Mathematical Logic Part Three
|
|
|
- Clifton Butler
- 5 years ago
- Views:
Transcription
1 Mathematical Logic Part hree
2 riday our Square! oday at 4:15PM, Outside Gates
3 Announcements Problem Set 3 due right now. Problem Set 4 goes out today. Checkpoint due Monday, October 22. Remainder due riday, October 26. Play around with propositional and firstorder logic!
4 What is irst-order Logic? irst-order logic is a logical system for reasoning about properties of objects. Augments the logical connectives from propositional logic with predicates that describe properties of objects, and functions that map objects to one another, quantifiers that allow us to reason about multiple objects simultaneously.
5 Quantifiers A statement of the form x. ψ asserts that for every choice of x in our domain, ψ is true. A statement of the form x. ψ asserts that for some choice of x in our domain, ψ is true. he syntax x S. φ x S. φ is allowed for quantifying over sets.
6 ranslating into irst-order Logic irst-order logic has great expressive power and is often used to formally encode mathematical definitions. Let's go provide rigorous definitions for the terms we've been using so far.
7 Set heory he union of two sets is the set containing all elements of both sets. S.. (Set(S) Set() x. (x S x S x ))
8 Set heory he intersection of two sets is the set containing all elements of both sets. S.. (Set(S) Set() x. (x S x S x ))
9 Relations R is a reflexive relation over A. a A. ara
10 Relations R is a symmetric relation over A. a A. b A. (arb bra)
11 Relations R is an antisymmetric relation over A. a A. b A. (arb bra a = b)
12 Relations R is a transitive relation over A. a A. b A. c A. (arb brc arc)
13 Negating Quantifiers We spent much of last lecture discussing how to negate propositional constructs. How do we negate quantifiers?
14 An Extremely Important able x. P(x) x. P(x) x. P(x) x. P(x) When is this true? or any choice of x, P(x) or some choice of x, P(x) or any choice of x, P(x) or some choice of x, P(x) When is this false? or some choice of x, P(x) or any choice of x, P(x) or some choice of x, P(x) or any choice of x, P(x)
15 An Extremely Important able x. P(x) x. P(x) x. P(x) x. P(x) When is this true? or any choice of x, P(x) or some choice of x, P(x) or any choice of x, P(x) or some choice of x, P(x) When is this false? x. P(x) x. P(x) x. P(x) x. P(x)
16 Negating irst-order Statements Use the equivalences to negate quantifiers. Mechanically: x. φ x. φ x. φ x. φ Push the negation across the quantifier. Change the quantifier from to or vice-versa. Use techniques from propositional logic to negate connectives.
17 Negating Quantifiers What is the negation of the following statement? x. y. (P(x) Q(y)) We can obtain it as follows: x. y. (P(x) Q(y)) x. y. (P(x) Q(y)) x. y. (P(x) Q(y)) x. y. (P(x) Q(y))
18 Analyzing Relations R is a binary relation over set A that is not reflexive a A. ara a A. ara Some a A is not related to itself by R.
19 Analyzing Relations R is a binary relation over A that is not antisymmetric x A. y A. (xry yrx x = y) x A. y A. (xry yrx x = y) x A. y A. (xry yrx x = y) x A. y A. (xry yrx (x = y)) x A. y A. (xry yrx x y) Some x A and y A are related to one another by R, but are not equal
20 A Useful Equivalence he following equivalences are useful when negating statements in first-order logic: (p q) p q (p q) p q hese identities are useful when negating statements involving quantifiers. is used in existentially-quantified statements. is used in universally-quantified statements.
21 Negating Quantifiers What is the negation of the following statement? x. y. (P(x) Q(y)) We can obtain it as follows: x. y. (P(x) Q(y)) x. y. (P(x) Q(y)) x. y. (P(x) Q(y)) x. y. (P(x) Q(y))
22 Using the predicates - ourn(), which states that is a tournament, - p, which states that p is a player in tournament, and - Beats(p 1, p 2 ), which states that p 1 beat p 2, Write a sentence in first-order logic that means Every tournament has a tournament winner.
23 Every tournament has a tournament winner.. (ourn() ( w. ( p. (p w (Beats(w, p) ( p'. (Beats(w, p') Beats(p', p)) ) ) ) ) ) ) here is some tournament graph where for each player w, there is some other player p who w didn't beat and for each player p' if that w beat p', then p' did not beat p.
24 Every tournament has a tournament winner.. (ourn() ( w. ( p. (p w (Beats(w, p) ( p'. (Beats(w, p') Beats(p', p)) ) ) ) ) ) ) here is some tournament graph where for each player w, there is some other player p who w didn't beat and for each player p' if that w beat p', then p' did not beat p.
