Artificial IntelligenceIntelligence

Ian Gentipg@cs.st-and.ac.uk

Search: 4Search Heuristics

Artificial IntelligenceIntelligence

Part I : Depth first search for SATPart II: Davis-Putnam AlgorithmPart III: Heuristics for SAT

Search 4

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Search: the story so far

Example Search problems, SAT, TSP, GamesSearch states, Search trees Don’t store whole search trees, just the frontierDepth first, breadth first, iterative deepeningBest FirstHeuristics for Eights PuzzleA*, Branch & Bound

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Example Search Problem: SAT

We need to define problems and solutionsPropositional Satisfiability (SAT)

really a logical problem -- I’ll present as a letters game

Problem is a list of words contains upper and lower case letters (order unimportant) e.g. ABC, ABc, AbC, Abc, aBC, abC, abc

Solution is choice of upper/lower case letter one choice per letter each word to contain at least one of our choices e.g. AbC is unique solution to above problem.

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Example Search Problem: SAT

We need to define problems and solutionsPropositional Satisfiability (SAT)

Now present it as a logical problem

Problem is a list of clauses contains literals

each literal a positive or negative variableliterals are e.g. +A, -B, +C, ….

Solution is choice of true or false for each variable one choice per letter each clause to contain at least one of our choices I.e. +A matches A = true, -A matches A = false

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It’s the same thing

Variables = letters literal = upper or lower case letterPositive = True = Upper caseNegative = False = Lower caseclause = wordproblem = problem

I reserve the right to use either or both versions confusingly

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Depth First Search for SAT

B(a B )

Im p oss ib le

b(a b )

Im p oss ib le

a(a )

C h oose B

B(A B C )

Im p oss ib le

b(A b C )

S o lu tion

C(A C )

C h oose B

cA c

im p oss ib le

A(A )

C h oose C

A B C , A B c , A b C , A b c , aB C , ab C , ab cs ta te = ()C h oose A

What heuristics should we use?

We need two kinds variable ordering

e.g. set A before B value ordering

e.g. set True before False

In Eights, only need value variable ordering irrelevant

In SAT, variable ordering vital value ordering less important

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Unit Propagation

One heuristic in SAT is vital to successWhen we have a unit clause …

e.g. +A we must set A = true if we set A = false the clause is unsatisfied, so is the whole

problem

A unit clause might be in the original problemor contain only one unset variable after simplification

e.g. clauses (aBC), (abc), set A = upper case, B = lower casewhat unit clause remains?

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Unit Propagation

e.g. clauses (aBC), (abc), set A = upper case, B = lower casewhat unit clause remains?

A = upper gives (BC), (bc)B = lower case satisfies (bc)

reduces (BC) to (C)

The unit clause is (C)We should set C = upper case

irrespective of other clauses in the problem

setting one unit clause can create a new one … leading to a cascade/chain reaction called unit propagation

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Depth First + Unit Propagation

Unit propagation is vital in SATWhenever there is a not-yet-satisfied unit clause

set the corresponding variable to True if literal positivefalse if literal negative

Use this to override all other heuristicsLater in lecture will think about other heuristics to

use as wellNext we will look at another algorithm

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Davis-Putnam

The best complete algorithm for SAT is Davis-Putnam first work by Davis-Putnam 1961 current version by Davis-Logemann-Loveland 1962 variously called DP/DLL/DPLL or just Davis-Putnam I will present a slight variant omitting “Pure literal” rule

A recursive algorithm Two stopping cases

an empty set of clauses is trivially satisfiable an empty clause is trivially unsatisfiable

there is no way to satisfy the clause

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Algorithm DPLL (clauses)

1. If clauses is empty clause set, Succeed2. If clauses contains an empty clause, Fail3. If clauses contains a unit clause (literal)

return result of DPLL(clauses[literal]) clauses[literal] means simplify clauses with value of literal

4. Else heuristically choose a variable u heuristically choose a value v 4.a. If DPLL(clauses[u:=v]) succeeds, Succeed 4.b. Else return result of DPLL(clauses[u:= not v])

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DPLL success

About 40 years old, DPLL is still the most successful complete algorithm for SAT

Intensive research on variants of DPLL in the 90s mostly very close to the 1962 version

Implementation can be very efficientMost work on finding good heuristics

Good heuristics should find solution quicklyor work out quickly that there is no solution

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It’s the same thing (again)

DPLL is just depth first search + unit propagationWe’ve now got three presentations of the same thing

search trees algorithm based on lists DPLL

Shows the general importance of depth first search

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Heuristics for DPLL

We need variable ordering heuristics can easily make the difference between success/failure

Tradeoff between simplicity and effectivenessThree very simple variable ordering heuristics

lexicographic: choose A before B before C before … random: choose a random variable first occurrence: choose first variable in first clause

Pros: all very easy to implementCons: ineffective except on very small or easy problems

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How can we design better heuristics

All the basic heuristics listed are unlikely to make the best choice except by good luck

We want to choose variables likely to finish search quickly

How can we design heuristics to do this? Pick variables occurring in lots of clauses? Prefer short clauses (AB) or long clauses (ABCDEFG) ? Pick variables occurring more often positively??

We need some design principles underlying our search

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Three Design Principles

The Constrainedness Hypothesis Choose variables which are more constrained than other

variables (e.g. pack suits before toothbrush for interview trip) Motivation: Most constrained first

attack the most difficult part of the problemit should either fail or succeed and make the rest easy

The Satisfaction Hypothesis Try to choose variables which seem likely to come closest to

satisfying the problem Motivation: we want to find a solution, so choose the variable

which comes as close to that as possible

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Three Design Principles

The simplification hypothesis Try to choose variables which will simplify the problem as

much as possible via unit propagation Motivation: search is exponential in the size of the problem

so making the problem small quickly minimizes search

Let’s look at 3 heuristics based on these principles not wildly different from each other often different principles give similar heuristics

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Most Constrained First

Short clauses are most constraining (A B) rules out 1/4 of all solutions (A B C D E) only rules out 1/32 of all solutions

Take account only of shortest clauses e.g. shortest clause in a problem may be of length 2

Several variants on this idea first occurrence in shortest clause most occurrences in shortest clauses (usually many such) first occurrence in all positive shortest clause

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Satisfaction Hypothesis

Try to satisfy as much as possible with next literalTake account of different lengths

clause of length i rules out a fraction 2-i of all solutions weight each clause by the number 2-i

For each literal, calculate weighted sum add the weight of each clause the literal appears in the larger this sum, the more difficulties are eliminated

This is the Jeroslow-Wang HeuristicVariable and value ordering

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Simplification Hypothesis

We want to simplify problem as much as possible I.e. get biggest possible cascade of unit propagation

One approach is to suck it and see make an assignment, see how much unit propagation

occurs, after testing all assignments, choose the one which

caused the biggest cascade exhaustive version is expensive (2n probes necessary) Successful variants probe a small number of promising

variables (e.g. from most constrained heuristic)

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Conclusions

Unit propagation vital to SATDavis Putnam (DP/DLL/DPLL) successful

= depth first + unit propagation

Need heuristics, especially variable ordering Three design principles helpNot yet clear which is the bestHeuristic design is still a black art