3/11/2002copyright brian williams1 propositional logic and satisfiability brian c. williams...

3/11/2002 copyright Brian Williams 1

Propositional Logic and Satisfiability

Brian C. Williams16.412/6.834 October 7th, 2002


Reading Assignments: Propositional Logic

• AIMA Ch. 6 – Propositional Logic


Outline

• Motivation• Propositional Logic

• Syntax• Semantics

• Propositional Satisfiability• Systematic methods• Characteristics of systematic search• Through local search

• Summary• Appendix: Reduction to Clausal Form

4copyright Brian Williams3/11/2002

How can Logic Help Us to Perform Diagnosis?

Helium tank

Fuel tankOxidizer tank

MainEngines

Flow1 = zeroPressure1 = nominal

Pressure2= nominal

Acceleration = zero


Consistency-based Diagnosis

• Given:• Observations “no thrust”• Device schematic• Component models

• Specifies all correct and faulty modes• Constraints describe behavior of each mode.

• Find:• Component mode assignment that is consistent with

the Obs, Schematic & Models.


• Candidate: Assignment to all component modes.

0

1

0

0

1

Or1

Or2

Or3

And1

And2

A

B

C

D

E

F

G

X

Y

Z

1And(i):• G(i):

Out(i) = In1(i) AND In2(i)

• U(i): 0

Diagnosis identifies consistent modes

Candidate = {G(And1) & G(And2) & G(Or1) & G(Or2) & G(Or3)}


Diagnosis identifies consistent modes

AND(i):• G(i):

Out(i) = In1(i) AND In2(i)

• U(i):

• Candidate: Assignment to all component modes.• Diagnosis D: Candidate consistent with model Phi and

observables OBS.

3

2

2

3

3

Or1

Or3

And1

A

B

C

D

E

F

G

X

Y

Z

10

12

Diagnosis = {G(And1) & U(And2) & G(Or1) & G(Or2) & G(Or3)}

Or2

8copyright Brian Williams3/11/2002

How do we write down the constraints?

Helium tank

Fuel tankOxidizer tank

MainEngines

Flow1 = zeroPressure1 = nominal

Pressure2= nominal

Acceleration = zero


What formal languages exist for describing constraints?

Logic:• Propositional logic truth of facts• First order logic facts,objects,relations• Temporal logic time, ….• Modal logics knowledge, belief

…• Probability ….


Logic in General

• Logics• formal languages for representing information such

that conclusions can be drawn.

• Syntax • defines the sentences in the language.

• Semantics • define the “meaning” of sentences;

• i.e., truth of a sentence in a world.


Outline

• Motivation

• Propositional Logic• Syntax• Semantics

• Propositional Satisfiability

• Summary

• Appendix: Reduction to Clausal Form


Propositional Logic:Syntax• Proposition

• Statement that is true or false• (valve v1)• (= voltage high)

• Propositional sentence (S)• S ::= proposition |

• (NOT S) |

• (OR S1 ... Sn) |

• (AND S1 ... Sn)

• Defined Constructs• (implies S1 S2) => ((not S1) OR S2)

• (IFF S1 S2) => (AND (IMPLIES S1 S2)(IMPLIES S2 S1))

• . . .


Engine Example: propositional logic sentence

(mode(E1) = ok implies

(thrust(E1) = on if and only if flow(V1) = on and flow(V2) = on)) and

(mode(E1) = ok or mode(E1) = unknown) and

not (mode(E1) = ok and mode(E1) = unknown)

E1

V1 V2


Propositional Logic: SemanticsA model assigns true/false to every proposition symbol Pi

• A = True, B = False, C = False

The truth of sentence S is defined as a composition of boolean operators applied to the model (S, S1 and S2 are sentences):

• Not S is True iff S is False

• S1 and S2 is True iff S1 is True and S2 is True

• S1 or S2 is True iff S1 is True or S2 is True

• S1 implies S2 is True iff S1 is False or S2 is True

• S1 iff S2 is True iff S1 implies S2 is True

and S2 implies S1 is True


Example: Determining the truth of a sentence


[(thrust(E1) = on if and only if (flow(V1) = on and flow(V2) = on)) and


not (mode(E1) = ok and mode(E1) = unknown)])

Model:

mode(E1) = ok is True

thrust(E1) = on is False

flow(V1) = on is True

flow(V2) = on is False

mode(E1) = unknown is False



(True implies

[(thrust(E1) = on if and only if (flow(V1) = on and flow(V2) = on)) and

(True or mode(E1) = unknown) and

not (True and mode(E1) = unknown)])

Model:








(True implies

[(False if and only if (True and False)) and

(True or False) and

not (True and False)])

Model:








(True implies


(True or False) and

not (True and False)])

Model:








(True implies


(True or False) and

not False])

Model:








(True implies


(True or False) and

True])

Model:








(True implies

[(False if and only if False) and

True and

True])

Model:








(True implies

[(False implies False ) and (False implies False )) and

True and

True])

Model:








(True implies

[(not False or False ) and (not False or False )) and

True and

True])

Model:








(True implies

[(True or False ) and (True or False )) and

True and

True])

Model:








(True implies

[(True and True) and

True and

True])

Model:








(True implies

[True and

True and

True])

Model:








(True implies

True)

Model:








(not True or

True)

Model:








(False or

True)

Model:








True!