25 Every tournament has a tournament winner.. (ourn() ( w. ( p. (p w ( Beats(w, p) ( p'. (Beats(w, p') Beats(p', p)) ) ) ) ) ) ) here is some tournament graph where for each player w, there is some other player p who w didn't beat and for each player p' if that w beat p', then p' did not beat p.
26 Every tournament has a tournament winner.. (ourn() ( w. ( p. (p w ( Beats(w, p) ( p'. (Beats(w, p') Beats(p', p)) ) ) ) ) ) ) here is some tournament where for each player w, there is some other player p who w didn't beat and for each player p' if w beat p', then p' did not beat p.
27 Uniqueness
28 Uniqueness Often, statements have the form there is a unique x such that Some sources use a uniqueness quantifier to express this:!n. P(n) However, it's possible to encode uniqueness using just the two quantifiers we've seen.!n. P(n) n. (P(n) m. (P(m) m = n)) In CS103, do not use the! quantifier. Just use and.
29 Summary of irst-order Logic Predicates allow us to reason about different properties of the same object. unctions allow us to transform objects into one another. Quantifiers allow us to reason about properties of some or all objects. here are many useful identities for negating first-order formulae.
30 SA Solving
31 P(x) Back to Propositional Logic...
32 Is his ormula Ever rue? p p
33 Is his ormula Ever rue? p p
34 Is his ormula Ever rue? (r s t) (s t r) (t r s) t s
35 Is his ormula Ever rue? (x 0 (x 1 x 0 )) (x 2 x 1 x 0 ) (x 1 x 1 )
36 Satisfiability A propositional logic formula φ is called satisfiable if there is some assignment to its variables that makes it evaluate to true. An assignment of true and false to the variables of φ that makes it evaluate to true is called a satisfying assignment. Similar terms: φ is tautological if every variable assignment is a satisfying assignment. φ is satisfiable if some variable assignment is a satisfying assignment. φ is unsatisfiable if no variable assignment is a satisfying assignment.
37 SA he boolean satisfiability problem (SA) is the following: Given a propositional logic formula φ, is φ satisfiable? Note: Goal is just to get a yes/no answer, not to actually find a satisfying assignment. It's possible to extract a satisfying assignment if you know one exists; you're asked to prove this in the problem set. Extremely important problem both in theory and in practice.
38 Applications of SA
39
40
41
42 Solving SA: ake One
43 A Simple Algorithm Given a formula φ, we can just build a truth table for φ and check all of the rows. If any of them evaluate to true, then φ is satisfiable. If none of them evaluate to true, then φ is unsatisfiable. So what might this look like?
44 he ruth able Algorithm p p p
45 he ruth able Algorithm p q (p q) q
46 q (r q) r p q p r
47 A Large Problem ruth tables can get very big very quickly! With n variables, there will be 2 n rows. Many real-word SA instances have hundreds of thousands of variables; this is completely infeasible!
48 Clause-Based Algorithms
49 Simplifying Our ormulas Arbitrary formulas in propositional logic can be complex. Lots of different connectives. Arbitrary nesting of formulas. Can be difficult to see how they all interrelate. Goal: Convert formulas into a simpler format.
50 Literals and Clauses A literal in propositional logic is a variable or its negation: x y But not x y. A clause is a many-way OR (disjunction) of literals. x y z x But not x (y z)
51 Conjunctive Normal orm A propositional logic formula φ is in conjunctive normal form (CN) if it is the many-way AND (conjunction) of clauses. (x y z) ( x y) (x y z w) x z But not (x (y z)) (x y) Only legal operators are,,. No nesting allowed.
52 he Structure of CN ( x y z ) ( x y z ) ( x y z ) Each clause must have at least one true literal in it.
53 he Structure of CN ( x y z ) ( x y z ) ( x y z ) We should pick at least one true literal from each clause
54 he Structure of CN ( x y z ) ( x y z ) ( x y z ) subject to the constraint that we never choose a literal and its negation
55 Getting to CN here are excellent algorithms for solving SA formulas in CN. How do we convert an arbitrary propositional logic formula into a formula in CN? Outline: urn any formula into a mezzanine format called NN. Convert NN formulas into CN formulas. Details at the end of these slides.
56 SA for CN ormulas: A Simple Backtracking Algorithm
57 Backtracking SA Solving If φ is an empty set of clauses, return true. All clauses are satisfied. If φ contains ( ), return false. Some clause is unsatisfiable. Otherwise: Choose some variable x. Return whether φ x is satisfiable or φ x is satisfiable, where φ x is φ with x set to false and φ x is φ with x set to true.
