# Constraint Programming. Master Parisien de Recherche en Informatique

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1 Constraint Programming Master Parisien de Recherche en Informatique 2008/2009 François Fages et Sylvain Soliman Project team Contraintes INRIA Paris-Rocquencourt fages François Fages 1

2 Constraint Programming Machine Computation on partial information structures. Von Neuman machine memory of values assignment variables Constraints machine memory of constraints mathematical variables V1 Xi in [3,15] Xi = Xj+2 Vi write Vi := Vj+1 Xi + 2*Xj >= b X_I = V constraint posting Vj read card(1, [X>=Y+5, Y>=X+3]) constraint entailment Xi >= 5? François Fages 2

3 The Paradigm of Constraint Logic Programming Program = Logical Formula Execution = Proof search Class of languages CLP(X ) parametrized by X : Axiomatization: Domain of discourse X, Model of the problem P Constraint satisfiability, Logical resolution principle Primitive Constraints over X U = R I Relations defined by logical formulae x, y path(x, y) edge(x, y) z(edge(x, z) path(z, y)) François Fages 3

4 Languages for defining new relations First-order logic predicate calculus x, y path(x, y) edge(x, y) z(edge(x, z) path(z, y)) Prolog/CLP(X ) clauses path(x,y):- edge(x,y). path(x,y):- edge(x,z), path(z,y). Concurrent constraint process languages CC(X ) Process A = c p(x) (A A) A + A ask(c) A xa path(x, Y ) :: edge(x, Y ) + Z(edge(X, Z) path(z, Y )) Constraint libraries in object-oriented/functional/imperative languages. François Fages 4

5 CLP(FD) N-queens Problem GNU-Prolog program: queens(n,l):- length(l,n), fd_domain(l,1,n), safe(l), fd_labeling(l,first_fail). safe([]). safe([x L]):- noattack(l,x,1), safe(l). noattack([],_,_). noattack([y L],X,I):- X#\=Y, X#\=Y+I, X+I#\=Y, I1 is I+1, noattack(l,x,i1). François Fages 5

6 Search space of all solutions François Fages 6

7 Successes in combinatorial search problems Job shop scheduling, resource allocation, graph coloring,... Decision Problems: existence of a solution (of given cost) in P: if algorithm of polynomial time complexity in NP: if non-deterministic algorithm of polynomial complexity. NP-complete if polynomial encoding of any other NP problem Optimisation Problems: computation of a solution of optimal cost NP-hard if the decision problem is NP-complete The size of the search space does not tell the complexity of the problem Sorting n elements in O(n log n), search space in!n... SAT over n Boolean in O(2 n ), search space in 2 n. François Fages 7

8 Hot Research Topics in Constraint Programming Combinatorial optimization problems Ressource allocation, scheduling, frequency allocation, geometrical placement problems,... Benchmarks (open shop 6x6, graph colouring, time tabling,...) New applications in Bioinformatics, Web, multimedia,... Algorithms Global constraints, hybridization of algorithms CP, MILP, local search Symmetry detection and breaking Search procedures, randomization, adaptive solving strategies Languages Modelling languages Extensible CP languages Automatic synthesis of constraint solvers François Fages 8

9 Panorama of CP Systems Prolog-based systems open source: CLP(H,FD) GNU-Prolog (Univ. Paris 5, INRIA) CC(H,FD) SWI-Prolog (CWI), Cia-Prolog (UPM),... LCC(H,FD) SiLCC (INRIA) commercial: CC(H,FD,Q,R) Sicstus-Prolog (SICS), Cosytec CHIP,... Java-based systems open source: Choco (EMN), CHR JCK toolkit (Univ. Ulm) commercial: JSolver, JConfigurator (ILOG), Koalog,... C++ based systems Ilog-Solver, Ilog-Scheduler, Cosytec Chip++ CAML-based system FaCiLe (Univ. Toulouse) François Fages 9

10 Workplan of the Lecture (1/2) 1. Introduction Constraint Programming Logical background 2. Constraint Solving Decidability of constraint languages and complete theories Constraint solving algorithms on H and R by rewriting Contraint propagation algorithms on FD Symmetry breaking 3. Constraint Logic Programs Operational semantics and examples CLP (FD) Fixed semantics and abstract interpretation Logical semantics, theorem proving and negation François Fages 10

11 Workplan of the Lecture (2/2) 4 Concurrent Constraint Languages Operational semantics and examples CC(FD) Denotational semantics of determinate and non-determinate CC programs Logical semantics in Linear Logic Phase model checking LL constraint systems and imperative features 17/11/2009 Small constraint programming project in GNU-Prolog 09/02/2010 Examination 16/02/2009 Start internships François Fages 11

12 Lecture notes Article Des contraintes au programme. Revue littraire Actes de Savoirs, PUF Chapter Programmation logique et contraintes Encyclopdie Vuibert d Informatique. Slides of lectures 1-4 Lecture notes on Constraint Logic Programming Slides of lectures 5-16 Krzysztof Apt, Principles of Constraint Programming, Cambridge University Press, François Fages 12

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