Global Grammar Constraints

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Global Grammar Constraints

• Toby Walsh
• National ICT Australia and
• University of New South Wales
• www.cse.unsw.edu.au/tw
• Joint work with Claude-Guy Quimper
• To be presented at CP06

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Global grammar constraints

• Often easy to specify a global constraint
• ALLDIFFERENT(X1,..Xn) iff
• Xi/Xj for iltj
• Difficult to build an efficient and effective
  propagator
• Especially if we want global reasoning

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Global grammar constraints

• Promising direction initiated by Beldiceanu,
  Carlsson, Pesant and Petit is to specify
  constraints via automata/grammar
• Sequence of variables
• string in some formal language
• Satisfying assignment
• string accepted by the
  grammar/automata

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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
• Encoding similar to that used by Beldiceanu et al
  for their automata with counters

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REGULAR constraint

• Deterministic finite automaton (DFA)
• ltQ,Sigma,T,q0,Fgt
• Q is finite set of states
• Sigma is alphabet (from which strings formed)
• T is transition function Q x Sigma -gt Q
• q0 is starting state
• F subseteq Q are accepting states
• DFAs accept precisely regular languages

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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

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REGULAR constraint

• REGULAR constraint can be encoded into ternary
  constraints
• Introduce Qi1
• state of the DFA after the ith transition
• Then post sequence of constraints
• C(Xi,Qi,Qi1) iff
• DFA goes from state Qi to Qi1 on symbol Xi

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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

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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
• Encoding provides access to states of automata
• Can be useful for expressing problems
• E.g. minimizing number of times we are in a
  particular state

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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

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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 ltlast value, length of current stretchgt
• Q0 ltdummy,0gt
• T(lta,qgt,a)lta,q1gt if q1ltlongest(a)
• T(lta,qgt,b)ltb,1gt if (a,b) in P and qgtshortest(a)
• All states are accepting

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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 (12)k 2 (12) 2 (12)k 0
• Where 0closed, 1production, 2maintenance
• Can use the same ternary encoding

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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

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Soft REGULAR constraint

• SoftREGULAR(A,X1,..Xn,N)
• Consider Hamming distance (edit distance similar
  though a little more complex)
• Qi1 is state of automaton after the ith
  transition
• Di1 is Hamming distance up to the ith variable
• Post sequence of constraints
• C(Xi,Qi,Qi1,Di,Di1) where
• Di1Di if T(Xi,Qi)Qi1 else Di11Di

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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

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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 gt (qi,initial value)
• T(qi,a)qj gt T((qi,b),a)(qj,b)
• Thereby pass along value taken by X1 so it can be
  checked on last transition

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Cyclic forms of REGULAR

• REGULARo(A,X1,..,Xn)
• Xi .. X1(in-1)mod n is accepted by A for each
  1ltiltn
• 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

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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

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Other generalizations of REGULAR

• REGULAR FIX(A,X1,..Xn,B1,..Bm) iff
• REGULAR(A,X1,..Xn) and Bi1 iff exists j. XjI
• Certain values must occur within the sequence
• For example, there must be a maintenance shift
• Unfortunately NP-hard to enforce GAC on this

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Other generalizations of REGULAR

• REGULAR FIX(A,X1,..Xn,B1,..Bm)
• Simple reduction from Hamiltonian path
• Automaton A accepts any walk on a graph
• nm and Bi1 for all i

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Chomsky hierarchy

• Regular languages
• Context-free languages
• Context-sensitive languages
• ..

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Chomsky hierarchy

• Regular languages
• GAC propagator in O(ndQ) time
• Conext-free languages
• GAC propagator in O(n3) time and O(n2) 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!

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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

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Context-free grammars

• CFG(G,X1,Xn)
• Consider a block stacking example
• S -gt NP P PN NPN
• N -gt n nN
• P -gt aa bb aPa bPb
• These rules give n w rev(w) n where w is (ab)
• Not expressible using a regular language
• Chomsky normal form
• Non-terminal -gt Terminal
• Non-terminal -gt Non-terminal Non-terminal

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CFG propagator

• Adapt CYK parser
• Works on Chomsky normal form
• Non-terminal -gt Terminal
• Non-terminal -gt Non-terminal Non-terminal
• Using dynamic programming to compute supports
• Bottom up
• Enforces GAC in Theta(n3) time
• Simultaneously and independently proposed by
  Sellmannn CP06

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CFG propagator

• Adapt Earley chart parser
• Carry support information
• Works on grammar in any form
• More top down
• Better on tightly restricted grammars
• Enforces GAC in O(n3) time
• Best case is better as not Theta(n3)

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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!

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Questions?