58 Analyzing Backtracking he backtracking solver works reasonably well on most inputs. Low memory usage just need to remember one potential path along the tree. or formulas with many satisfying assignments, typically finds one very quickly. But it has its weaknesses. Completely blind searching might miss obvious choices. In the worst-case, must explore the entire tree, which has 2 n leaves for n variables.
59 Adding Heuristics A heuristic is an approach to solving a problem that may or may not work correctly. Contrast with an algorithm, which has definitive guarantees on its behavior. he simplicity of CN makes it possible to add heuristics to our backtracking solver. What sorts of heuristics might we add?
60 Pure Literal Elimination A literal is called pure if its negation appears nowhere in the formula. Setting that literal to true will satisfy some number of clauses automatically and simplify the formula. Many formulas can be satisfied by iteratively applying pure literal elimination.
61 Unit Propagation A unit clause is a clause containing just one literal. or the formula to be true, that literal must be set to true. his might expose other unit clauses.
62 DPLL he DPLL algorithm is a modification of the simple backtracking search algorithm. Named for Davis, Putnam, Logemann, and Loveland, its inventors. Incorporates the two heuristics we just saw.
63 DPLL Simplify φ with unit propagation. Simplify φ with pure literal elimination. If φ is empty, return true. If φ contains ( ), return false. Otherwise: Choose some variable x. Return whether φ x is satisfiable or φ x is satisfiable, where φ x is φ with x set to false and φ x is φ with x set to true.
64 DPLL is Powerful DPLL was invented 50 years and three months ago, but is still the basis for most SA solvers. he two heuristics aggressively simplify many common cases. However, still has an exponential worstcase runtime.
65 How hard is SA? We'll see more on this later on...
66 An Important Milestone
67 Recap: Discrete Mathematics he past four weeks have focused exclusively on discrete mathematics: Induction Graphs Relations unctions he Pigeonhole Principle Logic Set heory Cardinality hese are the building blocks we will use throughout the rest of the quarter. hese are the building blocks you will use throughout the rest of your CS career.
68 Next Up: Computability heory It's time to switch gears and address the limits of what can be computed. We'll explore What is the formal definition of a computer? What might computers look like with various resource constraints? What problems can be solved by computers? What problems can't be solved by computers? Get ready to explore the boundaries of what computers could ever be made to do.
69 Next ime ormal Languages What is the mathematical definition of a problem? inite Automata What does a mathematical model of a computer look like?
70 Appendix: Getting to CN
71 Negation Normal orm A formula φ in propositional logic is in negation normal form (NN) iff he only connectives are,, and. is only applied directly to variables. Examples: (a (b c)) (d a) a b c Non-Examples: (a b) a ( b c) a b c
72 Getting to NN NN is a stepping stone toward CN: Only have,, and. All negations pushed onto variables. Build an algorithm to get from arbitrary propositional logic down to NN. Our conversion process will work as follows: Eliminate complex connectives. Simplify negations.
73 Eliminating Complex Connectives NN only allows,,. irst step: Replace other connectives with these three. Replace φ ψ with ( φ ψ). Can also replace,, and with formulas just using,, and ; you'll do this in the problem set. Result: A new formula that is logically equivalent to the original and not much bigger.
74 Eliminating Complex Connectives (p q r) (s t q) (p q r) (s t q) (p q r) ( s t q)
75 Simplifying Negations NN only allows negations in front of variables. Now that we have just,, and, repeatedly apply these rules to achieve this result: Replace φ with φ Replace (φ ψ) with φ ψ Replace (φ ψ) with φ ψ his process eventually terminates; the height of the negations keeps decreasing.
76 Simplifying Negations (p q r) ( s t q) (p q r) ( s t q) (p q r) ( s t q) (p q r) (s t q)
77 rom NN to CN Now that we can get to NN, let's get down to CN. Recall: CN is the conjunction of clauses: (x 1 x 2 x 3 ) (x 1 x 4 ) We'll use an inductive approach to convert NN into CN.
78 rom NN to CN Every NN formula is either A literal, he conjunction of two NN formulas: φ ψ he disjunction of two NN formulas: φ ψ Let's work through some examples and see if we can find a pattern.
