Counting Quantification

Counting quanticationFrom Wikipedia, the free encyclopedia

Contents

1 (, )-denition of limit 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Informal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Precise statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Comparison with innitesimal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Boolean satisability problem 52.1 Basic denitions and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Complexity and restricted versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Unrestricted satisability (SAT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 3-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Exactly-1 3-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.4 2-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.5 Horn-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.6 XOR-satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.7 Schaefers dichotomy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Extensions of SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Self-reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Algorithms for solving SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.9.1 SAT problem format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9.2 Online SAT solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9.3 Oine SAT solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9.4 SAT applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9.5 Conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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2.9.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9.7 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9.8 Evaluation of SAT solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Bounded quantier 153.1 Bounded quantiers in arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Bounded quantiers in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Branching quantier 174.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Relation to natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Conditional quantier 205.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Conjunctive normal form 216.1 Examples and Non-Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Conversion into CNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.5 Converting from rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Counting quantication 257.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Donkey sentence 268.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Discourse representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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9 Existential quantication 319.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.2.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2.3 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.3 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10 First-order logic 3510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 4210.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 4310.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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10.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

10.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

11 Game semantics 5611.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.2 Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.3 Intuitionistic logic, denotational semantics, linear logic, logical pluralism . . . . . . . . . . . . . . 5711.4 Quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.6.1 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.6.2 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

12 Generalized quantier 6012.1 Type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.2 Typed lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.3.2 Conservativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13 Lindstrm quantier 6413.1 Generalization of rst-order quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2 Expressiveness hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.3 As precursors to Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.4 Algorithmic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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13.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

14 Logic 6714.1 The study of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

14.1.1 Logical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.1.2 Deductive and inductive reasoning, and abductive inference . . . . . . . . . . . . . . . . . 6814.1.3 Consistency, validity, soundness, and completeness . . . . . . . . . . . . . . . . . . . . . . 6914.1.4 Rival conceptions of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.3 Types of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

14.3.1 Syllogistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.3.2 Propositional logic (sentential logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.3.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.3.4 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7214.3.5 Informal reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7214.3.6 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7314.3.7 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7314.3.8 Computational logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7314.3.9 Bivalence and the law of the excluded middle; non-classical logics . . . . . . . . . . . . . 7414.3.10 Is logic empirical?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.11 Implication: strict or material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.12 Tolerating the impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.13 Rejection of logical truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

15 Mathematical logic 8115.1 Subelds and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8115.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

15.2.1 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.2.2 19th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.2.3 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

15.3 Formal logical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.3.1 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.3.2 Other classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.3.3 Nonclassical and modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.3.4 Algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

15.4 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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15.5 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.6 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

15.6.1 Algorithmically unsolvable problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.7 Proof theory and constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.8 Connections with computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.9 Foundations of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

15.12.1 Undergraduate texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.12.2 Graduate texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.12.3 Research papers, monographs, texts, and surveys . . . . . . . . . . . . . . . . . . . . . . 9115.12.4 Classical papers, texts, and collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

15.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16 Mathematics 9516.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

16.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9616.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

16.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9916.2.1 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

16.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 10216.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10316.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

16.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10416.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10516.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

16.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10716.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10816.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10816.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11016.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11116.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

17 Maxima and minima 11317.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.2 Finding functional maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.4 Functions of more than one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11517.5 Maxima or minima of a functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11617.6 In relation to sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11617.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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17.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11717.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

18 Plural quantication 11818.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11818.2 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

18.2.1 Multigrade (variably polyadic) predicates and relations . . . . . . . . . . . . . . . . . . . 11818.2.2 Nominalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

18.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11918.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12018.3.2 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

18.4 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12118.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12118.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12118.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

19 Predicate logic 12319.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12319.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12319.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

20 Propositional formula 12520.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

20.1.1 Relationship between propositional and predicate formulas . . . . . . . . . . . . . . . . . 12620.1.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

20.2 An algebra of propositions, the propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . 12620.2.1 Usefulness of propositional formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12720.2.2 Propositional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12720.2.3 Truth-value assignments, formula evaluations . . . . . . . . . . . . . . . . . . . . . . . . 127

20.3 Propositional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12820.3.1 Connectives of rhetoric, philosophy and mathematics . . . . . . . . . . . . . . . . . . . . 12820.3.2 Engineering connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12820.3.3 CASE connective: IF ... THEN ... ELSE ... . . . . . . . . . . . . . . . . . . . . . . . . . 12820.3.4 IDENTITY and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

20.4 More complex formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13020.4.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13020.4.2 Axiom and denition schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13120.4.3 Substitution versus replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

20.5 Inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13120.6 Parsing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

20.6.1 Connective seniority (symbol rank) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13220.6.2 Commutative and associative laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

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20.6.3 Distributive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13320.6.4 De Morgans laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13320.6.5 Laws of absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13420.6.6 Laws of evaluation: Identity, nullity, and complement . . . . . . . . . . . . . . . . . . . . 13420.6.7 Double negative (Involution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

20.7 Well-formed formulas (ws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13420.7.1 Ws versus valid formulas in inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

20.8 Reduced sets of connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13520.8.1 The stroke (NAND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13520.8.2 IF ... THEN ... ELSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

20.9 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13720.9.1 Reduction to normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13720.9.2 Reduction by use of the map method (Veitch, Karnaugh) . . . . . . . . . . . . . . . . . . 138

20.10Impredicative propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13920.11Propositional formula with feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

20.11.1 Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14020.11.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

20.12Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14120.13Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14320.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

21 Quanticational variability eect 15121.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15121.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15121.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15121.4 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

22 Quantier (linguistics) 15322.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15322.2 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15322.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15422.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15422.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

23 Quantier (logic) 15523.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15523.2 Algebraic approaches to quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15523.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15623.4 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15723.5 Equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15723.6 Range of quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15823.7 Formal semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

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23.8 Paucal, multal and other degree quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16023.9 Other quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16023.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16123.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16123.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16123.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

24 Quantier rank 16324.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16324.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16324.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16424.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16424.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

25 Quantier variance 16525.1 Quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16525.2 Usage, not 'existence'? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16625.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16625.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

26 Two-variable logic 16826.1 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16826.2 Counting quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16826.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

27 Uniqueness quantication 16927.1 Proving uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16927.2 Reduction to ordinary existential and universal quantication . . . . . . . . . . . . . . . . . . . . 16927.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17027.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17027.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

28 Universal quantication 17128.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

28.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17228.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

28.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17228.2.2 Other connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17328.2.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17428.2.4 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

28.3 Universal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17428.4 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17528.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

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28.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17528.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

29 Unsatisable core 17629.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

30 Witness (mathematics) 17730.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17730.2 Henkin witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17730.3 Relation to game semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17730.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17730.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17830.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 179

30.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17930.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18430.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Chapter 1

(, )-denition of limit

x

y

c

L

Whenever a point x is within units of c, f(x) is within units of L

In calculus, the (, )-denition of limit ("epsilon-delta denition of limit) is a formalization of the notion of limit.It was rst given by Bernard Bolzano in 1817. Augustin-Louis Cauchy never gave an ( "; ) denition of limit inhis Cours d'Analyse, but occasionally used "; arguments in proofs. The denitive modern statement was ultimately

1

2 CHAPTER 1. (, )-DEFINITION OF LIMIT

provided by Karl Weierstrass.[1][2]

1.1 HistoryIsaac Newton was aware, in the context of the derivative concept, that the limit of the ratio of evanescent quantitieswas not itself a ratio, as when he wrote:

Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they canapproach so closely that their dierence is less than any given quantity...

