From Constraints to Finite Automata to Filtering Algorithms Mats Carlsson, SICS Nicolas Beldiceanu, EMN

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Example x + y = 92x + 4y = 24 x y

ESOP, March 29, 2004 Constraint Propagation Variables feature variable domain (finite set of integers) Propagators implement constraints Propagation loop execute propagators until simultaneous fixpoint

ESOP, March 29, 2004 Propagator Propagator p is a procedure (coroutine) implements constraintcon(p) its semantics (set of tuples) computes on set of variablesvar(p) Execution of propagator p filters domains of variables in var(p) signals failure signals entailment

ESOP, March 29, 2004 Propagators Are Intensional Propagators implement filtering aka: narrowing, domain reduction, value removal No extensional representation of con(p) impractical in most cases (space) Extensional representation of constraint can be provided by special propagator often: “element” constraint, “relation” constraint, …

ESOP, March 29, 2004 Propagation Events Normally, a propagator p is resumed whenever some value in a domain of var(p) has been removed. In some cases, some events (e.g. removing internal values) are irrelevant whilst other (bounds adjustments) are relevant.

ESOP, March 29, 2004 Idempotent Propagators A propagator is idempotent if it always computes a fixpoint. Most constraint programming systems can accommodate both idempotent and non- idempotent propagators.

ESOP, March 29, 2004 Implementing Propagators Implementation uses operations on variables reading domain information filtering domains (removing values) Variables are the only communication channels between propagators Algorithms for Domain filtering Failure detection Entailment detection

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Classes of Constraints Basic constraints Constraints for which the solver is complete x  D, x = v, x = y (variable aliasing) Primitive constraints (need propagators) Non-decomposable constraints x<y, x  y, x+y = z, x*y = z, … Global constraints (need propagators) Subsume a set of basic or primitive constraints, usually providing stronger consistency

ESOP, March 29, 2004 Support and Consistency Given: constraint C, variable x  var(C), its domain D(x), integer v. x=v has support for C iff v  D(x) C has a solution such that x=v C is hyperarc consistent iff  x  var(C)  v  D(x) x=v has support for C Maintaining hyperarc consistency may not be possible with polynomial algorithms (e.g. diophantine equations)

ESOP, March 29, 2004 Entailment A constraint con(p) is entailed if it holds for any combination of values in the current domains. Consequences for its propagator p: It has no more work to do It should not be resumed any more (up to backtracking) It is usually reponsible for detecting entailment

ESOP, March 29, 2004 Failure A constraint con(p) is false if it does not hold for any combination of values in the current domains. Consequences for its propagator p: It should signal inconsistency, e.g. by instigating backtracking It is reponsible for detecting failure

ESOP, March 29, 2004 Notation Vectors and subvectors X = (x 0,…,x n-1 ) X [0,r) = (x 0,…,x r-1 ), r  n Domain variables D(x), the domain of x (set of integers) min(x), lower bound of x, O(1) max(x), upper bound of x, O(1) prev(x,b) = max{y  D(x) | y<b}, O(d) next(x,b) = min{y  D(x) | y>b}, O(d)

ESOP, March 29, 2004 Constraint Signatures The constraint store is the set of all domains D(x) For alphabet A, constraint C, constraint store G, let S(C,G,A) be the signature of C wrt. G and A. The filtering algorithm is derived from a finite automaton for signatures.

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Definition: X  lex Y Let: X = (x 0,…,x n-1 ) Y = (y 0,…,y n-1 ) x i, y i domain variables or integers X  lex Y holds iff n=0, or x 0 <y 0, or x 0 =y 0  (x 1,…,x n-1 )  lex (y 1,…,y n-1 ).

ESOP, March 29, 2004 Signature: X  lex Y LetterCondition <max(x i )<min(y i ) =x i = y i, integers >min(x i )>max(y i )  max(x i )=min(y i )  min(x i )<max(y i )  min(x i )=max(y i )  max(x i )>min(y i ) ?otherwise $End of string

ESOP, March 29, 2004 Signature example: X  lex Y X Y S=  ?<>$

ESOP, March 29, 2004 Poset of signature letters <=> ?  E.g., a  becomes a < or a = in a ground store.

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Finite Automaton for X  lex Y 1234 T1T3T2 F1D1D3D2   $ $$ $$ $$     

ESOP, March 29, 2004 Success State T1 1 T1  $$ Enforce x i =y i in the leading prefix for C to hold. Afterwards, the leading prefix is ground and equal.

ESOP, March 29, 2004 Success State T2 124 T2      <  q Enforce x q <y q in order for there to be at least one < preceding the first >.

ESOP, March 29, 2004 Success State T3 123 T3  $$ $$     q Only enforce x q  y q, for < can appear in a later position.

