A rewritting method for Well-Founded Semantics with Explicit Negation Pedro Cabalar University of Corunna, SPAIN.

A rewritting method for Well-Founded Semantics with Explicit Negation

Pedro Cabalar

University of Corunna, SPAIN.

2

Introduction

• Logic programming (LP) semantics for default negation:

– Stable models [Gelfond&Lifschitz88]

– Well-Founded Semantics (WFS) [van Gelder et al. 91]

• Bottom-up computation for WFS [Brass et al. 01]

– More efficient than van Gelder’s alternated fixpoint

– Based on program transformations

3

Introduction

• Extended Logic Programming:default negation (not p) plus explicit negation ( ) :

– Answer Sets [Gelfond&Lifschitz91]

– WFS with explicit negation (WFSX) [Pereira&Alferes92]

p

• Our work: extend Brass et al’s method to WFSX

– Adding two natural transformations

– Helps to understand relation WFS vs. WFSX

Outline

Some LP definitions Brass et al’s method WFSX Coherence transformations Conclusions

Outline


6

Some LP definitions

• Logic program P: set of rules like a b , not c

c not b

b

• Reduct PI: we use I to interprete all ‘not p’. Example: take I={a,b}

7

Some LP definitions

• Logic program P: set of rules like a b , not c

c not b

b

• Reduct PI: we use I to interprete all ‘not p’. Example: take I={a,b}

(I) = least model of PI

• Stable model: any fixpoint I = (I)

• Well-founded model (WFM):

– Positive atoms I+ = least fixpoint of – Negative atoms I- = HB – greatest fixpoint of

l.f.p.g.f.p.+

-

HB

Outline


Outline


10

Brass et al’s method

• Trivial interpretation: a 3-valued interpretation where

– Positive atoms I+ = facts(P)

– Negative atoms I- = HB – heads(P)

• We exhaustively apply 5 program transformationsP N S F L

• The trivial interpretation of the final program will bethe WFM

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Brass et al’s method: an example

a not b , c d not g , e

b not a e not g , d

c f not d

d not c f g , not e

I+ = facts(P) = {c}

I- = HB – heads(P) = {g}

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b not a e not g , d

c f not d

d not c f g , not e

I+ = facts(P) = {c}


SSuccess: delete c from bodiesNegative reduction: delete rules with not c in the bodyN

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b not a e not g , d

c f not d

d not c f g , not e

I+ = facts(P) = {c}


PPositive reduction: delete not g from bodiesFailure: delete rules with g in the bodyF

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a not b d e

b not a e d

c f not d

I+ = facts(P) = {c}


Interesting property: exhausting {P,N,S,F} yields Fitting’s model… but for WFS we must get rid of positive cycles (d,e)

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a not b d e

b not a e d

c f not d

I+ = facts(P) = {c}


LPositive loop detection: delete rules with some p ()optimistic viewing: “what if all not’s happened to be true?”

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a not b d e

b not a e d

c f not d

I+ = facts(P) = {c}


LPositive loop detection: delete rules with some p ()() = {a, b, c, f }

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a not b d e

b not a e d

c f not d

I+ = facts(P) = {c}


LPositive loop detection: delete rules with some p ()() = {a, b, c, f } i.e. delete rules with some {d, e, g}

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a not b

b not a

c f not d

I+ = facts(P) = {c}

I- = HB – heads(P) = {g, e, d}

P

... we must go on until no new transformation is applicable.

Positive reduction: delete not d from bodies

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I+ = facts(P) = {c, f }

I- = HB – heads(P) = {g, e, d }

We can’t go on: ge get the WFM!

a not b

b not a

c f not d

Outline


Outline


22

WFSX

• Extended LP: two negations

not p “p is not known to be true”

“p is known to be false”p

• Objective literal L is any p or . We’ll denote L s.t. = pp p

• Answer sets: reject stable models containing both p and p

• WFS Coherence problem: should imply not pp

p not qq not pp

WFM+ = { }WFM- = { }

pq

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WFSX

• Given P we define its seminormal version Ps

p not qq not pp

p not q, not p q not p, not q not pp

P Ps

• The well-founded model is defined now as:

– Positive atoms I+ = least fixpoint of s

– Negative atoms I- = s(I+)

• In the example, we get I+ = { , q } I- = { p, }p q

Outline


Outline


26

Coherence transformations

• We begin redefining trivial interpretation ...

– I+ = facts(P) = { p }

– I- = HB – heads(P) = { , }a b

a not bb not a bpp

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• We begin redefining trivial interpretation ...

– I+ = facts(P) = { p }

– I- = HB – heads(P) { L | L facts(P) } = { , , }a b

a not bb not a bpp

p

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p not qq not pq p

p

I+ = { }I- = { p }

p

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p not qq not pq p

p

I+ = { }I- = { p }

p

RCoherence reduction: delete not p from bodiesCoherence Failure: delete rules with p in the bodyC

30


p not qq

p

I+ = { }I- = { }

p , q

NDelete rules containing not q in the body

p , q

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• Theorem 2: transformations {P,S,N,F,L,C,R} are sound w.r.t. WFSX

• Theorem 3: Let W be the WFM under WFS:(i) if W contradictory (p, p W+) then P contradictory in

WFSX(ii) the WFM under WFSX contains more or equal info than W

• The converse of (i) does not hold ...

• Corollary: when WFS leads to complete (and not contradictory) WFM it coincides with WFSX

a not a

a

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Theorem 4 (main result)

Given P ... P' where

x {P, S, N, F, L, C, R}

P' is the final program (free of contradictory facts)

The trivial interpretation of P' is the WFM of P under WFSX.

x x

Outline


Outline


35

Conclusions

• We added two natural transformations w.r.t. coherence:

"whenever L founded, L unfounded"

• Used and implemented for applying WFSX to causal theories of actions [Cabalar01]

• Can be used as slight efficiency improvement for answer sets?

• Explore a new semantics: Fitting's + coherence transformations

A rewritting method for Well-Founded Semantics with Explicit Negation Pedro Cabalar University of Corunna, SPAIN.

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