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

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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

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

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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}

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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

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

Outline

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

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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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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}

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

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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}

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

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Brass et al’s method: an example

a not b d e

b not a e d

c f not d

I+ = facts(P) = {c} I- = HB – heads(P) = {g}

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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Brass et al’s method: an example

a not b d e

b not a e d

c f not d

I+ = facts(P) = {c} I- = HB – heads(P) = {g}

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

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Brass et al’s method: an example

a not b d e

b not a e d

c f not d

I+ = facts(P) = {c} I- = HB – heads(P) = {g}

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

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Brass et al’s method: an example

a not b d e

b not a e d

c f not d

I+ = facts(P) = {c} I- = HB – heads(P) = {g}

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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Brass et al’s method: an example

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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Brass et al’s method: an example

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

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

Outline

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

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WFSX

• Extended LP: two negationsnot 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 ppp 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

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

Outline

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

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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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Coherence transformations

• 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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Coherence transformations

p not qq not pq pp

I+ = { }I- = { p }

p

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Coherence transformations

p not qq not pq pp

I+ = { }I- = { p }

p

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

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Coherence transformations

p not qq

p

I+ = { }I- = { }

p , q

N Delete rules containing not q in the body

p , q

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Coherence transformations

• 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 aa

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Coherence transformations

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

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

Outline

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

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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