79 Examples: NN to CN (x y z w) ( x y)
80 Examples: NN to CN x (y z) (x y) (x z)
81 y x (y z ) x z (x y) ( x z)
82 Examples: NN to CN (w x) (y z) ((w x) y) ((w x) z) (w y) (x y) (w z) (x z)
83 Examples: NN to CN (v w x) (y z) ((v w x) y) ((v w x) z) ((v w) y) (x y) ((v w) z) (x z) (v y) (w y) (x y) (v z) (w z) (y z)
84 Converting NN to CN Apply the following reasoning inductively: If the formula is a literal, do nothing. If the formula is φ ψ: Convert φ and ψ to CN, call it φ' and ψ'. Yield φ' ψ' If the formula is φ ψ: Convert φ and ψ to CN, call it φ' and ψ'. Repeatedly apply the distributive law x (y z) (x y) (x z) to φ' ψ' until simplified.
85 A Problem (a b) (c d) (e f) (g h) ((a c) (a d) (b c) (b d)) (e f) (g h) ((a c e) (a c f) (a d e) (a d f) ) ((b c e) (b c f) (b d e) (b d f)) (g h) (a c e g) (a c f g) (a d e g) (a d f g) (b c e g) (b c f g) (b d e g) (b d f g) (a c e h) (a c f h) (a d e h) (a d f h) (b c e h) (b c f h) (b d e h) (b d f h)
86 Exponential Blowup Our logic for eliminating can lead to exponential size increases. Not a problem with the algorithm; some formulas produce exponentially large CN formulas. We will need to find another approach.
87 Equivalence and Equisatisfiability Recall: wo logical formulas φ and ψ are are equivalent (denoted φ ψ) if they always take on the same truth values. wo logical formulas φ and ψ are equisatisfiable (denoted φ ψ) if φ is satisfiable iff ψ is satisfiable. Either φ and ψ are satisfiable, or φ and ψ are unsatisfiable. o solve SA for a formula φ, we can instead solve SA for an equisatisfiable ψ.
88 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b )
89 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b )
90 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a b c )
91 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a ) ( b ) ( c )
92 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q )
93 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a c )
94 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a ) ( c )
95 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
96 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
97 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
98 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
99 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
100 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
101 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
102 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
103 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
104 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
105 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
106 Equisatisfiably from NN to CN ( a b c ) ( a c ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
107 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
108 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b )
109 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b )
110 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) )
111 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q ) ( b q ) ( c q ) ( a q ) ( c q )
112 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r)
113 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) a b
114 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a ) ( b )
115 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
116 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
117 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
118 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
119 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
120 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
121 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
122 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
123 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
124 Equisatisfiably from NN to CN ( ( a q ) ( b q ) ( c q ) ( a q ) ( c q ) ) ( a b ) ( a q r) ( b q r ) ( c q r) ( a q r) ( c q r) ( a r ) ( b r )
125 Equisatisfiably from NN to CN If the formula is a literal, do nothing. If the formula is φ ψ: Convert φ and ψ to CN, call it φ' and ψ'. Yield φ' ψ' If the formula is φ ψ: Convert φ and ψ to CN, call it φ' and ψ'. Create a new variable q. Add q to each clause of φ'. Add q to each clause of ψ'. Yield φ' ψ'. Adds at most n new variables to each clause, where n is the number of clauses. Size increase at worst quadratic.
Announcements. Office Hours Swap: OH schedule has been updated to reflect this.
SA Solving Announements Offie Hours Swap: Zavain has offie hours from 4-6PM toay in builing 460, room 040A. Rose has offie hours tonight from 7-9PM in Gates B26B. Keith has offie hours hursay from 2-4PM
Outline for Today. Binary Relations. Equivalence Relations. Partial Orders. Quick Midterm Review
Binary Relations Outline for Today Binary Relations Studying connections between objects from a different perspective. Equivalence Relations Relations that break objects into groups. Partial Orders Relations
Mathematical Logic Part Three
Mathematical Logic Part Three Recap from Last Time What is First-Order Logic? First-order logic is a logical system for reasoning about properties of objects. Augments the logical connectives from propositional
