Global Constraints

Toby WalshNational ICT Australia and

University of New South Waleswww.cse.unsw.edu.au/~tw

Course outline● Introduction● All Different● Lex ordering● Value precedence● Complexity● GAC-Schema● Soft Global Constraints● Global Grammar Constraints● Roots Constraint● Range Constraint● Slide Constraint● Global Constraints on Sets

Global grammar constraints

● Often easy to specify a global constraint

– ALLDIFFERENT([X1,..Xn]) iff Xi=/=Xj for i<j

● Difficult to build an efficient and effective propagator– Especially if we want global reasoning

Global grammar constraints

● Promising direction initiated is to specify constraints via automata/grammar

– Sequence of variables = string in some formal language– Satisfying assignment = string accepted by the grammar/automata

Global constraints meets formal language theory

REGULAR constraint

● REGULAR(A,[X1,..Xn]) holds iff– X1 .. Xn is a string accepted by the deterministic finite

automaton A– Proposed by Pesant at CP 2004– GAC algorithm using dynamic programming– However, DP is not needed since simple ternary

encoding is just as efficient and effective

REGULAR constraint

● Deterministic finite automaton (DFA)– <Q,Sigma,T,q0,F>– Q is finite set of states– Sigma is alphabet (from which strings formed)– T is transition function: Q x Sigma -> Q– q0 is starting state– F subseteq Q are accepting states

● DFAs accept precisely regular languages– Regular language can be specified by rules of the form:NonTerminal -> Terminal | Terminal NonTerminal

REGULAR constraint

● DFAs accept precisely regular languages– Regular language can be specified by rules of the form:

NonTerminal -> Terminal NonTerminal -> Terminal NonTerminal

- Alternatively given by regular expressions- More limited than BNF which can express context-free

grammars

REGULAR constraint

● Deterministic finite automation (DFA)

5 tuple <Q,Sigma,T,q0,F> where

– Q is finite set of states– Sigma is alphabet (from which strings formed)– T is transition function: Q x Sigma -> Q– q0 is starting state– F subseteq Q are accepting states

REGULAR constraint

● Regular language– S -> 0 | 0A| AB | 1B | 1– A -> 0 | 0A– B -> 1 | 1B

● DFA– Q={q0,q1,q2,q3}– Sigma={0.1}– T(q0,0)=q0. T(q0,1)=q1– T(q1,0)=q2, T(q1,1)=q1– T(q2,0)=q2, T(q2,1)=q3– T(q3,0)=T(q3,1)=q3– F={q0,q1,q2}

REGULAR constraint

● Regular language– S -> A | AB | ABA | BA | B– A -> 0 | 0A– B -> 1 | 1B

● DFA– Q={q0,q1,q2,q3}– Sigma={0.1}– T(q0,0)=q0. T(q0,1)=q1– T(q1,0)=q2, T(q1,1)=q1– T(q2,0)=q2, T(q2,1)=q3– T(q3,0)=T(q3,1)=q3– F={q0,q1,q2}

This is the CONTIGUITYglobal constraint

REGULAR constraint

● Many global constraints are instances of REGULAR– AMONG– CONTIGUITY– LEX– PRECEDENCE– STRETCH– ..

● Domain consistency can be enforced in O(ndQ) time using dynamic programming

REGULAR constraint

● REGULAR constraint can be encoded into ternary constraints

● Introduce Qi+1– state of the DFA after the ith transition

● Then post sequence of constraints– C(Xi,Qi,Qi+1) iff DFA goes from state Qi to Qi+1 on symbol Xi

REGULAR constraint

● REGULAR constraint can be encoded into ternary constraints

● Constraint graph is Berge-acyclic– Constraints only overlap on one variable– Enforcing GAC on ternary constraints achieves GAC

on REGULAR in O(ndQ) time

REGULAR constraint

● PRECEDENCE([X1,..Xn]) iff– min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) for

all j<k● DFA has one state for each value plus a single

non-accepting state, fail– State represents largest value so far used

● T(Si,vj)=Si if j<=i● T(Si,vj)=Sj if j=i+1● T(Si,vj)=fail if j>i+1● T(fail,v)=fail

REGULAR constraint

● PRECEDENCE([X1,..Xn]) iff– min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) for

all j<k● DFA has one state for each value plus a single

non-accepting state, fail– State represents largest value so far used

● T(Si,vj)=Si if j<i● T(Si,vj)=Sj if j=i+1● T(Si,vj)=fail if j>i+1● T(fail,v)=fail● REGULAR encoding of this is just these transition

constraints (can ignore fail)

REGULAR constraint

● STRETCH([X1,..Xn]) holds iff– Any stretch of consecutive values is between

shortest(v) and longest(v) length– Any change (v1,v2) is in some permitted set, P– For example, you can only have 3 consecutive night

shifts and a night shift must be followed by a day off

REGULAR constraint

● STRETCH([X1,..Xn]) holds iff– Any stretch of consecutive values is between

shortest(v) and longest(v) length– Any change (v1,v2) is in some permitted set, P

● DFA– Qi is <last value, length of current stretch>– Q0= <dummy,0>– T(<a,q>,a)=<a,q+1> if q+1<=longest(a)– T(<a,q>,b)=<b,1> if (a,b) in P and q>=shortest(a)– All states are accepting