ESOP, March 29, 2004 Delay States 1234 T3T2 D1D3D2   $ $$ $$      q Not yet enough information to know what to do at position q. T1 $$

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Filtering Algorithms Non-incremental, O(n) Run finite automaton from scratch Consider all letters from scratch Incremental, amortized O(1) Deal with one letter change at a time Needs to know what letter has changed, in what state

ESOP, March 29, 2004 Incremental Restart T1T3T2 F1D1D3D2   $ $$ $$ $$      Resume in state 1.

ESOP, March 29, 2004 Incremental Restart T1T3T2 F1D1D3D2   $ $$ $$ $$      Resume in state 2.

ESOP, March 29, 2004 Incremental Restart T1T3T2 F1D1D3D2   $ $$ $$ $$      Resume in state 3 or 4, resp.

ESOP, March 29, 2004 Incremental Restart T1T3T2 F1D1D3D2   $ $$ $$ $$      If changed to =, no-op. Otherwise, resume in state 3 or 4, resp.

ESOP, March 29, 2004 Finite Automaton for X < lex Y 1234 T1T3T2 F1D1D3D2  $$ $$ $  $      

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Definition: lex_chain(X 0,…,X m-1 ) Let: X i = (x i0,…,x in-1 ) x ij domain variables or integers lex_chain(X 0,…,X m-1 ) holds iff X 0  lex …  lex X m-1

ESOP, March 29, 2004 Internal constraint: between(A,X,B) Preconditions: A = (a 0,…,a n-1 ), B = (b 0,…,b n-1 ) X = (x 0,…,x n-1 ) a i,b i integers; x i domain variables  i  [0,n) : a i  D(x i ), b i  D(x i ) Holds iff: A  lex X  lex B

ESOP, March 29, 2004 Signature: between(A,X,B) LetterCondition < a i <b i  next(x i,a i )  b i « a i <b i  next(x i,a i )<b i =min(x i )=a i =b i =max(x i ) # min(x i )  a i =b i  max(x i )  a i =b i > a i >b i  b i  min(x i )  max(x i )  a i » a i >b i  (min(x i )<b i  a i <max(x i )) $End of string

ESOP, March 29, 2004 Signature example: between(A,X,B) A5466 X B5544 S#<>»$ X’ {4,6}{3,4,6,7}

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Finite Automaton: between(A,X,B) 12 T1T2 F1 =#=# >»>» «$<«»#$ >= < State 1 denotes a prefix in which a i =b i. Hence we must enforce x i =a i =b i there.

ESOP, March 29, 2004 Success State T1: between(A,X,B) 1 T1 =#=# «$ Either q=n or x q =v has support for all a q  v  b q. Hence we enforce a q  x q  b q. q

ESOP, March 29, 2004 Success State T2: between(A,X,B) 12 T2 =#=# <«»#$ >= < We have: X [0,r) =A [0,r)  X [0,r) =B [0,r) Hence we enforce: a i  x i  b i for i  [0,r). Either r=n or x r =v has support for all v  b r  v  a r. Hence we enforce: x r  b r  x r  a r. r

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Feasible Upper Bound: Problem Given X and B, compute the lexicographically largest U such that: U  lex B  i  [0,n) : u i  D(x i ) Similarly for feasible lowest bound

ESOP, March 29, 2004 Feasible Upper Bound: Algorithm Compute  as the smallest i such that U [0,i)  B [0,i) Compute: u i = b i, if i <  u i = prev(x i,b i ), if i =  u i = max(x i ), if i >  Similarly for feasible lowest bound

ESOP, March 29, 2004 Filtering: lex_chain(X 0,…,X m-1 ) 1. Compute feasible upper bound B i for X i from i=m-1 down to i=0. 2. Compute feasible lower bound A i for X i from i=0 to i=m Enforce A i  lex X i  lex B i for all i. Arc-consistency in O(nmd) time.

ESOP, March 29, 2004 Outline Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X  lex Y Definition and signature Finite automaton Filtering algorithm Case Study: lex_chain(X 0,…,X m-1 ) Definition and signature Finite automaton Filtering algorithm Conclusion

ESOP, March 29, 2004 Results An approach to designing filtering algorithms by derivation from FAs on constraint signatures Case studies and hyperarc consistency algorithms for two constraints: X  lex Y, running in amortized O(1) time per propagation event lex_chain(X 0,…,X m-1 ), running in O(nmd) time per invocation

ESOP, March 29, 2004 Future Work What constraints are amenable to the approach? Where does the alphabet come from? Where does the automaton come from? Where do the pruning rules come from? How do we make the algorithms incremental?

ESOP, March 29, 2004 References and proofs SICS T : Revisiting the Lexicographic Ordering Constraint, Mats Carlsson, Nicolas Beldiceanu. SICS T : Arc-Consistency for a Chain of Lexicographic Ordering Constraints, Mats Carlsson, Nicolas Beldiceanu.

ESOP, March 29, 2004 Related Work Global constraints for lexicographic orderings. A. Frisch, B. Hnich, Z. Kızıltan, I. Miguel, T. Walsh. Proc. CP’2002. LNCS 2470, pp , Springer, 2002.