Model:







Outline

• Motivation• Propositional Logic

• Syntax• Semantics

• Propositional Satisfiability• Clausal Form• Systematic methods• Characteristics of systematic search• Through local search



Propositional SatisfiabilityPropositional Satisfiability

Find a model M that satisfies sentence T (called a theory):

• Reduce sentence T to clausal form.• Perform search similar to Forward Checking

Propositional satisfiability testing: 1990: 100 variables / 200 clauses (constraints) 1998: 10,000 - 100,000 vars / 10^6 clauses Novel applications: e.g. diagnosis, planning, software / circuit testing, machine learning, and protein folding


Propositional Clauses:A Simpler Form

• Literal: proposition or its negation• B, Not A

• Clause: disjunction of literals• (not A or B or E)

• Conjunctive Normal Form• Phi = (A or B or C) and

(not A or B or E) and (not B or C or D)

• Viewed as a set of clauses


Engine Example: Clausal Model


(thrust(E1) = on iff flow(V1) = on and flow(V2) = on)) and



not (mode(E1) = ok) or not (thrust(E1) = on) or flow(V1) = on;


not (mode(E1) = ok) or not (flow(V1) = on) or

not (flow(V2) = on) or thrust(E1) = on;

mode(E1) = ok or mode(E1) = unknown;

not (mode(E1) = ok) or not (mode(E1) = unknown);


Outline

• Propositional Satisfiability• Clausal form• Systematic methods

• Backtrack Search• Deduction by Resolution• DPLL: Combining search and deduction • Characteristics of DPLL

• local search using GSAT



Propositional Satisfiability by Enumeration

• Assign true or false to an unassigned proposition.

• Backtrack as soon as a clause is violated.

Example:• C1: Not A or B• C2: Not C or A• C3: Not B or C

A

F T

BF T

C

F T

C

F T

BF T

C C

F TF T


Backtrack Search Procedure

BT(phi,A)

Input: A cnf theory phi, an assignment A to propositions in phi

Output: A decision of whether phi is satisfiable.

1. If a clause is violated return(false);

2. Else if all propositions are assigned return(true);

3. Else Q = some unassigned proposition in phi;

4. Return (BT(phi, A[Q = True]) or

5. BT(phi, A[Q = False])


Outline






Inference: Resolution on Clauses

or B, not B or or

Example: Checking Satisfiability of C1 – C4:• C1: A• C2: Not A or B• C3: Not B or C• C4: Not C or Not A• C5: Not A or C Resolving C2 and C3 using B• C6: C Resolving C1 and C5 using A• C7: Not A Resolving C6 and C4 using C• C8: False Resolving C7 and C1 using A


Outline






How Do We Combine Resolution and Back Track Search?Backtrack Search• Assign true or false to an

unassigned proposition.• Backtrack as soon as a clause

is violated.• Satisfiable if assignment is

complete.


A

F T

BF T

C

F T

C

F T

BF T

C C

F TF T

Similar to Forward Checking: Perform limited form of inference apply inference rule after assigning each variable.


Perform limited inference:Unit Clause Resolution

Idea: Conclude propositions, not disjunctive clauses

• Unit clause: a clause with a single literal.

• Not A

• Unit resolution: resolution with unit clauses:• not A (A or B or C)

(B or C)

• Unit propagation: unit resolve until only a unit clause is left:

• (not A) (not B) (A or B or C)

C


Unit Propagation Examples

• C1: Not A or B• C2: Not C or A• C3: Not B or C• C4: A

C4A

TrueC1

B

TrueC3

C

True

Satisfied

Satisfied

Satisfied

Satisfied


Unit Propagation Examples

• C1: Not A or B• C2: Not C or A• C3: Not B or C• C4: A

• C4’: Not B

C1 C3C4

C1 C2C4’

