: Given ϕ Formula and α 2 AP (ϕ), does α = ϕ?

94 From Definition to an Evaluation Algorithm Truth Evaluation Problem: Given ϕ Formula and α 2 AP (ϕ), does α = ϕ? Eval(ϕ, α) if ϕ A then return α(a); if ϕ φ then return Eval (φ, α); if ϕ ψ φ then return Eval (ψ, α) Eval (φ, α);

95 From Definition to an Evaluation Algorithm Truth Evaluation Problem: Given ϕ Formula and α 2 AP (ϕ), does α = ϕ? Eval(ϕ, α) if ϕ A then return α(a); if ϕ φ then return Eval (φ, α); if ϕ ψ φ then return Eval (ψ, α) Eval (φ, α); Eval uses polynomial time and space.

96 Nothing More Than What We Already Know Recall the Example: Let φ = (A (B C)) Let α = {A 0, B 0, C 1}

97 Nothing More Than What We Already Know Recall the Example: Let φ = (A (B C)) Let α = {A 0, B 0, C 1} Eval(φ, α) = Eval(A, α) Eval(B C, α) = 0 Eval(B, α) Eval(C, α) = 0 (0 1) = 0 1 = 1

98 Nothing More Than What We Already Know Recall the Example: Let φ = (A (B C)) Let α = {A 0, B 0, C 1} Eval(φ, α) = Eval(A, α) Eval(B C, α) = 0 Eval(B, α) Eval(C, α) = 0 (0 1) = 0 1 = 1 Hence, α = φ.

99 Extending Truth Table p q (p (q p)) (p p) (p q)

100 Extending Truth Table p q r (p (q r)

101 Extending Truth Table p q r (p (q r)

102 Set of Assignment Intuition: a formula specifies a set of truth assignments. Function models: models : Formula 2 2Prop (a formula set of satisfying assignments) Recursive definition: models(a) = {α α(a) = 1}, A Prop models( ϕ) = 2 Prop models(ϕ) models(ϕ 1 ϕ 2 ) = models(ϕ 1 ) models(ϕ 2 ) models(ϕ 1 ϕ 2 ) = models(ϕ 1 ) models(ϕ 2 ) models(ϕ 1 ϕ 2 ) = (2 Prop models(ϕ 1 ) models(ϕ 2 )

103 Example models(a B) = {{10}, {01}, {11}} This is compatible with the recursive definition: models(a B) = models(a) models(b) = {{10}, {11}} {{01}, {11}} = {{10}, {01}, {11}}

104 Theorem Let ϕ Formula and α 2 Prop, then the following statements are equivalent: α = ϕ α models(ϕ)

105 Projected Assignment AP(ϕ): the Atomic Propositions in ϕ. Clearly AP(ϕ) Prop. Let α 1, α 2 2 Prop, Formula. Lemma: if α 1 AP(ϕ) = α 2 AP(ϕ), then α 1 = ϕ iff α 2 = ϕ Corollary: α = ϕ iff α AP(ϕ) = ϕ We will assume, for simplicity, that Prop = AP(ϕ).

106 Extension of = to Assignment Sets Let ϕ Formula Let T be a set of assignments, i.e., T 2 2Prop Definition. T = ϕ if T models(ϕ) i.e., = 2 2Prop Formula

107 Extension of = to Formulas = 2 Formula 2 Formula Definition. Let Γ 1, Γ 2 be propositional formulas. Γ 1 = Γ 2 iff models(γ 1 ) models(γ 2 ) iff for all α 2 Prop if α = Γ 1 then α = Γ 2 Examples: x 1 x 2 = x 1 x 2 x 1 x 2 = x 2 x 3

108 Classification of Formulas A formula ϕ is called valid if models(ϕ) = 2 Prop. (also called a tautology). A formula ϕ is called satisfiable if models(ϕ). A formula ϕ is called unsatisfiable if models(ϕ) = (also called a contradiction).