Mathematical Logic Part One
Mathematical Logic Part One An Important Question How do we formalize the logic we've been using in our proofs? Where We're Going Propositional Logic (oday) Basic logical connectives. ruth tables. Logical
COMP219: Artificial Intelligence. Lecture 20: Propositional Reasoning
COMP219: Artificial Intelligence Lecture 20: Propositional Reasoning 1 Overview Last time Logic for KR in general; Propositional Logic; Natural Deduction Today Entailment, satisfiability and validity Normal
Formal Verification Methods 1: Propositional Logic
Formal Verification Methods 1: Propositional Logic John Harrison Intel Corporation Course overview Propositional logic A resurgence of interest Logic and circuits Normal forms The Davis-Putnam procedure
Tutorial 1: Modern SMT Solvers and Verification
University of Illinois at Urbana-Champaign Tutorial 1: Modern SMT Solvers and Verification Sayan Mitra Electrical & Computer Engineering Coordinated Science Laboratory University of Illinois at Urbana
Propositional Logic. Methods & Tools for Software Engineering (MTSE) Fall Prof. Arie Gurfinkel
Propositional Logic Methods & Tools for Software Engineering (MTSE) Fall 2017 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
A brief introduction to Logic. (slides from
A brief introduction to Logic (slides from http://www.decision-procedures.org/) 1 A Brief Introduction to Logic - Outline Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity
SAT Solvers: Theory and Practice
Summer School on Verification Technology, Systems & Applications, September 17, 2008 p. 1/98 SAT Solvers: Theory and Practice Clark Barrett barrett@cs.nyu.edu New York University Summer School on Verification
Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask
Set 6: Knowledge Representation: The Propositional Calculus Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask Outline Representing knowledge using logic Agent that reason logically A knowledge based agent Representing
An Introduction to SAT Solving
An Introduction to SAT Solving Applied Logic for Computer Science UWO December 3, 2017 Applied Logic for Computer Science An Introduction to SAT Solving UWO December 3, 2017 1 / 46 Plan 1 The Boolean satisfiability
Lecture Notes on SAT Solvers & DPLL
15-414: Bug Catching: Automated Program Verification Lecture Notes on SAT Solvers & DPLL Matt Fredrikson André Platzer Carnegie Mellon University Lecture 10 1 Introduction In this lecture we will switch
Lecture 9: The Splitting Method for SAT
Lecture 9: The Splitting Method for SAT 1 Importance of SAT Cook-Levin Theorem: SAT is NP-complete. The reason why SAT is an important problem can be summarized as below: 1. A natural NP-Complete problem.
Mathematical Logic Part One
Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going Propositional Logic (oday) Basic logical connectives. ruth tables. Logical
Solvers for the Problem of Boolean Satisfiability (SAT) Will Klieber Aug 31, 2011
Solvers for the Problem of Boolean Satisfiability (SAT) Will Klieber 15-414 Aug 31, 2011 Why study SAT solvers? Many problems reduce to SAT. Formal verification CAD, VLSI Optimization AI, planning, automated
Chapter 7 Propositional Satisfiability Techniques
Lecture slides for Automated Planning: Theory and Practice Chapter 7 Propositional Satisfiability Techniques Dana S. Nau University of Maryland 12:58 PM February 15, 2012 1 Motivation Propositional satisfiability:
Lecture 2 Propositional Logic & SAT
CS 5110/6110 Rigorous System Design Spring 2017 Jan-17 Lecture 2 Propositional Logic & SAT Zvonimir Rakamarić University of Utah Announcements Homework 1 will be posted soon Propositional logic: Chapter
CS156: The Calculus of Computation
CS156: The Calculus of Computation Zohar Manna Winter 2010 It is reasonable to hope that the relationship between computation and mathematical logic will be as fruitful in the next century as that between
Chapter 7 Propositional Satisfiability Techniques
Lecture slides for Automated Planning: Theory and Practice Chapter 7 Propositional Satisfiability Techniques Dana S. Nau CMSC 722, AI Planning University of Maryland, Spring 2008 1 Motivation Propositional
EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS
EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS Lecture 10, 5/9/2005 University of Washington, Department of Electrical Engineering Spring 2005 Instructor: Professor Jeff A. Bilmes Logical Agents Chapter 7
Mathematical Logic Part One
Mathematical Logic Part One Announcements Problem Set 3 checkpoint due right now. Problem Set 2 due now with a late day. Solutions distributed at end of lecture. One inal Note on the Pigeonhole Principle
Satisfiability Modulo Theories
Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT
Advanced Topics in LP and FP
Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection
Logical Agents. Chapter 7
Logical Agents Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem
Announcements. Friday Four Square! Problem Set 8 due right now. Problem Set 9 out, due next Friday at 2:15PM. Did you lose a phone in my office?