A

True

B

True

C

True

A

False

B

False

C

False

C4A

True

Satisfied

Satisfied

Satisfied

Satisfied


C1 : ¬r q pC2: ¬ p ¬ t

r

true

q

false

p

t

procedure propagate(C) // C is a clause

if all literals in C are false except l, and l is unassigned

then assign true to l and

record C as a support for l and

for each clause C’ mentioning “not l”,

propagate(C’)

end propagate

Unit Propagation


C1 : ¬r q pC2: ¬ p ¬ t

r

true

q

false

p

t






propagate(C’)

end propagate

Unit Propagation


C1 : ¬r q p

r q

p

C2: ¬ p ¬ t

true false

true

t






propagate(C’)

end propagate

Unit Propagation


r q

p

true false

true






propagate(C’)

end propagate

C2: ¬ p ¬ t

t

C1 : ¬r q p

Unit Propagation


r q

p

C2: ¬ p ¬ t

true false

true

tfalse






propagate(C’)

end propagate

C1 : ¬r q p

Unit Propagation


Propositional Satisfiability by DPLL

Initially:• Unit propagate.

Repeat:• Assign true or false to an

unassigned proposition.• Unit propagate.• Backtrack as soon as a

clause is violated.• Satisfiable if assignment

is complete.

Propagate:C = FB = FA = F

A

F TPropagate:B = TC = TA = T



DPLL Procedure[Davis, Putnam Logmann, Loveland, 1962]

DPLL(phi,A) Input: A cnf theory phi, an assignment A to propositions in

phiOutput: A decision of whether phi is satisfiable.1. propagate(phi);2. If a clause is violated return(false);3. Else if all propositions are assigned return(true);4. Else Q = some unassigned proposition in phi;6. Return (DPLL(phi, A[Q = True]) or 7. DPLL(phi, A[Q = False])


Outline

• Motivation• Propositional Logic• Propositional Satisfiability

• Systematic methods• Characteristics of systematic search• Through local search



Hardness of 3SAT

02 3 4 5

Ratio of Clauses-to-Variables

6 7 8

1000

3000

DP

Calls

2000

4000

50 var 40 var 20 var


The 4.3 Point

0.02 3 4 5

Ratio of Clauses-to-Variables

6 7 8

0.2

0.6

Pro

bab

ilit

yD

P C

alls

0.4

50 var 40 var 20 var

50% sat

Mitchell, Selman, and Levesque 1991

0.8

1.0

0

1000

3000

2000

4000


Intuition

• At low ratios:• few clauses (constraints)• many assignments• easily found

• At high ratios:• many clauses• inconsistencies easily detected


Phase transition 2-, 3-, 4-, 5-, and 6-SATPhase transition 2-, 3-, 4-, 5-, and 6-SAT


Outline





GSAT Example

• C1: Not A or B• C2: Not C or Not A• C3: or B or Not C

C1, C2, C3 violated A

True

B

False

C

True

C3 violated

False

C2 violated

True

C1 violated

False

1. Pick random assignment

2. Check effect of flipping each assignment, counting violated clauses

3. Pick assignment with fewest violations.


GSAT Example


C1 violated A

True

B

False

C

False

Satisfied

False

Satisfied

True

C1,C2,C3 violated

True





GSAT Example


Satisfied A

True

B

True

C

False





Outline





Satisfiability Testing ProceduresSatisfiability Testing Procedures

• Reduce to CNF (Clausal Form) then:• Systematic, complete procedures

• Depth-first backtrack search (Davis, Putnam, & Loveland 1961)

• unit propagation, shortest clause heuristic• State-of-the-art implementations:

• ntab (Crawford & Auton 1997)• itms (Nayak & Williams 1997)

• many others! See SATLIB 1998 / Hoos & Stutzle

• Stochastic, incomplete procedures• GSAT (Selman et. al 1993)

• Walksat (Selman & Kautz 1993)

• greedy local search + noise to escape local minima


SummaryLogical agents apply inference to a knowledge base to derive new information

and make decisions.

Basic concepts in logic

• Syntax: formal structure of sentences.

• Semantics: truth of sentence wrt models.

• Entailment: necessary truth of a sentence given another.

• Inference: deriving sentences from others.

• Soundness: derivations produce only entailed sentences.

• Completeness: derivations can produce all entailed sentences.

• Enumeration, Resolution, DPLL and DPLL+TMS are sound and complete methods for propositional logic.

• Local repair methods (e.g., GSAT) are sound but incomplete.