Occasionally Newton explained limits in terms similar to the epsilon-delta denition.[3] Augustin-Louis Cauchy gavea denition of limit in terms of a more primitive notion he called a variable quantity. He never gave an epsilon-delta denition of limit (Grabiner 1981). Some of Cauchys proofs contain indications of the epsilon, delta method.Whether or not his foundational approach can be considered a harbinger of Weierstrasss is a subject of scholarlydispute. Grabiner feels that it is, while Schubring (2005) disagrees.[1] Nakane concludes that Cauchy andWeierstrassgave the same name to dierent notions of limit.[4]

1.2 Informal statementLet f be a function. To say that

limx!c f(x) = L

means that f(x) can be made as close as desired to L by making the independent variable x close enough, but notequal, to the value c.How close is close enough to c" depends on how close one wants to make f(x) to L. It also of course depends onwhich function f is and on which number c is. Therefore let the positive number (epsilon) be how close one wishesto make f(x) to L; strictly one wants the distance to be less than . Further, if the positive number is how close onewill make x to c, and if the distance from x to c is less than (but not zero), then the distance from f(x) to L will beless than . Therefore depends on . The limit statement means that no matter how small is made, can be madesmall enough.The letters and can be understood as error and distance, and in fact Cauchy used as an abbreviation forerror in some of his work.[1] In these terms, the error () in the measurement of the value at the limit can be madeas small as desired by reducing the distance () to the limit point.This denition also works for functions with more than one argument. For such functions, can be understood as theradius of a circle or a sphere or some higher-dimensional analogy centered at the point where the existence of a limitis being proven, in the domain of the function and, for which, every point inside maps to a function value less than away from the value of the function at the limit point.

1.3 Precise statementThe ("; ) denition of the limit of a function is as follows:[5]

Let f : D ! R be a function dened on a subset D R , let c be a limit point of D , and let L be a real number.Then

the function f has a limit L at c

is dened to mean

for all " > 0 , there exists a > 0 such that for all x in D that satisfy 0 < jx cj < , the inequalityjf(x) Lj < " holds.

1.4. WORKED EXAMPLE 3

Symbolically:

limx!c f(x) = L () (8" > 0)(9 > 0)(8x 2 D)(0 < jx cj < ) jf(x) Lj < ")

1.4 Worked exampleLet us prove the statement that

limx!5

(3x 3) = 12:

This is easily shown through graphical understandings of the limit, and as such serves as a strong basis for introductionto proof. According to the formal denition above, a limit statement is correct if and only if conning x to units ofc will inevitably conne f(x) to " units of L . In this specic case, this means that the statement is true if and onlyif conning x to units of 5 will inevitably conne

3x 3

to " units of 12. The overall key to showing this implication is to demonstrate how and " must be related to eachother such that the implication holds. Mathematically, we want to show that

0 < jx 5j < ) j(3x 3) 12j < ":

Simplifying, factoring, and dividing 3 on the right hand side of the implication yields

jx 5j < "/3;which immediately gives the required result if we choose

= "/3:

Thus the proof is completed. The key to the proof lies in the ability of one to choose boundaries in x , and thenconclude corresponding boundaries in f(x) , which in this case were related by a factor of 3, which is entirely due tothe slope of 3 in the line

y = 3x 3:

1.5 ContinuityA function f is said to be continuous at c if it is both dened at c and its value at c equals the limit of f as x approachesc:

limx!c f(x) = f(c):

If the condition 0 < |x c| is left out of the denition of limit, then requiring f(x) to have a limit at c would be thesame as requiring f(x) to be continuous at c.f is said to be continuous on an interval I if it is continuous at every point c of I.

4 CHAPTER 1. (, )-DEFINITION OF LIMIT

1.6 Comparison with innitesimal denitionKeisler proved that a hyperreal denition of limit reduces the quantier complexity by two quantiers.[6] Namely,f(x) converges to a limit L as x tends to a if and only if for every innitesimal e, the value f(x + e) is innitelyclose to L; see microcontinuity for a related denition of continuity, essentially due to Cauchy. Innitesimal cal-culus textbooks based on Robinson's approach provide denitions of continuity, derivative, and integral at standardpoints in terms of innitesimals. Once notions such as continuity have been thoroughly explained via the approachusing microcontinuity, the epsilon-delta approach is presented as well. Karel Hrbacek argues that the denitions ofcontinuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the - methodin order to cover also non-standard values of the input[7] Baszczyk et al. argue that microcontinuity is useful indeveloping a transparent denition of uniform continuity, and characterize the criticism by Hrbacek as a dubiouslament.[8] Hrbacek proposes an alternative non-standard analysis, which is unlike Robinsons having many levelsof innitesimals, so that limits at one level can be dened in terms of innitesimals at the next level.[9]

1.7 See also Continuous function Limit of a sequence List of calculus topics

1.8 Notes[1] Grabiner, Judith V. (March 1983), Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus (PDF),

The American Mathematical Monthly (Mathematical Association of America) 90 (3): 185194, doi:10.2307/2975545,JSTOR 2975545, archived from the original on 2009-05-03, retrieved 2009-05-01

[2] Cauchy, A.-L. (1823), Septime Leon - Valeurs de quelques expressions qui se prsentent sous les formes indtermines11 ;10; : : : Relation qui existe entre le rapport aux dirences nies et la fonction drive, Rsum des leons donnes lcole royale polytechnique sur le calcul innitsimal, Paris, archived from the original on 2009-05-03, retrieved 2009-05-01, p. 44.. Accessed 2009-05-01.

[3] Pourciau, B. (2001), Newton and the Notion of Limit, Historia Mathematica 28 (1), doi:10.1006/hmat.2000.2301

[4] Nakane, Michiyo. Did Weierstrasss dierential calculus have a limit-avoiding character? His denition of a limit in style. BSHM Bull. 29 (2014), no. 1, 5159.

[5] Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math. p. 83. ISBN 978-0070542358.

