Constraint Logic Programming in CLP(B oolean N aturals R …web.ncf.ca/an386/publications/presentations/CLPBNR_NRC.pdf · Constraint Interval Arithmetic page 2 Computing Research

page 1Constraint Interval Arithmetic

BNRComputing Research Laboratory

National Research Council

Constraint Logic Programmingin CLP(Boolean Naturals Reals)

André Vellino

Computing Research LaboratoryBell-Northern Research

Ottawa

November 1993




Outline of the Talk

• What is Logic Programming?

• History and Motivation for CLP(BNR)

• Constraint Interval Arithmetic

• Applications and Plans for the Future




What is Logic Programming?(a) The expression of computer programs in an efficientlyprovable subset of first-order predicate calculus;

(b) A non-deterministic, relational and recursive language thatincorporates unification and a depth-first search algorithm;

(c) A language with no types, no assignment statements nocontrol constructs and no data-structures;

(d) A paradigm for which the equation:

Algorithms + Data Structures = Programs

is replaced by:

Knowledge + Question = Answer(s)




Prolog Programs

Answers found_in(ottawa, ontario). found_in(montreal, quebec). found_in(toronto, ontario). found_in(ontario, canada). found_in(quebec, canada). found_in(boston, mass). found_in(mass, usa). found_in(canada, north_america). found_in(usa, north_america). found_in(ottawa, canada). found_in(ottawa, north_america). found_in(montreal, canada). found_in(montreal, north_america). found_in(toronto, canada). found_in(toronto, north_america). found_in(ontario, north_america). found_in(quebec, north_america). found_in(boston, usa). found_in(boston, north_america). found_in(mass, north_america).

Knowledgelocated_in(ottawa,ontario).located_in(montreal,quebec).located_in(toronto,ontario).located_in(ontario,canada).located_in(quebec,canada).located_in(boston,mass).located_in(mass,usa).located_in(canada,north_america).located_in(usa,north_america).

found_in(A,B) :- located_in(A,B).found_in(X,Z) :- located_in(X,Y), found_in(Y,Z).

+Question?- found_in(Containee, Container).

=




Call Processing in Prolog

idling

calling

talking

ringing

dialansweroriginate

originate offhookanswer

release onhook

react(idling, dial, calling, originate ).react(idling, originate , ringing, none).react(ringing, offhook , talking, answer ).react(calling, answer , talking, none).react(talking, onhook , idling, release ).react(talking, release , idling, none).

react(_, onhook , idling, release ).react(_, release , idling, none).




Outline of the Talk








A Brief History ofCLP(BNR)

XMS-Prolog interpreter 1985-1987

Constraint Interval Arithmetic talk by John Cleary 1987

BNR-Prolog interpreter for Macintosh 1988

Panels UI toolkit 1989

Dassault Electronique develop InterLog 1990

BNR-Prolog compiler for Sun Workstations 1991

Constraints on Integers and Boolean CLP(BNR)-1992

Ports to DEC, HP 300-400, HP 700-1993




Related Systems

• CHIP -- Constraint Handling in Prolog (ECRC)

• CHARME / Decision Power

• CLP(R) IBM -- Yorktown Heights

• Prolog-III -- Marseille

• ILOG-Solver -- C++ library

• InterLog -- Dassault Electronique

• Screamer -- Non-determinsitic Common-LISP

•




Why Constraints?Software Development:

FasterSimpler programs;

Correctness easier to ascertain;More maintainable;Eliminates many difficult algorithm design problems;

One implementation can be used for different purposes;

Real World Modelling:

Can use uncertain or fuzzy data and tolerancesCan make effective use of redundant informationCan use approximation information without error

Sensitivity analysis is trivial




Applications of CLP(BNR)• manufacturing specifications (Dassault Electronique);

• temporal reasoning (BNR);

• analysis of gear ratios (Ford);

• hardware configuration (BNR)

• FPGA-gate allocation (BNR);

• hardware timing analysis (BNR);

• Others ... operations research;performance modeling (Carleton U.);x-ray crystallography




Theoretical Motivations

• Commutative

• Persistence

• Monotone

p(X) & q(X) q(X) & p(X)

P & ....& Q & P

"Ask a vaguer question get a vaguer answer"

P & ....& Q

("Ask a more precise question get a more precise answer")




Impure Logic Programs

• Side-effects

• Filters

• Arithmetic

Input/Output, Database Updates

var(X) (monotone but not persistent)nonvar(X) (does not commute)