N P NP Completeness Announcements Friday Four Square! Today at 4:15PM, outside Gates. Problem Set 8 due right now. Problem Set 9 out, due next Friday at 2:15PM. Explore P, NP, and their connection. Did
Overview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR Last time Expert Systems and Ontologies oday Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof theory Natural
Inf2D 06: Logical Agents: Knowledge Bases and the Wumpus World
Inf2D 06: Logical Agents: Knowledge Bases and the Wumpus World School of Informatics, University of Edinburgh 26/01/18 Slide Credits: Jacques Fleuriot, Michael Rovatsos, Michael Herrmann Outline Knowledge-based
LOGIC PROPOSITIONAL REASONING
LOGIC PROPOSITIONAL REASONING WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität Linz Version 2018.1
Propositional Logic: Evaluating the Formulas
Institute for Formal Models and Verification Johannes Kepler University Linz VL Logik (LVA-Nr. 342208) Winter Semester 2015/2016 Propositional Logic: Evaluating the Formulas Version 2015.2 Armin Biere
Planning as Satisfiability
Planning as Satisfiability Alan Fern * Review of propositional logic (see chapter 7) Planning as propositional satisfiability Satisfiability techniques (see chapter 7) Combining satisfiability techniques
Proof Methods for Propositional Logic
Proof Methods for Propositional Logic Logical equivalence Two sentences are logically equivalent iff they are true in the same models: α ß iff α β and β α Russell and Norvig Chapter 7 CS440 Fall 2015 1
The Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
Intelligent Agents. Pınar Yolum Utrecht University
Intelligent Agents Pınar Yolum p.yolum@uu.nl Utrecht University Logical Agents (Based mostly on the course slides from http://aima.cs.berkeley.edu/) Outline Knowledge-based agents Wumpus world Logic in
NP-Completeness Part II
NP-Completeness Part II Please evaluate this course on Axess. Your comments really do make a difference. Announcements Problem Set 8 due tomorrow at 12:50PM sharp with one late day. Problem Set 9 out,
Propositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) This lecture topic: Propositional Logic (two lectures) Chapter 7.1-7.4 (previous lecture, Part I) Chapter 7.5 (this lecture, Part II) (optional: 7.6-7.8)
More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences. Computability and Logic
More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences Computability and Logic Equivalences Involving Conditionals Some Important Equivalences Involving Conditionals
Computational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
Propositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short
NP-Completeness Part II
NP-Completeness Part II Recap from Last Time NP-Hardness A language L is called NP-hard iff for every L' NP, we have L' P L. A language in L is called NP-complete iff L is NP-hard and L NP. The class NPC
First Order Logic: Syntax and Semantics
irst Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Logic Recap You should already know the basics of irst Order Logic (OL) It s a prerequisite of this course!
COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning
COMP9414, Monday 26 March, 2012 Propositional Logic 2 COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning Overview Proof systems (including soundness and completeness) Normal Forms
Deliberative Agents Knowledge Representation I. Deliberative Agents
Deliberative Agents Knowledge Representation I Vasant Honavar Bioinformatics and Computational Biology Program Center for Computational Intelligence, Learning, & Discovery honavar@cs.iastate.edu www.cs.iastate.edu/~honavar/
Logical Agents. Chapter 7
Logical Agents Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem
15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
Price: $25 (incl. T-Shirt, morning tea and lunch) Visit:
Three days of interesting talks & workshops from industry experts across Australia Explore new computing topics Network with students & employers in Brisbane Price: $25 (incl. T-Shirt, morning tea and
Mathematical Logic Part One
Mathematical Logic Part One Announcements Problem Session tonight from 7:00 7:50 in 380-380X. Optional, but highly recommended! Problem Set 3 Checkpoint due right now. 2 Handouts Problem Set 3 Checkpoint
Lecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology
Recap from Last Time
NP-Completeness Recap from Last Time Analyzing NTMs When discussing deterministic TMs, the notion of time complexity is (reasonably) straightforward. Recall: One way of thinking about nondeterminism is
ECE473 Lecture 15: Propositional Logic
ECE473 Lecture 15: Propositional Logic Jeffrey Mark Siskind School of Electrical and Computer Engineering Spring 2018 Siskind (Purdue ECE) ECE473 Lecture 15: Propositional Logic Spring 2018 1 / 23 What
Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
CS156: The Calculus of Computation Zohar Manna Autumn 2008
Page 3 of 52 Page 4 of 52 CS156: The Calculus of Computation Zohar Manna Autumn 2008 Lecturer: Zohar Manna (manna@cs.stanford.edu) Office Hours: MW 12:30-1:00 at Gates 481 TAs: Boyu Wang (wangboyu@stanford.edu)
Propositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