[6] Keisler, H. Jerome (2008), Quantiers in limits (PDF), Andrzej Mostowski and foundational studies, IOS, Amsterdam,pp. 151170

[7] Hrbacek, K. (2007), Stratied Analysis?", in Van Den Berg, I.; Neves, V., The Strength of Nonstandard Analysis, Springer

[8] Baszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), Ten misconceptions from the history of analysis and their debunk-ing, Foundations of Science, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8

[9] Hrbacek, K. (2009). Relative set theory: Internal view. Journal of Logic and Analysis 1.

1.9 Bibliography Grabiner, Judith V. The origins of Cauchys rigorous calculus. MIT Press, Cambridge, Mass.-London, 1981. Schubring, Gert (2005), Conicts Between Generalization, Rigor, and Intuition: Number Concepts Underlyingthe Development of Analysis in 17th19th Century France and Germany, Springer, ISBN 0-387-22836-5

Chapter 2

Boolean satisability problem

3SAT redirects here. For the Central European television network, see 3sat.

In computer science, the Boolean Satisability Problem (sometimes called Propositional Satisability Problemand abbreviated as SATISFIABILITY or SAT) is the problem of determining if there exists an interpretation thatsatises a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can beconsistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is thecase, the formula is called satisable. On the other hand, if no such assignment exists, the function expressed by theformula is identically FALSE for all possible variable assignments and the formula is unsatisable. For example, theformula "a AND NOT b" is satisable because one can nd the values a = TRUE and b = FALSE, which make (aAND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisable.SAT is one of the rst problems that was proven to be NP-complete. This means that all problems in the complexityclass NP, which includes a wide range of natural decision and optimization problems, are at most as dicult to solve asSAT. There is no known algorithm that eciently solves SAT, and it is generally believed that no such algorithm exists;yet this belief has not been proven mathematically, and resolving the question whether SAT has an ecient algorithmis equivalent to the P versus NP problem, which is the most famous open problem in the theory of computing.Despite the fact that no algorithms are known that solve SAT eciently, correctly, and for all possible input instances,many instances of SAT that occur in practice, such as in articial intelligence, circuit design and automatic theoremproving, can actually be solved rather eciently using heuristical SAT-solvers. Such algorithms are not believed to beecient on all SAT instances, but experimentally these algorithms tend to work well for many practical applications.

2.1 Basic denitions and terminologyA propositional logic formula, also calledBoolean expression, is built from variables, operators AND (conjunction,also denoted by ), OR (disjunction, ), NOT (negation, ), and parentheses. A formula is said to be satisable ifit can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables. The Booleansatisability problem (SAT) is, given a formula, to check whether it is satisable. This decision problem is ofcentral importance in various areas of computer science, including theoretical computer science, complexity theory,algorithmics, cryptography and articial intelligence.There are several special cases of the Boolean satisability problem in which the formulas are required to have aparticular structure. A literal is either a variable, then called positive literal, or the negation of a variable, thencalled negative literal. A clause is a disjunction of literals (or a single literal). A clause is called Horn clause if itcontains at most one positive literal. A formula is in conjunctive normal form (CNF) if it is a conjunction of clauses(or a single clause). For example, "x1" is a positive literal, "x2" is a negative literal, "x1 x2" is a clause, and "(x1 x2) (x1 x2 x3) x1" is a formula in conjunctive normal form, its 1st and 3rd clause are Horn clauses, butits 2nd clause is not. The formula is satisable, choosing x1=FALSE, x2=FALSE, and x3 arbitrarily, since (FALSE FALSE) (FALSE FALSE x3) FALSE evaluates to (FALSE TRUE) (TRUE FALSE x3) TRUE, and in turn to TRUE TRUE TRUE (i.e. to TRUE). In contrast, the CNF formula a a, consisting oftwo clauses of one literal, is unsatisable, since for a=TRUE and a=FALSE it evaluates to TRUE TRUE (i.e. toFALSE) and FALSE FALSE (i.e. again to FALSE), respectively.

5

6 CHAPTER 2. BOOLEAN SATISFIABILITY PROBLEM

For some versions of the SAT problem, it is useful to dene the notion of a generalized conjunctive normal formformula, viz. as a conjunction of arbitrarilymany generalized clauses, the latter being of the formR(l1,...,ln) for someboolean operator R and (ordinary) literals li. Dierent sets of allowed boolean operators lead to dierent problemversions. As an example, R(x,a,b) is a generalized clause, and R(x,a,b) R(b,y,c) R(c,d,z) is a generalizedconjunctive normal form. This formula is used below, with R being the ternary operator that is TRUE just if exactlyone of its arguments is.Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent conjunc-tive normal form, which may, however, be exponentially longer. For example, transforming the formula (x1y1) (x2y2) ... (xnyn) into conjunctive normal form yields (x1x2xn) (y1x2xn) (x1y2xn) (y1y2xn) ... (x1x2yn) (y1x2yn) (x1y2yn) (y1y2yn); while the former is adisjunction of n conjunctions of 2 variables, the latter consists of 2n clauses of n variables.

2.2 Complexity and restricted versions

2.2.1 Unrestricted satisability (SAT)Main article: CookLevin theorem

SAT was the rst known NP-complete problem, as proved by Stephen Cook at the University of Toronto in 1971[1]and independently by Leonid Levin at the National Academy of Sciences in 1973.[2] Until that time, the concept ofan NP-complete problem did not even exist. The proof shows how every decision problem in the complexity classNP can be reduced to the SAT problem for CNF[note 1] formulas, sometimes called CNFSAT. A useful property ofCooks reduction is that it preserves the number of accepting answers. For example, deciding whether a given graphhas a 3-coloring is another problem in NP; if a graph has 17 valid 3-colorings, the SAT formula produced by theCookLevin reduction will have 17 satisfying assignments.NP-completeness only refers to the run-time of the worst case instances. Many of the instances that occur in practicalapplications can be solved much more quickly. See Algorithms for solving SAT below.SAT is trivial if the formulas are restricted to those in disjunctive normal form, that is, they are disjunction ofconjunctions of literals. Such a formula is indeed satisable if and only if at least one of its conjunctions is satisable,and a conjunction is satisable if and only if it does not contain both x and NOT x for some variable x. This can bechecked in linear time. Furthermore, if they are restricted to being in full disjunctive normal form, in which everyvariable appears exactly once in every conjunction, they can be checked in constant time (each conjunction representsone satisfying assignment). But it can take exponential time and space to convert a general SAT problem to disjunctivenormal form; for an example exchange "" and "" in the above exponential blow-up example for conjunctive normalforms.