Prolog Arithmetic

?- X is 2 *3.X = 6

?- 4 is X * X.NO

?- X is 2 * 3, Y is X * 2.X=6 Y=12

?- Y is X * 2, X is 2 * 3.NO

Functional

Not Relational

Not Logical(not commutative)




Conventional Arithmetic

Round off errors in floating point arithmetic causesprograms to produce incorrect results. e.g. 1.21 is 1.1 * 1.1 is false. (A + B) * C = (A * C) + (B * C) is not always true

Thus, it is never safe to combine programs that dosymbolic algebra with programs that evaluate themnumerically.




Constraint Arithmetic on Intervals

Intervals Correctness of Arithmetic Precision problems Mixing of symbolic and numeric methods

Constraints Commutativity of operations Interchangability of unknowns




Outline of the Talk








CLP(BNR)

A system which takes a set of relations over

Booleans Naturals and Reals, a set of initial bounds

on all the quantities and derives a set of posteriori

bounds on all the quantities using a local propagation,

arc-consistency based, fixed-point iteration.




Interval ArithmeticAn interval represents a real number that lies between twofloating point bounds.

This overcomes the problem of incorrect arithmetic, becauseround off errors can be accommodated.

X [2.5, 3.5]Y [2.6, 4.2]X+Y [5.0999,7.70001]




Interval Constraint Network

X

Y

Z

X + 3 == U

U

Y * U == V

V + X == Z V

Z == Y * (X + 3) + X X + 3 == UY * U == VV + X == Z

Source Code Constraint Network




Equality is Intersection?- X:real(2,4),Y:real(3,5), X == Y.X = [3,4], Y = [3,4]

?- X:real(2,4),Y:real(3,5), X == Y, X >= 3.5, Y =< 3.7.X = [3.5,3.7], Y = [3.5,3.7]

X Y




InequalityX =< Y




Relational Addition

?- X:real(3, 7), Y:real(2, 8), Z:real(4, 6), X + Y == Z.X = [3, 4]Y = [2, 3]Z = [5, 6]

Evaluating a relational expression on intervalsmay narrow them all.

Interpretation: ∀x ∈ X, ∃ y ∈Y, ∃ z ∈ Z such that x + y = z

∀ y ∈Y, ∃x ∈ X, ∃ z ∈ Z such that x + y = z

∀ z ∈ Z ∃x ∈ X, ∃ y ∈Y, such that x + y = z




Limitations of Local Propagation

?- [X,Y,Z]: real(-1,1), X + Y + Z == 0, X + Y + Z == 1.

Some constraint networks are not congenial to thenarrowing of intervals. For example the question

succeeds even though the two equations are inconsistent.

The inconsistency is detected numerically only whenfurther narrowing is attempted.




Redundant Constraints

Sometimes redundant constraints can help, e.g.

?- [X,Y,Z]: real(-1,1), T : real, X + Y + Z == T, T == 0, T == 1.

is false.




Searching for Solutions

?- [X,Y]:real, Y >= 3, Y =< 4,Y == X * X.

X = [-2, 2]

To narrow for more precise solutions,we need to search the solution spacealgorithmically (e.g. divide and conquer)

-2.0 -1.0 1.0 2.0

2.0

4.0

-2.0

Y

?- [X,Y]:real, Y >= 3, Y =< 4,Y == X * X, X >= 0 ; X =< 0.

X = [1.7320, 2.0001]X = [-2.0001, -1.7320]




Searching for SolutionsSince interval arithmetic is integrated in a Prolog systemit is possible to write your own relaxation algorithm.

For example, you can recursively constrain a giveninterval to be alternately less than or greater than itsmidpoint and thus 'scan' the interval for solutions.

e.g.

?- X:real(0, 1), 0 == 35 * X ** 256 - 14 * X ** 17 + X, solve(X).

X = [0.0, 0.0] ;X = [0.84794,0.84795] ;X = [0.99584,0.99585]




Integers and Booleans?- I:integer, Y:real(3.5,8.26), I**2==Y. ?- [I:integer, Y : real(3.5, 8.26), I**2 == Y]

where [I : integer(-2, 2), Y: 4].

?- B:boolean, 1 == (B and (C or ~D)) . ?- [1 : boolean, 1 == (1 and (C or ~ D))] where [C : boolean, D : boolean].