Logical Agents. Outline
Logical Agents *(Chapter 7 (Russel & Norvig, 2004)) Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability
WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008
WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008 WHAT I LL TALK ABOUT Propositional Logic Terminology, Satisfiability, Decision Procedure First-Order Logic Terminology, Background Theories Satisfiability
Introduction to Artificial Intelligence. Logical Agents
Introduction to Artificial Intelligence Logical Agents (Logic, Deduction, Knowledge Representation) Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU Winter Term 2004/2005 B. Beckert: KI für IM p.1 Outline Knowledge-based
Propositional Logic: Methods of Proof. Chapter 7, Part II
Propositional Logic: Methods of Proof Chapter 7, Part II Inference in Formal Symbol Systems: Ontology, Representation, ti Inference Formal Symbol Systems Symbols correspond to things/ideas in the world
Reading 11 : Relations and Functions
CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Reading 11 : Relations and Functions Instructor: Beck Hasti and Gautam Prakriya In reading 3, we described a correspondence between predicates
Lecture 9: Search 8. Victor R. Lesser. CMPSCI 683 Fall 2010
Lecture 9: Search 8 Victor R. Lesser CMPSCI 683 Fall 2010 ANNOUNCEMENTS REMEMBER LECTURE ON TUESDAY! EXAM ON OCTOBER 18 OPEN BOOK ALL MATERIAL COVERED IN LECTURES REQUIRED READINGS WILL MOST PROBABLY NOT
Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas
Journal of Artificial Intelligence Research 1 (1993) 1-15 Submitted 6/91; published 9/91 Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas Enrico Giunchiglia Massimo
Propositional and Predicate Logic. jean/gbooks/logic.html
CMSC 630 February 10, 2009 1 Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, 1986. 2003 revised edition available on line at http://www.cis.upenn.edu/
Logic and Inferences
Artificial Intelligence Logic and Inferences Readings: Chapter 7 of Russell & Norvig. Artificial Intelligence p.1/34 Components of Propositional Logic Logic constants: True (1), and False (0) Propositional
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 2: Propositional Logic
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 2: Propositional Logic Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson Propositional Logic 1 / 33 Propositional
Mathematical Logic Part Three
Mathematical Logic Part Three Recap from Last Time What is First-Order Logic? First-order logic is a logical system for reasoning about properties of objects. Augments the logical connectives from propositional
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 8. Satisfiability and Model Construction Davis-Putnam-Logemann-Loveland Procedure, Phase Transitions, GSAT Joschka Boedecker and Wolfram Burgard and Bernhard Nebel
Propositional and First Order Reasoning
Propositional and First Order Reasoning Terminology Propositional variable: boolean variable (p) Literal: propositional variable or its negation p p Clause: disjunction of literals q \/ p \/ r given by
Announcements. CS243: Discrete Structures. Propositional Logic II. Review. Operator Precedence. Operator Precedence, cont. Operator Precedence Example
Announcements CS243: Discrete Structures Propositional Logic II Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Tuesday 09/11 Weilin s Tuesday office hours are
CSE507. Introduction. Computer-Aided Reasoning for Software. Emina Torlak courses.cs.washington.edu/courses/cse507/17wi/
Computer-Aided Reasoning for Software CSE507 courses.cs.washington.edu/courses/cse507/17wi/ Introduction Emina Torlak emina@cs.washington.edu Today What is this course about? Course logistics Review of
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson SAT Solving 1 / 36 Review: Propositional
Mathematical Logic Part Two
Mathematical Logic Part Two Outline for Today Recap from Last Time The Contrapositive Using Propositional Logic First-Order Logic First-Order Translations Recap from Last Time Recap So Far A propositional
Binary Relations Part Two
Binary Relations Part Two Outline for Today Recap from Last Time Where are we, again? A Fundamental Theorem What do equivalence relations do? Strict Orders Representing prerequisites. Hasse Diagrams Drawing
Normal Forms of Propositional Logic
Normal Forms of Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 12, 2017 Bow-Yaw Wang (Academia Sinica) Normal Forms of Propositional Logic September
Automata Theory for Presburger Arithmetic Logic
Automata Theory for Presburger Arithmetic Logic References from Introduction to Automata Theory, Languages & Computation and Constraints in Computational Logic Theory & Application Presented by Masood
Comp487/587 - Boolean Formulas
Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested
7 LOGICAL AGENTS. OHJ-2556 Artificial Intelligence, Spring OHJ-2556 Artificial Intelligence, Spring
109 7 LOGICAL AGENS We now turn to knowledge-based agents that have a knowledge base KB at their disposal With the help of the KB the agent aims at maintaining knowledge of its partially-observable environment