2.2.2 3-satisabilityLike the satisability problem for arbitrary formulas, determining the satisability of a formula in conjunctive nor-mal form where each clause is limited to at most three literals is NP-complete also; this problem is called 3-SAT,3CNFSAT, or 3-satisability. To reduce the unrestricted SAT problem to 3-SAT, transform each clause "l1 ... ln" to a conjunction of n 2 clauses "(l1 l2 x2) (x2 l3 x3) (x3 l4 x4) ... (xn ln xn ) (xn ln l)", where x2,...,xn are fresh variables not occurring elsewhere. Although the two formulasare not logically equivalent, they are equisatisable. The formula resulting from transforming all clauses is at most 3times as long as its original, i.e. the length growth is polynomial.[3]

3-SAT is one of Karps 21 NP-complete problems, and it is used as a starting point for proving that other problemsare also NP-hard.[note 2] This is done by polynomial-time reduction from 3-SAT to the other problem. An exampleof a problem where this method has been used is the clique problem: given a CNF formula consisting of c clauses,the corresponding graph consists of a vertex for each literal, and an edge between each two non-contradicting[note 3]literals from dierent clauses, cf. picture. The graph has a c-clique if and only if the formula is satisable.[4]

There is a simple randomized algorithm due to Schning (1999) that runs in time (4/3)n where n is the number ofclauses and succeeds with high probability to correctly decide 3-SAT.[5]

The exponential time hypothesis asserts that no algorithm can solve 3-SAT in time exp(o(n)).

2.2. COMPLEXITY AND RESTRICTED VERSIONS 7

~x ~y ~y

x

y

x

y

~x

y

The 3-SAT instance (xxy) (xyy) (xyy) reduced to a clique problem. The green vertices form a 3-clique andcorrespond to the satisfying assignment x=FALSE, y=TRUE.

Selman, Mitchell, and Levesque (1996) give empirical data on the diculty of randomly generated 3-SAT formulas,depending on their size parameters. Diculty is measured in number recursive calls made by a DPLL algorithm.[6]

3-satisability can be generalized to k-satisability (k-SAT, also k-CNF-SAT), when formulas in CNF are consid-ered with each clause containing up to k literals. However, since for any k3, this problem can neither be easier than3-SAT nor harder than SAT, and the latter two are NP-complete, so must be k-SAT.Some authors restrict k-SAT to CNF formulas with exactly k literals. This doesn't lead to a dierent complexityclass either, as each clause "l1 ... lj" with j


Left: Schaefers reduction of a 3-SAT clause xyz. The result of R is TRUE (1) if exactly one of its arguments is TRUE, and FALSE(0) otherwise. All 8 combinations of values for x,y,z are examined, one per line. The fresh variables a,...,f can be chosen to satisfyall clauses (exactly one green argument for each R) in all lines except the rst, where xyz is FALSE. Right: A simpler reductionwith the same properties.

One-in-three 3-SAT, together with its positive case, is listed asNP-complete problem LO4 in the standard reference,Computers and Intractability: A Guide to the Theory of NP-Completeness by Michael R. Garey and David S. Johnson.One-in-three 3-SAT was proved to be NP-complete by Thomas J. Schaefer as a special case of Schaefers dichotomytheorem, which asserts that any problem generalizing Boolean satisability in a certain way is either in the class P oris NP-complete.[7]

Schaefer gives a construction allowing an easy polynomial-time reduction from 3-SAT to one-in-three 3-SAT. Let "(xor y or z)" be a clause in a 3CNF formula. Add six fresh boolean variables a, b, c, d, e, and f, to be used to simulatethis clause and no other. Then the formula R(x,a,d) R(y,b,d) R(a,b,e) R(c,d,f) R(z,c,FALSE) is satisable bysome setting of the fresh variables if and only if at least one of x, y, or z is TRUE, see picture (left). Thus any 3-SATinstance with m clauses and n variables may be converted into an equisatisable one-in-three 3-SAT instance with5m clauses and n+6m variables.[8] Another reduction involves only four fresh variables and three clauses: R(x,a,b) R(b,y,c) R(c,d,z), see picture (right).

2.2.4 2-satisabilityMain article: 2-satisability

SAT is easier if the number of literals in a clause is limited to at most 2, in which case the problem is called 2-SAT.This problem can be solved in polynomial time, and in fact is complete for the complexity class NL. If additionallyall OR operations in literals are changed to XOR operations, the result is called exclusive-or 2-satisability, whichis a problem complete for the complexity class SL = L.

2.2.5 Horn-satisabilityMain article: Horn-satisability

The problem of deciding the satisability of a given conjunction of Horn clauses is called Horn-satisability, orHORN-SAT. It can be solved in polynomial time by a single step of the Unit propagation algorithm, which producesthe single minimal model of the set of Horn clauses (w.r.t. the set of literals assigned to TRUE). Horn-satisabilityis P-complete. It can be seen as Ps version of the Boolean satisability problem.Horn clauses are of interest because they are able to express implication of one variable from a set of other variables.Indeed, one such clause x1 ... xn y can be rewritten as x1 ... xn y, that is, if x1,...,xn are all TRUE,then y needs to be TRUE as well.A generalization of the class of Horn formulae is that of renamable-Horn formulae, which is the set of formulae thatcan be placed in Horn form by replacing some variables with their respective negation. For example, (x1 x2) (x1 x2 x3) x1 is not a Horn formula, but can be renamed to the Horn formula (x1 x2) (x1 x2 y3) x1 by introducing y3 as negation of x3. In contrast, no renaming of (x1 x2 x3) (x1 x2 x3) x1leads to a Horn formula. Checking the existence of such a replacement can be done in linear time; therefore, thesatisability of such formulae is in P as it can be solved by rst performing this replacement and then checking thesatisability of the resulting Horn formula.

2.3. EXTENSIONS OF SAT 9

2.2.6 XOR-satisability

Another special case is the class of problems where each clause contains XOR (i.e. exclusive or) rather than (plain)OR operators.[note 5] This is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod2, and can be solved in cubic time by Gaussian elimination;[9] see the box for an example. This recast is based onthe kinship between Boolean algebras and Boolean rings, and the fact that arithmetic modulo two forms a nite eld.Since a XOR b XOR c evaluates to TRUE if and only if exactly 1 or 3 members of {a,b,c} are TRUE, each solutionof the 1-in-3-SAT problem for a given CNF formula is also a solution of the XOR-3-SAT problem, and in turn eachsolution of XOR-3-SAT is a solution of 3-SAT, cf. picture. As a consequence, for each CNF formula, it is possibleto solve the XOR-3-SAT problem dened by the formula, and based on the result infer either that the 3-SAT problemis solvable or that the 1-in-3-SAT problem is unsolvable.Provided that the complexity classes P and NP are not equal, neither 2-, nor Horn-, nor XOR-satisability is NP-complete, unlike SAT. The assumption that P and NP are not equal is currently not proven.

2.2.7 Schaefers dichotomy theorem

Main article: Schaefers dichotomy theorem

The restrictions above (CNF, 2CNF, 3CNF, Horn, XOR-SAT) bound the considered formulae to be conjunctionsof subformulae; each restriction states a specic form for all subformulae: for example, only binary clauses can besubformulae in 2CNF.Schaefers dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these sub-formulae, the corresponding satisability problem is in P or NP-complete. The membership in P of the satisabilityof 2CNF, Horn, and XOR-SAT formulae are special cases of this theorem.