NFA constraint

● Automaton does not need to be deterministic● Non-deterministic finite automaton (NFA) still

only accept regular languages– But may require exponentially fewer states– Important as O(ndQ) running time for propagator– E.g. 0* (1|2)* 2 (1|2)^k 0*– Where 0=closed, 1=production, 2=maintenance

● Can use the same ternary encoding

Soft REGULAR constraint

● May wish to be “near” to a regular string● Near could be

– Hamming distance– Edit distance

● SoftREGULAR(A,[X1,..Xn],N) holds iff– X1..Xn is at distance N from a string accepted by the

finite automaton A– Can encode this into a sequence of 5-ary constraints

Soft REGULAR constraint

● SoftREGULAR(A,[X1,..Xn],N)– Consider Hamming distance (edit distance similar

though a little more complex)– Qi+1 is state of automaton after the ith transition– Di+1 is Hamming distance up to the ith variable– Post sequence of constraints

● C(Xi,Qi,Qi+1,Di,Di+1) where● Di+1=Di if T(Xi,Qi)=Qi+1 else Di+1=1+Di

Soft REGULAR constraint

● SoftREGULAR(A,[X1,..Xn],N)– To propagate– Dynamic programming

● Pass support along sequence – Just post the 5-ary constraints

● Accept less than GAC– Tuple up the variables

Cyclic forms of REGULAR

● REGULAR+(A,[X1,..,Xn])– X1 .. XnX1 is accepted by A– Can convert into REGULAR by increasing states by

factor of d where d is number of initial symbols– qi => (qi,d)– T(qi,a)=qj => T((qi,d),a)=(qj,d)– T(q0,d)=qk => T(q0,d)=(qk,d)– Thereby pass along value taken by X1 so it can be

checked on last transition

Cyclic forms of REGULAR

● REGULARo(A,[X1,..,Xn])– Xi .. X1+(i+n-1)mod n is accepted by A for each

1<=i<=n– Can decompose into n instances of the REGULAR

constraint– However, this hinders propagation

● Suppose A accepts just alternating sequences of 0 and 1● Xi in {0,1} and REGULARo(A,[X1,X2.X3])

– Unfortunately enforcing GAC on REGULARo is NP-hard

Cyclic forms of REGULAR

● REGULARo(A,[X1,..,Xn])– Reduction from Hamiltonian cycle– Consider polynomial sized automaton A1 that accepts

any sequence in which the 1st character is never repeated

– Consider polynomial sized automaton A2 that accepts any walk in a graph

● T(a,b)=b iff (a,b) in edges of graph– Consider polynomial sized automaton A1 intersect A2– This accepts only those strings corresponding to

Hamiltonian cycles

Other generalizations of REGULAR

● REGULAR FIX(A,[X1,..Xn],[B1,..Bm]) iff– REGULAR(A,[X1,..Xn]) and Bi=1 iff exists j. Xj=I– Certain values must occur within the sequence– For example, there must be a maintenance shift– Unfortunately NP-hard to enforce GAC on this

Other generalizations of REGULAR

● REGULAR FIX(A,[X1,..Xn],[B1,..Bm])– Simple reduction from Hamiltonian path– Automaton A accepts any walk on a graph– n=m and Bi=1 for all i

Chomsky hierarchy

● Regular languages● Context-free languages● Context-sensitive languages● ..

Chomsky hierarchy

● Regular languages– GAC propagator in O(ndQ) time

● Conext-free languages– GAC propagator in O(n^3) time and O(n^2) space– Asymptotically optimal as same as parsing!

● Conext-sensitive languages– Checking if a string is in the language PSPACE-

complete– Undecidable to know if empty string in grammar and

thus to detect domain wipeout and enforce GAC!

Context-free grammars

● Possible applications– Hierarchy configuration– Bioinformatics– Natual language parsing– …

● CFG(G,[X1,…Xn]) holds iff– X1 .. Xn is a string accepted by the context free

grammar G

Context-free grammars

● CFG(G,[X1,…Xn])– Consider a block stacking example– S -> NP | P | PN | NPN– N -> n | nN– P -> aa | bb | aPa | bPb– These rules give n* w rev(w) n* where w is (a|b)*– Not expressible using a regular language

● Chomsky normal form– Non-terminal -> Terminal– Non-terminal -> Non-terminal Non-terminal

Context-free grammars

● CFG(G,[X1,…Xn])– Example with X1 in {n,a}, X2 in {b}, X3 in {a,b} and

X4 in {n,a}– Enforcing GAC on CFG prunes X3=a– Only supports are nbbn and abba

CFG propagator

● Adapt CYK parser ● Works on Chomsky normal form

– Non-terminal -> Terminal– Non-terminal -> Non-terminal Non-terminal

● Using dynamic programming– Computes V[i,j], set of possible parsings for the ith to

the jth symbols

CFG propagator

● Adapt CYK parser which uses dynamic programming– For i=1 to n do– V[i,1]:={A | A->a in G, a in dom(Xi)– For j=2 to n do– For i=1 to n-j+1 do– V[i,j]:={}– For k=1 to j-1 do– V[i,j]:=V[i,j] u {A|A->BC in G, – B in V[i,k], C in V[i+k,j-k]}– If not(S in V[1,n]) then “unsat”

Conclusions

● Global grammar constraints– Specify wide range of global constraints– Provide efficient and effective propagators

automatically– Nice marriage of formal language theory and constraint

programming!