Packet #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
Introduction to Artificial Intelligence Propositional Logic & SAT Solving. UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010
Introduction to Artificial Intelligence Propositional Logic & SAT Solving UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010 Today Representation in Propositional Logic Semantics & Deduction
Mathematical Logic Part Three
Mathematical Logic Part Three Recap from Last Time What is First-Order Logic? First-order logic is a logical system for reasoning about properties of objects. Augments the logical connectives from propositional
Logical Inference. Artificial Intelligence. Topic 12. Reading: Russell and Norvig, Chapter 7, Section 5
rtificial Intelligence Topic 12 Logical Inference Reading: Russell and Norvig, Chapter 7, Section 5 c Cara MacNish. Includes material c S. Russell & P. Norvig 1995,2003 with permission. CITS4211 Logical
Logic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz ) 2 Knowledge Representation represent knowledge about the world in a manner that
Chapter 3 Deterministic planning
Chapter 3 Deterministic planning In this chapter we describe a number of algorithms for solving the historically most important and most basic type of planning problem. Two rather strong simplifying assumptions
Class Assignment Strategies
Class Assignment Strategies ì Team- A'ack: Team a'ack ì Individualis2c: Search for possible ì Poli2cal: look at others and make decision based on who is winning, who is loosing, and conversa;on ì Emo2on
3 Propositional Logic
3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists
Kecerdasan Buatan M. Ali Fauzi
Kecerdasan Buatan M. Ali Fauzi Artificial Intelligence M. Ali Fauzi Logical Agents M. Ali Fauzi In which we design agents that can form representations of the would, use a process of inference to derive
CS 380: ARTIFICIAL INTELLIGENCE PREDICATE LOGICS. Santiago Ontañón
CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS Santiago Ontañón so367@drexeledu Summary of last day: Logical gents: The can reason from the knowledge they have They can make deductions from their perceptions,
Propositional Calculus
Propositional Calculus Dr. Neil T. Dantam CSCI-498/598 RPM, Colorado School of Mines Spring 2018 Dantam (Mines CSCI, RPM) Propositional Calculus Spring 2018 1 / 64 Calculus? Definition: Calculus A well
Logical Agents. Santa Clara University
Logical Agents Santa Clara University Logical Agents Humans know things Humans use knowledge to make plans Humans do not act completely reflexive, but reason AI: Simple problem-solving agents have knowledge
Notes. Corneliu Popeea. May 3, 2013
Notes Corneliu Popeea May 3, 2013 1 Propositional logic Syntax We rely on a set of atomic propositions, AP, containing atoms like p, q. A propositional logic formula φ Formula is then defined by the following
CS 380: ARTIFICIAL INTELLIGENCE
CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS 11/8/2013 Santiago Ontañón santi@cs.drexel.edu https://www.cs.drexel.edu/~santi/teaching/2013/cs380/intro.html Summary of last day: Logical gents: The can
Discrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland
Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................
Knowledge representation DATA INFORMATION KNOWLEDGE WISDOM. Figure Relation ship between data, information knowledge and wisdom.
Knowledge representation Introduction Knowledge is the progression that starts with data which s limited utility. Data when processed become information, information when interpreted or evaluated becomes
CS 188: Artificial Intelligence Spring 2007
CS 188: Artificial Intelligence Spring 2007 Lecture 8: Logical Agents - I 2/8/2007 Srini Narayanan ICSI and UC Berkeley Many slides over the course adapted from Dan Klein, Stuart Russell or Andrew Moore
Conjunctive Normal Form and SAT
Notes on Satisfiability-Based Problem Solving Conjunctive Normal Form and SAT David Mitchell mitchell@cs.sfu.ca October 4, 2015 These notes are a preliminary draft. Please use freely, but do not re-distribute
CS 173 Lecture 2: Propositional Logic
CS 173 Lecture 2: Propositional Logic José Meseguer University of Illinois at Urbana-Champaign 1 Propositional Formulas A proposition is a statement that is either true, T or false, F. A proposition usually
Example. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives
Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.
Artificial Intelligence
Artificial Intelligence Propositional Logic Marc Toussaint University of Stuttgart Winter 2016/17 (slides based on Stuart Russell s AI course) Motivation: Most students will have learnt about propositional
The Wumpus Game. Stench Gold. Start. Cao Hoang Tru CSE Faculty - HCMUT
The Wumpus Game Stench Stench Gold Stench Start 1 The Wumpus Game Stench in the square containing the wumpus and in the directly adjacent squares in the squares directly adjacent to a pit Glitter in the
CS 771 Artificial Intelligence. Propositional Logic
CS 771 Artificial Intelligence Propositional Logic Why Do We Need Logic? Problem-solving agents were very inflexible hard code every possible state E.g., in the transition of 8-puzzle problem, knowledge