2.3 Extensions of SATAn extension that has gained signicant popularity since 2003 is Satisability modulo theories (SMT) that canenrich CNF formulas with linear constraints, arrays, all-dierent constraints, uninterpreted functions,[10] etc. Suchextensions typically remain NP-complete, but very ecient solvers are now available that can handle many such kindsof constraints.The satisability problem becomes more dicult if both for all () and there exists () quantiers are allowedto bind the Boolean variables. An example of such an expression would be "x y z (x y z) (x y z)";it is valid, since for all values of x and y, an appropriate value of z can be found, viz. z=TRUE if both x and y areFALSE, and z=FALSE else. SAT itself (tacitly) uses only quantiers. If only quantiers are allowed instead, theso-called tautology problem is obtained, which is co-NP-complete. If both quantiers are allowed, the problem iscalled the quantied Boolean formula problem (QBF), which can be shown to be PSPACE-complete. It is widelybelieved that PSPACE-complete problems are strictly harder than any problem in NP, although this has not yet beenproved.A number of variants deal with the number of variable assignments making the formula TRUE. Ordinary SATasks if there is at least one such assignment. MAJ-SAT, which asks if the majority of all assignments make theformula TRUE, is complete for PP, a probabilistic class. The problem of how many variable assignments satisfy aformula, not a decision problem, is in #P.UNIQUE-SAT is the problem of determining whether a formula has exactlyone assignment; it is complete for US, the complexity class describing problems solvable by a non-deterministicpolynomial time Turing machine that accepts when there is exactly one nondeterministic accepting path and rejectsotherwise. When the input is restricted to formulas having at most one satisfying assignment (or none), the problemis called UNAMBIGOUS-SAT. A solving algorithm for UNAMBIGOUS-SAT is allowed to exhibit any behavior,including endless looping, on a formula having several satisfying assignments. Although this problem seems easier,Valiant and Vazirani have shown[11] that if there is a practical (i.e. randomized polynomial-time) algorithm to solveit, then all problems in NP can be solved just as easily.The maximum satisability problem, an FNP generalization of SAT, asks for the maximum number of clauseswhich can be satised by any assignment. It has ecient approximation algorithms, but is NP-hard to solve exactly.Worse still, it is APX-complete, meaning there is no polynomial-time approximation scheme (PTAS) for this problem


unless P=NP.Other generalisations include satisability for rst- and second-order logic, constraint satisfaction problems, 0-1 in-teger programming.

2.4 Self-reducibilityThe SAT problem is self-reducible, that is, each algorithm which correctly answers if an instance of SAT is solvablecan be used to nd a satisfying assignment. First, the question is asked on the given formula . If the answer is no,the formula is unsatisable. Otherwise, the question is asked on the partly instantiated formula {x1=TRUE}, i.e. with the rst variable x1 replaced by TRUE, and simplied accordingly. If the answer is yes, then x1=TRUE,otherwise x1=FALSE. Values of other variables can be found subsequently in the same way. In total, n+1 runs of thealgorithm are required, where n is the number of distinct variables in .This property of self-reducibility is used in several theorems in complexity theory:

NP P/poly PH = 2 (KarpLipton theorem)NP BPP NP = RPP = NP FP = FNP

2.5 Algorithms for solving SATSince the SAT problem is NP-complete, only algorithms with exponential worst-case complexity are known for it.In spite of this state of the art in complexity theory, ecient and scalable algorithms for SAT were developed overthe last decade and have contributed to dramatic advances in our ability to automatically solve problem instancesinvolving tens of thousands of variables and millions of constraints (i.e. clauses).[12] Examples of such problemsin electronic design automation (EDA) include formal equivalence checking, model checking, formal vericationof pipelined microprocessors,[10] automatic test pattern generation, routing of FPGAs,[13] planning, and schedulingproblems, and so on. A SAT-solving engine is now considered to be an essential component in the EDA toolbox.There are two classes of high-performance algorithms for solving instances of SAT in practice: the Conict-DrivenClause Learning algorithm, which can be viewed as a modern variant of the DPLL algorithm (well known imple-mentations include Cha[14] and GRASP[15]) and stochastic local search algorithms, such as WalkSAT.A DPLL SAT solver employs a systematic backtracking search procedure to explore the (exponentially sized) spaceof variable assignments looking for satisfying assignments. The basic search procedure was proposed in two seminalpapers in the early 1960s (see references below) and is now commonly referred to as the DavisPutnamLogemannLoveland algorithm (DPLL or DLL).[16][17] Theoretically, exponential lower bounds have been proved for theDPLL family of algorithms.In contrast, randomized algorithms like the PPSZ algorithm by Paturi, Pudlak, Saks, and Zane set variables in arandom order according to some heuristics, for example bounded-width resolution. If the heuristic can't nd thecorrect setting, the variable is assigned randomly. The PPSZ algorithm has a runtime of O(20:386n) for 3-SAT witha single satisfying assignment. Currently this is the best known runtime for this problem. In the setting with manysatisfying assignments the randomized algorithm by Schning has a better bound.[5][18]

Modern SAT solvers (developed in the last ten years) come in two avors: conict-driven and look-ahead.Conict-driven solvers augment the basic DPLL search algorithm with ecient conict analysis, clause learning,non-chronological backtracking (a.k.a. backjumping), as well as two-watched-literals unit propagation, adaptivebranching, and random restarts. These extras to the basic systematic search have been empirically shown to beessential for handling the large SAT instances that arise in electronic design automation (EDA). Look-ahead solvershave especially strengthened reductions (going beyond unit-clause propagation) and the heuristics, and they are gen-erally stronger than conict-driven solvers on hard instances (while conict-driven solvers can be much better on largeinstances which actually have an easy instance inside).Modern SAT solvers are also having signicant impact on the elds of software verication, constraint solving inarticial intelligence, and operations research, among others. Powerful solvers are readily available as free and opensource software. In particular, the conict-driven MiniSAT, which was relatively successful at the 2005 SAT com-petition, only has about 600 lines of code. A modern Parallel SAT solver is ManySAT. It can achieve super linear

2.6. SEE ALSO 11

speed-ups on important classes of problems. An example for look-ahead solvers is march_dl, which won a prize atthe 2007 SAT competition.Certain types of large random satisable instances of SAT can be solved by survey propagation (SP). Particularlyin hardware design and verication applications, satisability and other logical properties of a given propositionalformula are sometimes decided based on a representation of the formula as a binary decision diagram (BDD).Almost all SAT solvers include time-outs, so they will terminate in reasonable time even if they cannot nd a solution.Dierent SAT solvers will nd dierent instances easy or hard, and some excel at proving unsatisability, and othersat nding solutions. All of these behaviors can be seen in the SAT solving contests.[19]

2.6 See also Unsatisable core Satisability Modulo Theories Counting SAT KarloZwick algorithm Circuit satisability

2.7 Notes[1] The SAT problem for arbitrary formulas is NP-complete, too, since it is easily shown to be in NP, and it cannot be easier

than SAT for CNF formulas.

