Lparse Programs Revisited: Semantics and Representation of

Aggregates

Guohua Liu and Jia-Huai You

University of Alberta

Canada

Outline Stable models of weight-constraint programs - closely related to answer sets of Son-

Pontelli-Tu for LPs with constraint atoms - what about other lparse-stable models?

Some of them may be “circular” - a translation to avoid circularity

Can we represent commonly used aggregates by weight constraints?

Lparse programs

Weight constraint W :

Weight constraint rule:

where each is a weight constraint.

uwbwbwawalmn

bmbana ]not ,...,not ,,...,[ 11 11

nWWW ,...,10

iW

Semantics

The reduct of a weight constraint W w.r.t. M is the constraint

where

MW

Mb

b

i

iwll '

],...,[ '11 nana wawal

Semantics

The reduct of a weight constraint W w.r.t. M is the constraint

where The reduct is:

MW

Mb

b

i

iwll '

],...,[ '11 nana wawal

MP

}1 ,),(,)(

,,...,|,...,{

0

101

iallforuMWwMWlitp

PWWWWWpP

i

nM

nMM

Lparse-stable models may be “circular”

Consider the one-rule program:

a [not a = 1] 0

Both M1 = {} and M2= {a} are lparse-stable models.

In M2, we need assume a in order to derive a. The weight constraint in the body is actually

monotone.

Equivalent Expressions

This weight constraint is equivalent to each of the following (satisfaction-preserving):

a count({x | x D}) = 1 where D = {a}

a ({a}, {{a}}) (body is an abstract constraint atom)

Is non-minimal the culprit ?

Not always. Consider program P a [not a = 1] 0 f not f, not a

{a} is now minimal, but still circular.

Strongly satisfiable weight constraints

Notation: W is a weight constraint; M is a set of atoms

A weight constraint W is strongly satisfiable by M iff

M |= W implies, for any ,

W is strongly satisfiable by any M if -lit(W) contains no negative literal - upper-bound free

)}( |{)( WlitbnotMbWM iib

uVMWw )\,()(WMV b

Theorem

Let P be an lparse program and M a set of atoms. Suppose all the weight constraints appearing in the body of any rule in P are strongly satisfiable by M.

Then, M is an lparse-stable model of P iff M is an answer set (in the sense of Son et al.) for P.

Weak notion of non-circularity(unfoundedness)

Definition (essentially that of Calimeri et al. IJCAI-05)

An lparse-stable model M of a program P is circular if

there is a non-empty set s.t. , M \U does

not satisfy the body of any rule r in P, where

U)(rhead

MU

Weak notion of non-circularity(unfoundedness)

Definition (essentially that of Calimeri et al. IJCAI-05) An lparse-stable model M of a program P is circular if there is a non-empty set s.t. , M \U does not satisfy the body of any rule r in P, where

Example a b 2 [a=1, not b =1] b [a=1, not b = 1] 1{a,b} is an lparse-stable model, but not an answer set,

and it is not circular by the above definition.

U)(rhead

MU

Transformation to strongly satisfiable programs

Let l {W }u denote a weight constraint. Transform it to the conjunction of

1. l {W }2. l’ {W}where

Example: [not a =1]0 transformed to [not a =1] and 1 [not a =1]

m

ib

n

ia ii

wwul11

'

Representation of Aggregates

xpxaggr Result op )})(|({

},,,,,{

Take the form

where aggr is from {Sum,Count,Avg,Min,Max} op is from Result is a numeric constant

Linear size encoding sum and account: straightforward Encode by

max and min: more complex, but can be done

What cannot be encoded? Aggregate expressions involving Product constraint:

])(,...,)([ 0 11 kaapkaap nn

kxpxavg )})(|({

kxpxTimes )})(|({

Experiments

Programs Argument Size Smodels Smodels-A Company Contr 20 0.03 0.09 Company Contr 40 0.18 0.36 Company Contr 80 0.87 2.88 Company Contr 120 2.40 12.14 Employee Raise 15/5 0.01 0.69 Employee Raise 21/15 0.05 4.65 Employee Raise 24/20 0.05 5.55 Party Invit. 80 0.02 0.05 Party Invit. 160 0.07 0.10 NM1 125 0.61 0.21 NM1 150 0.75 0.29 NM2 125 0.65 2.24 NM2 150 1.08 3.36

Experiments (2) Execution Time Instance Size

C T Smodels DLV-A Smodels DLV-A

4 3 0.1 0.01 293 248 4 4 0.2 0.01 544 490 5 5 0.58 0.02 1213 1346 5 10 0.35 0.31 6500 7559 5 15 1.24 1.88 18549 22049 5 20 3.35 7.08 40080 47946 5 25 8.19 64.29 73765 88781 5 30 16.42 152.45 12230 14767

Conclusion Lparse semantics is closely related to answer sets

by Son et al. The gap can be closed by a simple transformation.

Lparse programs are already an effective representation language for aggregates. It only needs a simple frond end.

More efficient implementation of weight constraints

is needed.