?- [A,B]:boolean, [X,Y]:real, J:integer,((X**2 + 2.5*J =< log(Y)) and A) == ~ B.

Disjunctive Scheduling Constraints

(Start1 + Dur1 =< Start2) xor (Start2 + Dur2 =< Start1) == 1




Formal Properties

• Commutative

• Persistent

• Monotone

"Once P implies henceforth P"

"Ask a more precise question get a more precise answer"

"The order of constraints doesn't matter"




Outline of the Talk








Configuration in CLP(BNR)

Problem

Configuration of Northern Telecom DMS switching equipment. Different shelves can contain line cards subject to capacity constraints;

Shelves have requirement and exclusion constraints.

Problem: how to distribute components to minimize number of shelves




Problem Specificationbin_types( [blue,red,green]).commodities( [glass, plastic, steel, wood, copper]).

requires(wood,plastic).excludes(glass, copper).excludes(copper,plastic).

capacity( red, wood, 1 ).capacity( green, wood, 2 ).

capacity(red, steel,0).capacity(red,plastic,0).capacity(blue,wood,0).capacity(blue,plastic,0).capacity(green,glass,0).capacity(green,steel,0).

capacity( red, 3).capacity( blue, 1).capacity( green, 4).




Automatic Translation into CLP(BNR)bin([Blue, Green, Red],[Glass,Plastic,Steel,Wood,Copper],Total) :-

% declarations[Blue,Green,Red] : boolean,[Glass,Plastic,Steel,Wood,Copper] : integer(0,_),BinSize:integer(1,4),

% global constraintsRed + Green + Blue == 1,BinSize == Red*3 + Green*4 + Blue*1, Total is (Glass + Plastic + Steel + Wood + Copper),BinSize >= Total,

% requires/excludes constraintsGlass exclusive Copper, Copper exclusive Plastic, if (Wood >= 1) then (Plastic >= 1), if Green then (Glass + Steel == 0), if Green then (Wood =< 2), if Blue then ((Wood + Plastic) == 0), if Red then ((Steel + Wood) == 0), if Red then (Wood =< 1).

X exclusive Y :- X * Y == 0.if A then B :- A =< B.




FPGA Gate Arrays

Problem: layout the functional components of a gate array so that:

(a) only one register can be enabled on the data bus(b) each function has only 4 inputs(c) arithmetic and compare functions must be routed locally(d) minimize data-bus traffic to/from registers....

Solution space large (10!)3-5 orders of magnitude faster than Test & Generate algorithmTypical configuration constraint problemDrawback: no explanation facility




Hardware Timing AnalysisProblem: to determine the “timing correctness” of a circuit without simulating its functional behaviour

Answer: represent circuit component delays, rising and falling edges of clock etc. as intervals

represent composite delays as arithmetic functions ofcomponent delays

set up circuits as sets of simultaneous arithmetic relations

constitutes a formal verification (monotonic)

Results: 5 pages of CLP(BNR), 3400 gates in circuit linear (average) performance without correlated delays 120s (HP700) quadratic (average) performance with correlated delays




Comparison w/ Other CLPs

• can express & solve systems of non-linear constraints• sound arithmetic ∴ o.k. to do symbolic manipulation• not very useful for under-constrained linear systems• allows for disjunctive scheduling with Booleans• finite-domain problems expressible with integer-intervals

CLP(BNR)

CHIP--restricted to integers and finite domains

CLP(R) -- restricted to linear equations, limited by floating point errors

Prolog-III -- restricted to linear equations and rational numbers

ILOG-Solver -- C++ library w/ clumsy non-determinism and control

Screamer -- Non-determinsitic Common-LISP




Future Plans:Constraint Programming Research with Carleton University

• Continue CLP(BNR) research as applied to: resource-allocation,scheduling,network configuration problems;

• 5 Professors (& ~10 Grad students) involved: Majumdar, Woodside, Pagurek, Chinneck and Karam;

• 5 Applied Projects:Performance Analysis and Engineering;Task and Resource Allocation for Concurrent Software;Network Diagnosis;Configuration Management;Task Management for Multiprocessor Systems;

• 2 Research Projects:Global Optimization;Flexible Scheduling.

Documents

Constraint Logic Programming in CLP(B oolean N aturals R …web.ncf.ca/an386/publications/presentations/CLPBNR_NRC.pdf · Constraint Interval Arithmetic page 2 Computing Research