[2] i.e. at least as hard as every other problem in NP. A problem is NP-complete if and only if it is in NP and is NP-hard.

[3] i.e. such that one literal isn't the negation of the other

[4] viz. all maxterms that can be built with d1,...,dk, except "d1...dk"

[5] Formally, generalized conjunctive normal forms with a ternary boolean operator R are employed, which is TRUE just if 1or 3 of its arguments is. An input clause with more than 3 literals can be transformed into an equisatisable conjunction ofclauses 3 literals similar to above; i.e. XOR-SAT can be reduced to XOR-3-SAT.

2.8 References[1] Cook, Stephen A. (1971). The Complexity of Theorem-Proving Procedures (PDF). Proceedings of the 3rd Annual ACM

Symposium on Theory of Computing: 151158. doi:10.1145/800157.805047.

[2] Levin, Leonid (1973). Universal search problems (Russian: , Universal'nye perebornyezadachi)". Problems of Information Transmission (Russian: , Problemy Peredachi In-formatsii) 9 (3): 115116. (pdf) (Russian), translated into English by Trakhtenbrot, B. A. (1984). A survey of Rus-sian approaches to perebor (brute-force searches) algorithms. Annals of the History of Computing 6 (4): 384400.doi:10.1109/MAHC.1984.10036.

[3] Alfred V. Aho, John E. Hopcroft, Jerey D. Ullman (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley.; here: Thm.10.4

[4] Aho, Hopcroft, Ullman[3] (1974); Thm.10.5

[5] Schning, Uwe (Oct 1999). A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems. Proc. 40th Ann.Symp. Foundations of Computer Science (PDF). pp. 410414. doi:10.1109/SFFCS.1999.814612.

[6] Bart Selman, David Mitchell, Hector Levesque (1996). Generating Hard Satisability Problems. Articial Intelligence81: 1729. doi:10.1016/0004-3702(95)00045-3.

[7] Schaefer, Thomas J. (1978). The complexity of satisability problems (PDF). Proceedings of the 10th Annual ACMSymposium on Theory of Computing. San Diego, California. pp. 216226.


[8] (Schaefer, 1978), p.222, Lemma 3.5

[9] Moore, Cristopher; Mertens, Stephan (2011), TheNature of Computation, OxfordUniversity Press, p. 366, ISBN9780199233212.

[10] R. E. Bryant, S. M. German, andM. N. Velev, Microprocessor Verication Using Ecient Decision Procedures for a Logicof Equality with Uninterpreted Functions, in Analytic Tableaux and Related Methods, pp. 113, 1999.

[11] Valiant, L.; Vazirani, V. (1986). NP is as easy as detecting unique solutions (PDF). Theoretical Computer Science 47:8593. doi:10.1016/0304-3975(86)90135-0.

[12] Ohrimenko, Olga; Stuckey, Peter J.; Codish, Michael (2007), Propagation = Lazy Clause Generation, Principles andPractice of Constraint Programming CP 2007, Lecture Notes in Computer Science 4741, pp. 544558, doi:10.1007/978-3-540-74970-7_39, modern SAT solvers can often handle problems with millions of constraints and hundreds of thousandsof variables.

[13] Gi-Joon Nam; Sakallah, K. A.; Rutenbar, R. A. (2002). A new FPGA detailed routing approach via search-basedBoolean satisability (PDF). IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 21 (6):674. doi:10.1109/TCAD.2002.1004311.

[14] Moskewicz, M. W.; Madigan, C. F.; Zhao, Y.; Zhang, L.; Malik, S. (2001). Cha: Engineering an Ecient SAT Solver(PDF). Proceedings of the 38th conference on Design automation (DAC). p. 530. doi:10.1145/378239.379017. ISBN1581132972.

[15] Marques-Silva, J. P.; Sakallah, K. A. (1999). GRASP: a search algorithm for propositional satisability (PDF). IEEETransactions on Computers 48 (5): 506. doi:10.1109/12.769433.

[16] Davis, M.; Putnam, H. (1960). A Computing Procedure for Quantication Theory. Journal of the ACM 7 (3): 201.doi:10.1145/321033.321034.

[17] Davis, M.; Logemann, G.; Loveland, D. (1962). A machine program for theorem-proving (PDF). Communications ofthe ACM 5 (7): 394397. doi:10.1145/368273.368557.

[18] An improved exponential-time algorithm for k-SAT, Paturi, Pudlak, Saks, Zani

[19] The international SAT Competitions web page. Retrieved 2007-11-15.

References are ordered by date of publication:

Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A9.1: LO1 LO7, pp. 259 260.

Marques-Silva, J.; Glass, T. (1999). Combinational equivalence checking using satisability and recursivelearning (PDF). Design, Automation and Test in Europe Conference and Exhibition, 1999. Proceedings (Cat.No. PR00078). p. 145. doi:10.1109/DATE.1999.761110. ISBN 0-7695-0078-1.

Clarke, E.; Biere, A.; Raimi, R.; Zhu, Y. (2001). Bounded Model Checking Using Satisability Solving.Formal Methods in System Design 19: 7. doi:10.1023/A:1011276507260.

Giunchiglia, E.; Tacchella, A. (2004). Giunchiglia, Enrico; Tacchella, Armando, eds. Theory and Appli-cations of Satisability Testing. Lecture Notes in Computer Science 2919. doi:10.1007/b95238. ISBN978-3-540-20851-8.

Babic, D.; Bingham, J.; Hu, A. J. (2006). B-Cubing: New Possibilities for Ecient SAT-Solving (PDF).IEEE Transactions on Computers 55 (11): 1315. doi:10.1109/TC.2006.175.

Rodriguez, C.; Villagra, M.; Baran, B. (2007). Asynchronous team algorithms for Boolean Satisability(PDF). 2007 2nd Bio-InspiredModels of Network, Information and Computing Systems. p. 66. doi:10.1109/BIMNICS.2007.4610083.

Carla P. Gomes, Henry Kautz, Ashish Sabharwal, Bart Selman (2008). Satisability Solvers. In Frank VanHarmelen, Vladimir Lifschitz, Bruce Porter. Handbook of knowledge representation. Foundations of ArticialIntelligence 3. Elsevier. pp. 89134. doi:10.1016/S1574-6526(07)03002-7. ISBN 978-0-444-52211-5.

2.9. EXTERNAL LINKS 13

2.9 External links

2.9.1 SAT problem formatA SAT problem is often described in the DIMACS-CNF format: an input le in which each line represents a singledisjunction. For example, a le with the two lines1 5 4 0 1 5 3 4 0represents the formula "(x1 x5 x4) (x1 x5 x3 x4)".Another common format for this formula is the 7-bit ASCII representation "(x1 | ~x5 | x4) & (~x1 | x5 | x3 | x4)".