109 Characteristics of Formulas A formula ϕ is valid iff ϕ is unsatisfiable. ϕ is satisfiable iff ϕ is not valid.

110 Characteristics of Formulas We can write = ϕ when ϕ is valid. = ϕ when ϕ is not valid. = ϕ when ϕ is satisfiable. = ϕ when ϕ is unsatisfiable

111 Examples (p q) (p q) is valid (p q) p is satisfiable (p q) p is unsatisfiable

112 Equivalences = A 1 A = A 0 0 = A A = A (B C) (A B) (A C) = (A B) ( A B) = (A B) ( A B)

113 Minimal Set of Binary Operators Recall the question: what is the minimal set of operators necessary? A: Through such equivalences all Boolean operators can be written with a single operator ( ). Indeed, typically industrial circuits only use one type of logical gate. We will see how two are enough: and Or: = (A B) ( A B) Implies: = (A B) ( A B) Equivalence: = (A B) (A B) (B A)

114 Decision Problem The decision problem: Given a propositional formula φ, is φ satisfiable? An algorithm that always terminates with a correct answer to this problem is called a decision procedure for propositional logic.

115 Normal Forms

116 Definitions A literal is either an atom or a negation of an atom. Let φ = (A B). Then: Atoms: AP(φ) = {A, B} Literals: lit(φ) = {A, B} Equivalent formulas can have different literals φ = (A B) = A B Now lit(φ) = { A, B}

117 Definitions A term is a conjunction of literals Example: (A B C) A clause is a disjunction of literals Example: (A B C)

118 Negation Normal Form (NNF) A formula is said to be in Negation Normal Form (NNF) if it only contains,, connectives and only atoms can be negated. Examples: (A B) is not in NNF A B is in NNF

119 Coverting to NNF Every formula can be converted to NNF in linear time: Eliminate all connectives other than,, Use De Morgan and double-negation rules to push negations to the right Example: (A B) Eliminate : ( A B) Push negation using De Morgan: ( A B) Use Double negation rule: (A B)

120 Disjunctive Normal Form (DNF) A formula is said to be in Disjunctive Normal Form (DNF) if it is a disjunction of terms. In other words, it is a formula of the form ( l i,j ) i j where l i,j is the j-th literal in the i-th term. Examples (A B C) ( A D) (B) is in DNF.

121 Disjunctive Normal Form (DNF) A formula is said to be in Disjunctive Normal Form (DNF) if it is a disjunction of terms. In other words, it is a formula of the form ( l i,j ) i j where l i,j is the j-th literal in the i-th term. Examples (A B C) ( A D) (B) is in DNF. DNF is a special case of NNF.

122 Coverting to DNF Every formula can be converted to DNF in exponential time and space: Convert to NNF Distribute disjunctions following the rule: = A (B C) ((A B) (A C)) Example: (A B) ( C D) ((A B) ( C)) ((A B) D) (A C) (B C) (A D) (B D)

123 Coverting to DNF Every formula can be converted to DNF in exponential time and space: Convert to NNF Distribute disjunctions following the rule: = A (B C) ((A B) (A C)) Example: (A B) ( C D) ((A B) ( C)) ((A B) D) (A C) (B C) (A D) (B D) Q:How many clauses would the DNF have had we started from a conjunction of n clauses?

124 Satisfiability of DNF Is the following DNF formula satisfiable? (x 1 x 2 x 1 ) (x 2 x 1 ) (x 2 x 3 x 3 )

125 Satisfiability of DNF Is the following DNF formula satisfiable? (x 1 x 2 x 1 ) (x 2 x 1 ) (x 2 x 3 x 3 ) What is the complexity of satisfiability of DNF formulas?

126 Conjunctive Normal Form (CNF) A formula is said to be in Conjunctive Normal Form (CNF) if it is a conjunction of clauses. In other words, it is a formula of the form ( l i,j ) i j where l i,j is the j-th literal in the i-th term. Examples (A B C) ( A D) (B) is in CNF

127 Conjunctive Normal Form (CNF) A formula is said to be in Conjunctive Normal Form (CNF) if it is a conjunction of clauses. In other words, it is a formula of the form ( l i,j ) i j where l i,j is the j-th literal in the i-th term. Examples (A B C) ( A D) (B) is in CNF CNF is a special case of NNF.

128 Coverting to CNF Every formula can be converted to CNF:

129 Coverting to CNF Every formula can be converted to CNF: in exponential time and space with the same set of atoms

130 Coverting to CNF Every formula can be converted to CNF: in exponential time and space with the same set of atoms in linear time and space if new variables are added.

131 Coverting to CNF Every formula can be converted to CNF: in exponential time and space with the same set of atoms in linear time and space if new variables are added. In this case the original and converted formulas are equi-satisfiable. This technique is called Tseitin s encoding.

