Constraint Based Systems

Reading: Chapter 6 Chess article (see links)

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Agenda

Finishing up search for machine translation

Constraint satisfaction

Hill climbing function HillClimbing(problem, initial-state, queuing-fn)node ← MakeNode(initial-state(problem));while T do

next ← Best(SearchOperator-fn(node,cost-fn));if(IsBetter-fn(next, node)) then continue;else if(GoalTest(node)) then return node;else exit;

end whilereturn Failure; MT (Germann et al., ACL-2001)

node ← targetGloss(sourceSentence);while T do next ← Best( LocallyModifiedTranslationOf(node)); if(IsBetter(next, node)) then continue; else print node; exit;end while

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Types of changes

Translate one or two words (j1e1j2e2)

Translate and insert (j e1 e2)

Remove word of fertility 0 (i)

Swap segments (i1 i2 j1 j2)

Join words (i1 i2)

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Example

Total of 77,421 possible translations attempted

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How to search better?

MakeNode(initial-state(problem))

RemoveFront(Q)

SearchOperator-fn(node, cost-fn);

queuing-fn(problem, Q, (Next,Cost));

Example 1: Greedy Search MakeNode(initial-state(problem))

Machine Translation (Marcu and Wong, EMNLP-2002)

node ← targetGloss(sourceSentence);while T do next ← Best( LocallyModifiedTranslationOf(node)); if(IsBetter(next, node)) then continue; else print node; exit;end while

20.5

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21.5

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22.5

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23.5

IBM J M, p(E | F) gloss J M, p(E, F) gloss

IBM JM, p(E | F) gloss JM, p(E, F) gloss

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Climbing the wrong peak

What sentence is more grammatical?1. better bart than madonna , i say 2. i say better than bart madonna ,

Can you make a sentence with these words? a and apparently as be could dissimilar firing identical neural really so things thought two

Model validation

Model stress-testing

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Language-model stress-testing

Input: bag of words Output: best sequence according to a

linear combination of an ngram LM syntax-based LM (Collins, 1997)

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Size: 10-25 words long

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

ID E Search Errors Model Errors

NGLM NGLM+SBLM NGLM+SBLM*

Best searched• 51.6: and so could really be a neural apparently thought things as dissimilar firing two identicalOriginal word order• 64.3: could two things so apparently dissimilar as a thought and neural firing really be identical

0.00%10.00%20.00%30.00%40.00%50.00%60.00%70.00%80.00%90.00%

ID E Search Errors Model Errors

NGLM NGLM+SBLM NGLM+SBLM*

Best searched• 32.3: i say better than bart madonna ,Original word order• 41.6: better bart than madonna, i say

Size: 3-7 words long

SBLM*: trained on an additional 160k WSJ sentences.

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Cryptograms

QFL HCVPS

PX V ANSWLCEZK NCJVS; PQ XQVCQX QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ VSW WNLX SNQ XQNT ZSQPK RNZ JLQ QN QFL NEEPGL

CNHLCQ ECNXQ

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Cryptograms

QFL HCVPSTHE BRAIN

PX V ANSWLCEZK NCJVS; PQ XQVCQXIS A WONDERFUL ORGAN; IT STARTS

QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ THE MINUTE YOU GET UP IN THE MORNING

VSW WNLX SNQ XQNT ZSQPK RNZ JLQ QN AND DOES NOT STOP UNTIL YOU GET TO

QFL NEEPGLTHE OFFICE

CNHLCQ ECNXQROBERT FROST

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Constraint Satisfaction Problems

Standard representation allows a general-purpose approach to search

Set of variables, X1, X2 … Xn Each variable has domain Di of possible values

Set of constraints, C1, C2, … Cn

State = assignment of values to some or all variables

Xi=v,i , Xj=vj

Solution = complete assignment that satisfies all constraints

Some solutions maximize an objective function

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Cryptograms

QFL HCVPS

PX V ANSWLCEZK NCJVS; PQ XQVCQX QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ VSW WNLX SNQ XQNT ZSQPK RNZ JLQ QN QFL NEEPGL

CNHLCQ ECNXQ

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Cryptograms

QFL HCVPSTHE BRAIN

PX V ANSWLCEZK NCJVS; PQ XQVCQXIS A WONDERFUL ORGAN; IT STARTS

QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ THE MINUTE YOU GET UP IN THE MORNING

VSW WNLX SNQ XQNT ZSQPK RNZ JLQ QN AND DOES NOT STOP UNTIL YOU GET TO

QFL NEEPGLTHE OFFICE

CNHLCQ ECNXQROBERT FROST

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CSP algorithm

Depth-first search often used

Initial state: the empty assignment {}; all variables are unassigned

Successor fn: assign a value to any variable, provided no conflicts w/constraints

All CSP search algorithms generate successors by considering possible assignments for only a single variable at each node in the search tree

Goal test: the current assignment is complete Path cost: a constant cost for every step

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Local search

Complete-state formulation Every state is a compete assignment that

might or might not satisfy the constraints

Hill-climbing methods are appropriate

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Dimensions of CSPs

Discrete/finite domains

Map coloring 8 queens

Nonlinear constraints No general solution

Unary constraints Absolute constraints

Infinite domains Scheduling calendars

(discrete) Schedule experiments

for Hubble Linear constraints

Binary/N-ary constraints Preferences

Prof. McKeown prefers after 1pm

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M GWD’X URNMRPR MD QD QCXRLNMCR, QNXFWZOF M QH ULMDOMDO Q KFQDOR WC ZDGRLARQL.

