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.

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.

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

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)

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 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

Model:

(not True or

Model:

(False or

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

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.

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)

Unit Propagation Examples

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

TrueC1

TrueC3

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’

Satisfied

C1 : ¬r q pC2: ¬ 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

propagate(C’)

end propagate

Unit Propagation

C1 : ¬r q p

C2: ¬ p ¬ t

true false

propagate(C’)

end propagate

Unit Propagation

C1 : ¬r q p

C2: ¬ p ¬ t

true false

propagate(C’)

end propagate

Unit Propagation

true false

propagate(C’)

end propagate

C2: ¬ p ¬ t

C1 : ¬r q p

Unit Propagation

C2: ¬ p ¬ t

true false

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

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

50 var 40 var 20 var

The 4.3 Point

0.02 3 4 5

Ratio of Clauses-to-Variables

50 var 40 var 20 var

50% sat

Mitchell, Selman, and Levesque 1991

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

C3 violated

C2 violated

C1 violated

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

Satisfied

C1,C2,C3 violated

GSAT Example

Satisfied A

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))

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