132 Converting to CNF: the Exponential Way CNF(φ){ case } φ is a literal: return φ φ is ϕ 1 ϕ 2 : return CNF(ϕ 1 ) CNF(ϕ 2 ) φ is ϕ 1 ϕ 2 : return Dist(CNF(ϕ 1 ), CNF(ϕ 2 )) Dist(ϕ 1, ϕ 2 ){ case } ϕ 1 is ψ 11 ψ 12 : return Dist(ψ 11, ϕ 2 ) Dist(ψ 12, ϕ 2 ) ϕ 2 is ψ 21 ψ 22 : return Dist(ϕ 1, ψ 21 ) Dist(ϕ 1, ψ 22 )

133 Converting to CNF: the Exponential Way Consider the formula φ = (x 1 y 1 ) (x 2 y 2 )

134 Converting to CNF: the Exponential Way Consider the formula φ = (x 1 y 1 ) (x 2 y 2 ) CNF(φ) = (x 1 x 2 ) (x 1 y 2 ) (y 1 x 2 ) (y 1 y 2 )

135 Converting to CNF: the Exponential Way Consider the formula φ = (x 1 y 1 ) (x 2 y 2 ) CNF(φ) = (x 1 x 2 ) (x 1 y 2 ) (y 1 x 2 ) (y 1 y 2 ) Now consider: φ n = (x 1 y 1 ) (x 2 y 2 )... (x n y n )

136 Converting to CNF: the Exponential Way Consider the formula φ = (x 1 y 1 ) (x 2 y 2 ) CNF(φ) = (x 1 x 2 ) (x 1 y 2 ) (y 1 x 2 ) (y 1 y 2 ) Now consider: φ n = (x 1 y 1 ) (x 2 y 2 )... (x n y n ) Q: How many clauses CNF(φ n ) returns?

137 Converting to CNF: the Exponential Way Consider the formula φ = (x 1 y 1 ) (x 2 y 2 ) CNF(φ) = (x 1 x 2 ) (x 1 y 2 ) (y 1 x 2 ) (y 1 y 2 ) Now consider: φ n = (x 1 y 1 ) (x 2 y 2 )... (x n y n ) Q: How many clauses CNF(φ n ) returns? A: 2 n

138 Tseitin s Encoding Consider the formula (A (B C))

139 Tseitin s Encoding Consider the formula (A (B C)) The parse tree:

140 Tseitin s Encoding Consider the formula (A (B C)) The parse tree: Associate a new auxiliary variable with each gate. Add constraints that define these new variables. Finally, enforce the root node.

141 (a 1 (A a 2 )) (a 2 (B C)) (a 1 ) Tseitin s Encoding

142 Tseitin s Encoding (a 1 (A a 2 )) (a 2 (B C)) (a 1 ) Each such constraint has a CNF representation with 3 or 4 clauses.

143 Tseitin s Encoding (a 1 (A a 2 )) (a 2 (B C)) (a 1 ) Each such constraint has a CNF representation with 3 or 4 clauses. First: (a 1 A) (a 1 a 2 ) ( a 1 A a 2 ) Second: ( a 2 B) ( a 2 C) (a 2 B C)

144 Tseitin s Encoding φ n = (x 1 y 1 ) (x 2 y 2 )... (x n y n )

145 Tseitin s Encoding φ n = (x 1 y 1 ) (x 2 y 2 )... (x n y n ) With Tseitin s encoding we need: n auxiliary variables a 1,..., a n. Each adds 3 constraints. Top clause: (a 1... a n )

146 Tseitin s Encoding φ n = (x 1 y 1 ) (x 2 y 2 )... (x n y n ) With Tseitin s encoding we need: n auxiliary variables a 1,..., a n. Each adds 3 constraints. Top clause: (a 1... a n ) Hence, we have 3n + 1 clauses, instead of 2 n. 3n variables rather than 2n.

147 SAT Problem and SAT Solver SAT problem is: Given a Boolean formula in CNF, asking whether there exists an assignment to each variable so that the value of the formula is true.

148 SAT Problem and SAT Solver SAT problem is: Given a Boolean formula in CNF, asking whether there exists an assignment to each variable so that the value of the formula is true. It is a NPC problem, which means that there is only exponential algorithm so far. A SAT solver is a tool that solves the SAT problem.

149 SAT Problem and SAT Solver SAT problem is: Given a Boolean formula in CNF, asking whether there exists an assignment to each variable so that the value of the formula is true. It is a NPC problem, which means that there is only exponential algorithm so far. A SAT solver is a tool that solves the SAT problem. However, SAT solver is to be said as the most successful formal tools, which can handle 100,000 variables with millions of clauses in less than one sec.