AWWGS QNNRD

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M GWD’X URNMRPR MD QD A BCD’E A AD DQCXRLNMCR, QNXFWZOF M QH E A E

ULMDOMDO Q KFQDOR WC AD AD D C ZDGRLARQL.

DB AWWGS QNNRD

B D

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M GWD’X URNMRPR MD QD A BCD’I A AD DQCXRLNMCR, QNXFWZOF M QH I A I

ULMDOMDO Q KFQDOR WC AD AD D C ZDGRLARQL.

DB AWWGS QNNRD

B D

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M GWD’X URNMRPR MD QD A BCD’O A AD DQCXRLNMCR, QNXFWZOF M QH O A O

ULMDOMDO Q KFQDOR WC AD AD D C ZDGRLARQL.

DB AWWGS QNNRD

B D

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M GWD’X URNMRPR MD QD A BCD’U A AD DQCXRLNMCR, QNXFWZOF M QH U A U

ULMDOMDO Q KFQDOR WC AD AD D C ZDGRLARQL.

DB AWWGS QNNRD

B D

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M GWD’X URNMRPR MD QD A BCE’ A AE EQCXRLNMCR, QNXFWZOF M QH A

ULMDOMDO Q KFQDOR WC AE AE E C ZDGRLARQL.

EB AWWGS QNNRD

B E

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What additional knowledge about language would be helpful?

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General purpose methods for efficient implementation

Which variable should be assigned next?

in what order should its values be tried?

Can we detect inevitable failure early?

Can we take advantage of problem structure?

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Order

Choose the most constrained variable first

The variable with the fewest remaining values

Minimum Remaining Values (MRV) heuristic

What if there are >1? Tie breaker: Most constraining variable Choose the variable with the most

constraints on remaining variables

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Order on value choice

Given a variable, chose the least constraining value

The value that rules out the fewest values in the remaining variables

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M GWD’X URNMRPR MD QD I I I AQCXRLNMCR, QNXFWZOF M QHA I A I A

ULMDOMDO Q KFQDOR WC I I A A ZDGRLARQL. A

AWWGS QNNRD A

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M GWD’X URNMRPR MD QD I N I IN ANQCXRLNMCR, QNXFWZOF M QHA I A I A

ULMDOMDO Q KFQDOR WC IN IN A AN ZDGRLARQL. N A

AWWGS QNNRD A N

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Forward Checking

Keep track of remaining legal values for unassigned variables

Terminate search when any variable has no legal values

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M GWD’X URNMRPR MD QD I I I AQCXRLNMCR, QNXFWZOF M QHA I A I A

ULMDOMDO Q KFQDOR WC I I A A ZDGRLARQL. A

When M I selected D (F N S T) When QA selected D (N S T)

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M GWD’X URNMRPR MD QD I DON’T EL I E E IN ANQCXRLNMCR, QNXFWZOF M QHA FTERL I FE ALTHO GH I AM

ULMDOMDO Q KFQDOR WC R IN G I NG A HANGE OFZDGRLARQL. NDERWEAR

AWWGS QNNRD WOODY ALLEN

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M GWD’X URNMRPR MD QD I DON’T CEL I E E IN ANQCXRLNMCR, QNXFWZOF M QHA FTERL I FE ALTHO GH I AM

ULMDOMDO Q KFQDOR WCCR IN G I NG A HANGE OFZDGRLARQL. NDERWEAR

Select UC -> the domain of P is empty. Requires backtracking even before taking the next branch. Avoids selection Z, K next before discovering the error.

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M GWD’X URNMRPR MD QD I DON’T BEL I E VE IN ANQCXRLNMCR, QNXFWZOF M QHA FTERL I FE ALTHO UGH I AM

ULMDOMDO Q KFQDOR WCBR IN G I NG A CHANGE OFZDGRLARQL.UNDERWEAR AWWGS QNNRD

WOODY ALLEN

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Speed-ups

Problem Backtracking

BT+MRV ForwardChecking

FV+MRV

USA (>1000K) (>1000K) 2K 60

N-Queens (>40000K) 13,500K (>40000K) 817K

Zebra 3,859K 1K 35K .5K

Random1 415K 3K 26K 2K

Random2 942K 27K 77K 15K

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Other types of improvements

Constraint propagation Propagate implications of a constraint on

one variable onto other variables. Not only values, but constraints between other variables

Intelligent backtracking Conflict-directed backtracking: Save

possible conflicts for a variable value on assignment.

Back jump to assignment of values

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Crossword Puzzles