Appendix: Reduction to Clauses

Note: You are responsible for reading this material


Engine Example: Mapping to a Clausal Model


(thrust(E1) = on iff flow(V1) = on and flow(V2) = on)) and





not (mode(E1) = ok) or not (flow(V1) = on) or

not (flow(V2) = on) or thrust(E1) = on;

mode(E1) = ok or mode(E1) = unknown;

not (mode(E1) = ok) or not (mode(E1) = unknown);


Reducing Propositional Formula to Clauses (CNF)

• Eliminate IFF and Implies

• E1 iff E2 => (E1 implies E2) and (E2 implies E1)

• E1 implies E2 => not E1 or E2


Engine Example: Eliminate IFF


(thrust(E1) = on iff (flow(V1) = on and flow(V2) = on))) and




Engine Example: Eliminate IFF


(thrust(E1) = on iff (flow(V1) = on and flow(V2) = on))) and




((thrust(E1) = on implies (flow(V1) = on and flow(V2) = on)) and

((flow(V1) = on and flow(V2) = on) implies thrust(E1) = on))) and




Engine Example: Eliminate Implies


((thrust(E1) = on implies (flow(V1) = on and flow(V2) = on)) and

((flow(V1) = on and flow(V2) = on) implies thrust(E1) = on))) and




Engine Example: Eliminate Implies

(mode(E1) = ok implies ((thrust(E1) = on implies (flow(V1) = on and flow(V2) = on)) and ((flow(V1) = on and flow(V2) = on) implies thrust(E1) = on))) and(mode(E1) = ok or mode(E1) = unknown) andnot (mode(E1) = ok and mode(E1) = unknown)

(not (mode(E1) = ok) or ((not (thrust(E1) = on) or (flow(V1) = on and flow(V2) = on)) and (not (flow(V1) = on and flow(V2) = on)) or thrust(E1) = on))) and(mode(E1) = ok or mode(E1) = unknown) andnot (mode(E1) = ok and mode(E1) = unknown)



• Move negations in to propositions (De Morgan)

• Not (E1 and E2) => (not E1) or (not E2)

• Not (E1 or E2) => (not E1) and (not E2)

• Not (not E1) => E1


Engine Example: Move Negations In

(not (mode(E1) = ok) or

((not (thrust(E1) = on) or (flow(V1) = on and flow(V2) = on)) and

(not (flow(V1) = on and flow(V2) = on)) or thrust(E1) = on))) and




Engine Example: Move Negations In

(not (mode(E1) = ok) or ((not (thrust(E1) = on) or (flow(V1) = on and flow(V2) = on)) and (not (flow(V1) = on and flow(V2) = on)) or thrust(E1) = on))) and(mode(E1) = ok or mode(E1) = unknown) andnot (mode(E1) = ok and mode(E1) = unknown)

(not (mode(E1) = ok) or ((not (thrust(E1) = on) or (flow(V1) = on and flow(V2) = on)) and (not (flow(V1) = on) or not (flow(V2) = on)) or thrust(E1) = on) ) and(mode(E1) = ok or mode(E1) = unknown) and(not (mode(E1) = ok) or not (mode(E1) = unknown)))



• Move conjunction out (Distribution)• E1 or (E2 and E3) =>(E1 or E2) and (E1 or E3)


Engine Example: Move Conjunctions Out


((not (thrust(E1) = on) or (flow(V1) = on and flow(V2) = on)) and

(not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on))) and


(not (mode(E1) = ok) or not (mode(E1) = unknown))



(not (mode(E1) = ok) or ((not (thrust(E1) = on) or (flow(V1) = on and flow(V2) = on)) and (not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on))) and(mode(E1) = ok or mode(E1) = unknown) and(not (mode(E1) = ok) or not (mode(E1) = unknown))

(not (mode(E1) = ok) or (((not (thrust(E1) = on) or flow(V1) = on) and (not (thrust(E1) = on) or flow(V2) = on)) and (not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on))) and(mode(E1) = ok or mode(E1) = unknown) and(not (mode(E1) = ok) or not (mode(E1) = unknown))




(((not (thrust(E1) = on) or flow(V1) = on) and

(not (thrust(E1) = on) or flow(V2) = on)) and

(not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on))) and


(not (mode(E1) = ok) or not (mode(E1) = unknown))



(not (mode(E1) = ok) or (((not (thrust(E1) = on) or flow(V1) = on) and (not (thrust(E1) = on) or flow(V2) = on)) and (not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on))) and(mode(E1) = ok or mode(E1) = unknown) and(not (mode(E1) = ok) or not (mode(E1) = unknown))

(not (mode(E1) = ok) or not (thrust(E1) = on) or flow(V1) = on) and (not (mode(E1) = ok) or not (thrust(E1) = on) or flow(V2) = on)) and (not (mode(E1) = ok) or not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on) and(mode(E1) = ok or mode(E1) = unknown) and(not (mode(E1) = ok) or not (mode(E1) = unknown))

3/11/2002copyright brian williams1 propositional logic and satisfiability brian c. williams...

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