1 Constraint Programming Maurizio Gabbrielli Universita’ di Bologna Slides by: K. Marriott

2 Overview u Constraints (what they are) u Constraint solvers u Finite constraints domains: CSP u Constraint Logic Programming u Modeling with Constraints u Practical problems

3 Overview: part 2 u Concurrent constraint programming (ccp) u Timed extensions of ccp (tccp) u A proof system for (timed) ccp u Expressive power u Constraint Handling Rules (CHR)

4 Constraints u What are constraints? u Modelling problems u Constraint solving u Tree constraints u Other constraint domains u Properties of constraint solving

5 Constraints Variable: a place holder for values Function Symbol: mapping of values to values Relation Symbol: relation between values

6 Constraints Primitive Constraint: constraint relation with arguments Constraint: conjunction of primitive constraints (full FOL formula in some cases)

7 Satisfiability Valuation: an assignment of values (of a suitable domain) to variables Solution: valuation which satisfies constraint

8 Satisfiability Satisfiable: constraint has a solution Unsatisfiable: constraint does not have a solution satisfiable unsatisfiable

9 Constraints as Syntax u Constraints are strings of symbols u Brackets don't matter (don't use them) u Order does matter (!, so not really FOL) u Some algorithms will depend on order

10 Equivalent Constraints Two different constraints can represent the same information Two constraints are equivalent if they have the same set of solutions

11 Modelling with constraints u Constraints describe idealized behaviour of objects in the real world

12 Modelling with constraints start foundations interior walls exterior walls chimney roof doors tiles windows

13 Constraint Satisfaction u Given a constraint C two questions u satisfaction: does it have a solution? u solution: give me a solution, if it has one? u The first is more basic u A constraint solver answers the satisfaction problem

14 Constraint Satisfaction u How do we answer the question? u Simple approach try all valuations.

15 Constraint Satisfaction u The enumeration method wont work for Reals (why not?) u A smarter version will be used for finite domain constraints u How do we solve Real constraints u Remember Gauss-Jordan elimination from high school

16 Gauss-Jordan elimination u Choose an equation c from C u Rewrite c into the form x = e u Replace x everywhere else in C by e u Continue until u all equations are in the form x = e u or an equation is equivalent to d = 0 (d != 0) u Return true in the first case else false

17 Gauss-Jordan Example 1 Replace X by 2Y+Z-1 Replace Y by -1 Return false

18 Gauss-Jordan Example 2 Replace X by 2Y+Z-1 Replace Y by -1 Solved form : constraints in this form are satisfiable

19 Solved Form u Non-parametric variable: appears on the left of one equation. u Parametric variable: appears on the right of any number of equations. u Solution: choose parameter values and determine non-parameters

20 Tree Constraints u Tree constraints represent structured data u Tree constructor: character string u cons, node, null, widget, f u Constant: constructor or number u Tree: u A constant is a tree u A constructor with a list of > 0 trees is a tree u Drawn with constructor above children

21 Tree Examples order(part(77665, widget(red, moose)), quantity(17), date(3, feb, 1994)) cons(red,cons(blue,con s(red,cons(…))))

22 Tree Constraints u Height of a tree: u a constant has height 1 u a tree with children t1, …, tn has height one more than the maximum of trees t1,…,tn u Finite tree: has finite height u Examples: height 4 and height

23 Terms u A term is a tree with variables replacing subtrees u Term: u A constant is a term u A variable is a term u A constructor with a list of > 0 terms is a term u Drawn with constructor above children u Term equation: s = t (s,t terms)

24 Term Examples order(part(77665, widget(C, moose)), Q, date(3, feb, Y)) cons(red,cons(B,cons(r ed,L)))

25 Tree Constraint Solving u Assign trees to variables so that the terms are identical u cons(R, cons(B, nil)) = cons(red, L) u Similar to Gauss-Jordan u Starts with a set of term equations C and an empty set of term equations S u Continues until C is empty or it returns false

26 Tree Constraint Solving u unify u unify(C) u Remove equation c from C u case x=x: do nothing u case f(s1,..,sn)=g(t1,..,tn): return false u case f(s1,..,sn)=f(t1,..,tn): u add s1=t1,.., sn=tn to C u case t=x (x variable): add x=t to C u case x=t (x variable): add x=t to S u substitute t for x everywhere else in C and S

27 Tree Solving Example CS Like Gauss-Jordan, variables are parameters or non-parameters. A solution results from setting parameters (I.e T) to any value.

28 One extra case u Is there a solution to X = f(X) ? u NO! u if the height of X in the solution is n u then f(X) has height n+1 u Occurs check: u before substituting t for x u check that x does not occur in t

29 Other Constraint Domains u There are many u Boolean constraints u Sequence constraints u Blocks world u Many more, usually related to some well understood mathematical structure

30 Boolean Constraints Used to model circuits, register allocation problems, etc. An exclusive or gate Boolean constraint describing the xor circuit

31 Boolean Constraints Constraint modelling the circuit with faulty variables Constraint modelling that only one gate is faulty Observed behaviour: Solution:

32 Boolean Solver let m be the number of primitive constraints in C for i := 1 to n do generate a random valuation over the variables in C if the valuation satisfies C then return true endif endfor return unknown

33 Boolean Constraints u Something new? u The Boolean solver can return unknown u It is incomplete (doesnt answer all questions) u It is polynomial time, where a complete solver is exponential (unless P = NP) u Still such solvers can be useful!

34 Blocks World Constraints Objects in the blocks world can be on the floor or on another object. Physics restricts which positions are stable. Primitive constraints are e.g. red(X), on(X,Y), not_sphere(Y). floor Constraints don't have to be mathematical

35 Blocks World Constraints A solution to a Blocks World constraint is a picture with an annotation of which variable is which block Y X Z

36 Solver Definition u A constraint solver is a function solv which takes a constraint C and returns true, false or unknown depending on whether the constraint is satisfiable u if solv(C) = true then C is satisfiable u if solv(C) = false then C is unsatisfiable

37 Properties of Solvers u We desire solvers to have certain properties u well-behaved: u set based: answer depends only on set of primitive constraints u monotonic: is solver fails for C1 it also fails for C1 /\ C2 u variable name independent: the solver gives the same answer regardless of names of vars

38 Properties of Solvers u The most restrictive property we can ask u complete: A solver is complete if it always answers true or false. (never unknown)

39 Constraints Summary u Constraints are pieces of syntax used to model real world behaviour u A constraint solver determines if a constraint has a solution u Real arithmetic and tree constraints u Properties of solver we expect (well- behavedness)