BCSAT is a tool that converts input les in human-readable format to the DIMACS-CNF format.

2.9.2 Online SAT solvers BoolSAT Solves formulas in the DIMACS-CNF format or in a more human-friendly format: a and not b ora. Runs on a server.

minisat-in-your-browser Solves formulas in the DIMACS-CNF format. Runs on the browser. SATRennesPA - Solves formulas written in a user-friendly way. Runs on a server. Logictools - Provides dierent solvers in javascript for learning, comparison and hacking. Runs on the browser.

2.9.3 Oine SAT solvers MiniSAT DIMACS-CNF format. Lingeling won a gold medal in a 2011 SAT competition.

PicoSAT an earlier solver from the Lingeling group. Sat4j DIMACS-CNF format. Java source code available. Glucose DIMACS-CNF format. RSat won a gold medal in a 2007 SAT competition. UBCSAT. Supports unweighted and weighted clauses, both in the DIMACS-CNF format. C source codehosted on GitHub.

CryptoMiniSat won a gold medal in a 2011 SAT competition. C++ source code hosted on GitHub. Tries toput many useful features of MiniSat 2.0 core, PrecoSat ver 236, and Glucose into one package, adding manynew features

Spear Supports bit-vector arithmetic. Can use the DIMACS-CNF format or the Spear format. HyperSAT Written to experiment with B-cubing search space pruning. Won 3rd place in a 2005 SATcompetition. An earlier and slower solver from the developers of Spear.

BASolver ArgoSAT Fast SAT Solver based on genetic algorithms. zCha not supported anymore. BCSAT human-readable boolean circuit format (also converts this format to the DIMACS-CNF format andautomatically links to MiniSAT or zCha).


2.9.4 SAT applications WinSAT v2.04: A Windows-based SAT application made particularly for researchers.

2.9.5 ConferencesInternational Conference on Theory and Applications of Satisability Testing:

SAT 2013 SAT 2012 SAT 2011 SAT 2010 SAT 2009 SAT 2008 SAT 2007

2.9.6 Publications Journal on Satisability, Boolean Modeling and Computation Survey Propagation

2.9.7 Benchmarks Forced Satisable SAT Benchmarks SATLIB Software Verication Benchmarks Fadi Aloul SAT Benchmarks

SAT solving in general:

http://www.satlive.org http://www.satisfiability.org

2.9.8 Evaluation of SAT solvers Yearly evaluation of SAT solvers SAT solvers evaluation results for 2008 International SAT Competitions History

More information on SAT:

SAT and MAX-SAT for the Lay-researcher

This article includes material from a column in the ACM SIGDA e-newsletter by Prof. Karem SakallahOriginal text is available here

Chapter 3

Bounded quantier

This article is about bounded quantication in mathematical logic. For bounded quantication in type theory, seeBounded quantication.

In the study of formal theories in mathematical logic, bounded quantiers are often added to a language in additionto the standard quantiers "" and "". Bounded quantiers dier from "" and "" in that bounded quantiersrestrict the range of the quantied variable. The study of bounded quantiers is motivated by the fact that determiningwhether a sentence with only bounded quantiers is true is often not as dicult as determining whether an arbitrarysentence is true.Examples of bounded quantiers in the context of real analysis include "x>0, "y

16 CHAPTER 3. BOUNDED QUANTIFIER

polynomial. Consequently, all predicates denable by a bounded formula are Kalmr elementary, context-sensitive,and primitive recursive.In the arithmetical hierarchy, an arithmetical formula which contains only bounded quantiers is called00 ,00 , and00 . The superscript 0 is sometimes omitted.

3.2 Bounded quantiers in set theorySuppose that L is the language h2; : : : ;=i of the ZermeloFraenkel set theory, where the ellipsis may be replacedby term-forming operations such as a symbol for the powerset operation. There are two bounded quantiers: 8x 2 tand 9x 2 t . These quantiers bind the set variable x and contain a term t which may not mention x but which mayhave other free variables.The semantics of these quantiers is determined by the following rules:

9x 2 t (), 9x(x 2 t ^ )

8x 2 t (), 8x(x 2 t! )A ZF formula which contains only bounded quantiers is called 0 ,0 , and 0 . This forms the basis of the Levyhierarchy, which is dened analogously with the arithmetical hierarchy.Bounded quantiers are important in Kripke-Platek set theory and constructive set theory, where only 0 separationis included. That is, it includes separation for formulas with only bounded quantiers, but not separation for otherformulas. In KP the motivation is the fact that whether a set x satises a bounded quantier formula only depends onthe collection of sets that are close in rank to x (as the powerset operation can only be applied nitely many times toform a term). In constructive set theory, it is motivated on predicative grounds.

3.3 See also Subtyping bounded quantication in type theory System F

Chapter 4

Branching quantier

In logic a branching quantier,[1] also called a Henkin quantier, nite partially ordered quantier or evennonlinear quantier, is a partial ordering[2]

hQx1 : : : Qxniof quantiers for Q{,}. It is a special case of generalized quantier. In classical logic, quantier prexes arelinearly ordered such that the value of a variable ym bound by a quantier Qm depends on the value of the variables

y1,...,y-

bound by quantiers

Qy1,...,Qy-

preceding Qm. In a logic with (nite) partially ordered quantication this is not in general the case.Branching quantication rst appeared in a 1959 conference paper of Leon Henkin.[3] Systems of partially orderedquantication are intermediate in strength between rst-order logic and second-order logic. They are being used as abasis for Hintikkas and Gabriel Sandus independence-friendly logic.

4.1 Denition and propertiesThe simplest Henkin quantier QH is

(QHx1; x2; y1; y2)(x1; x2; y1; y2) 8x19y18x29y2

(x1; x2; y1; y2)

It (in fact every formula with a Henkin prex, not just the simplest one) is equivalent to its second-order Skolemization,i.e.

9f9g8x18x2(x1; x2; f(x1); g(x2))It is also powerful enough to dene the quantier QN (i.e. there are innitely many) dened as

(QNx)(x) 9a(QHx1; x2; y1; y2)[a ^ (x1 = x2 $ y1 = y2) ^ ((x1)! ((y1) ^ y1 6= a))]Several things follow from this, including the nonaxiomatizability of rst-order logic with QH (rst observed byEhrenfeucht), and its equivalence to the 11 -fragment of second-order logic (existential second-order logic)thelatter result published independently in 1970 by Herbert Enderton[4] and W. Walkoe.[5]

The following quantiers are also denable by QH .[2]

17

18 CHAPTER 4. BRANCHING QUANTIFIER

Rescher: The number of s is less than or equal to the number of s

(QLx)(x; x) Card(fx : xg) Card(fx : xg) (QHx1x2y1y2)[(x1 = x2 $ y1 = y2)^(x1 ! y1)]

Hrtig: The s are equinumerous with the s

(QIx)(x; x) (QLx)(x; x) ^ (QLx)( x; x)

Chang: The number of s is equinumerous with the domain of the model

(QCx)(x) (QLx)(x = x; x)The Henkin quantier QH can itself be expressed as a type (4) Lindstrm quantier.[2]

4.2 Relation to natural languagesHintikka in a 1973 paper[6] advanced the hypothesis that some sentences in natural languages are best understood interms of branching quantiers, for example: some relative of each villager and some relative of each townsman hateeach other is supposed to be interpreted, according to Hintikka, as:[7][8]

8x19y18x29y2

[(V (x1) ^ T (x2))! (R(x1; y1) ^R(x2; y2) ^H(y1; y2) ^H(y2; y1))]

which is known to have no rst-order logic equivalent.[7]

The idea of branching is not necessarily restricted to using the classical quantiers as leaves. In a 1979 paper,[9]Jon Barwise proposed variations of Hintikka sentences (as the above is sometimes called) in which the inner quan-tiers are themselves generalized quantiers, for example: Most villagers and most townsmen hate each other.[7]Observing that 11 is not closed under negation, Barwise also proposed a practical test to determine whether natu-ral language sentences really involve branching quantiers, namely to test whether their natural-language negationinvolves universal quantication over a set variable (a 11 sentence).[10]

Hintikkas proposal was met with skepticism by a number of logicians because some rst-order sentences like the onebelow appear to capture well enough the natural language Hintikka sentence.

[8x19y18x29y2 (x1; x2; y1; y2)] ^ [8x29y28x19y1 (x1; x2; y1; y2)] where(x1; x2; y1; y2) denotes (V (x1) ^ T (x2))! (R(x1; y1) ^R(x2; y2) ^H(y1; y2) ^H(y2; y1))

Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trainedin logic found that they are more likely to assign models matching the bidirectional rst-order sentence rather thanbranching-quantier sentence to several natural-language constructs derived from the Hintikka sentence. For instancestudents were shown undirected bipartite graphswith squares and circles as verticesand asked to say whethersentences like more than 3 circles and more than 3 squares are connected by lines were correctly describing thediagrams.[7]

4.3 See also Game semantics Dependence logic Independence-friendly logic (IF logic) Mostowski quantier Lindstrm quantier Nonrstorderizability

4.4. REFERENCES 19

4.4 References[1] Stanley Peters; Dag Westersthl (2006). Quantiers in language and logic. Clarendon Press. pp. 6672. ISBN 978-0-19-

929125-0.

[2] Antonio Badia (2009). Quantiers in Action: Generalized Quantication in Query, Logical andNatural Languages. Springer.p. 7476. ISBN 978-0-387-09563-9.

[3] Henkin, L. Some Remarks on Innitely Long Formulas. Innitistic Methods: Proceedings of the Symposium on Founda-tions of Mathematics, Warsaw, 29 September 1959, Panstwowe Wydawnictwo Naukowe and Pergamon Press, Warsaw,1961, pp. 167-183. OCLC 2277863

[4] Jaakko Hintikka and Gabriel Sandu, Game-theoretical semantics, inHandbook of logic and language, ed. J. van Benthemand A. ter Meulen, Elsevier 2011 (2nd ed.) citing Enderton, H.B., 1970. Finite partially-ordered quantiers. Z. Math.Logik Grundlag. Math. 16, 393397 doi:10.1002/malq.19700160802.

[5] Blass, A.; Gurevich, Y. (1986). Henkin quantiers and complete problems. Annals of Pure and Applied Logic 32:1. doi:10.1016/0168-0072(86)90040-0. citing W. Walkoe, Finite partially-ordered quantication, J. Symbolic Logic 35(1970) 535-555. JSTOR 2271440

[6] Hintikka, J. (1973). Quantiers vs. Quantication Theory. Dialectica 27 (34): 329358. doi:10.1111/j.1746-8361.1973.tb00624.x.

[7] Gierasimczuk, N.; Szymanik, J. (2009). Branching Quantication v. Two-way Quantication. Journal of Semantics 26(4): 367. doi:10.1093/jos/p008.

[8] Sher, G. (1990). Ways of branching quantifers. Linguistics and Philosophy 13 (4): 393422. doi:10.1007/BF00630749.

[9] Barwise, J. (1979). On branching quantiers in English. Journal of Philosophical Logic 8: 4780. doi:10.1007/BF00258419.

[10] Hand, Michael (1998). The Journal of Symbolic Logic 63 (4). Association for Symbolic Logic. pp. 16111614. JSTOR2586678.

4.5 External links Game-theoretical quantier at PlanetMath.

Chapter 5

Conditional quantier

In logic, a conditional quantier is a kind of Lindstrm quantier (or generalized quantier) QA that, relative to aclassical model A, satises some or all of the following conditions ("X" and "Y" range over arbitrary formulas in onefree variable):(The implication arrow denotes material implication in the metalanguage.) The minimal conditional logicM is char-acterized by the rst six properties, and stronger conditional logics include some of the other ones. For example, thequantier A, which can be viewed as set-theoretic inclusion, satises all of the above except [symmetry]. Clearly[symmetry] holds for A while e.g. [contraposition] fails.A semantic interpretation of conditional quantiers involves a relation between sets of subsets of a given structurei.e. a relation between properties dened on the structure. Some of the details can be found in the article Lindstrmquantier.Conditional quantiers are meant to capture certain properties concerning conditional reasoning at an abstract level.Generally, it is intended to clarify the role of conditionals in a rst-order language as they relate to other connectives,such as conjunction or disjunction. While they can cover nested conditionals, the greater complexity of the formula,specically the greater the number of conditional nesting, the less helpful they are as a methodological tool for un-derstanding conditionals, at least in some sense. Compare this methodological strategy for conditionals with that ofrst-degree entailment logics.

5.1 ReferencesSerge Lapierre. Conditionals and Quantiers, in Quantiers, Logic, and Language, Stanford University, pp. 237253, 1995.

20

Chapter 6

Conjunctive normal form

In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction ofclauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs. As a normal form, it is usefulin automated theorem proving. It is similar to the product of sums form used in circuit theory.All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one-literal clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form (DNF), the onlypropositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used aspart of a literal, which means that it can only precede a propositional variable or a predicate symbol.In automated theorem proving, the notion "clausal normal form" is often used in a narrower sense, meaning a par-ticular representation of a CNF formula as a set of sets of literals.

6.1 Examples and Non-ExamplesAll of the following formulas in the variables A, B, C, D, and E are in conjunctive normal form:

:A ^ (B _ C) (A _B) ^ (:B _ C _ :D) ^ (D _ :E) A _B A ^B

The

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