inconsistency tolerance in sneps

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Inconsistency Tolerance in SNePS

Stuart C. Shapiro Department of Computer Science and Engineering,

and Center for Cognitive Science

University at Buffalo, The State University of New York

201 Bell Hall, Buffalo, NY 14260-2000

[email protected]

http://www.cse.buffalo.edu/~shapiro/

June, 2003 S. C. Shapiro 2

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Acknowledgements

• João Martins

• Frances L. Johnson

• Bharat Bhushan

• The SNePS Research Group

• NSF, Instituto Nacional de Investigação Cientifica, Rome Air Development Center, AFOSR, U.S. Army CECOM


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OutlineIntroductionSome Rules of Inference~I and Belief RevisionCredibility Ordering and Automatic BRReasoning in Different ContextsDefault Reasoning by Preferential OrderingSummary


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SNePS

• A logic- and network-based

• Knowledge representation

• Reasoning

• And acting• System [Shapiro & Group ’02]

This talk will ignore network and acting aspects.


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Logic

• Based on R, the logic of relevant implication

[Anderson & Belnap ’75; Martins & Shapiro ’88, Shapiro ’92]


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Supported wffs

P{… <origin tag, origin set> …}

hyp hypothesisder derived

Set of hypothesesFrom which Phas been derived.

Origin set tracks relevance and ATMS assumptions.


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Rules of Inference:Hypothesis

Hyp: P {<hyp,{P}>}

: whale(Willy) and free(Willy). wff3: free(Willy) and whale(Willy) {<hyp,{wff3}>}


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Rules of Inference:&E

&E: From A and B {<t,s>}

infer A {<der,s>} or B {<der,s>}

wff3: free(Willy) and whale(Willy) {<hyp,{wff3}>}

: free(Willy)? wff2: free(Willy) {<der,{wff3}>}


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Rules of Inference:andorE

The os is the union of os's of parents


wff6:all(x)(andor(0,1){manatee(x), dolphin(x), whale(x)})

{<hyp,{wff6}>}

: dolphin(Willy)?

wff9: ~dolphin(Willy) {<der,{wff3,wff6}>}

At most 1


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Rules of Inference:=>E

The origin set is the union of os's of parents.

Since wff10: all(x)(whale(x) => mammal(x)) {<hyp,{wff10}>}

and wff1: whale(Willy){<der,{wff3}>}

I infer wff11: mammal(Willy) {<der,{wff3,wff10}>}


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Rules of Inference:=>I

origin set is diff of os's of parents.

wff12: all(x)(orca(x) => whale(x)) {<hyp,{wff12}>}

: orca(Keiko) => mammal(Keiko)?

Let me assume that wff13: orca(Keiko) {<hyp,{wff13}>}

Since wff12: all(x)(orca(x) => whale(x)) {<hyp,{wff12}>}and wff13: orca(Keiko){<hyp,{wff13}>}

I infer whale(Keiko) {<der,{wff12,wff13}>}


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Rules of Inference:=>I (cont’d)

origin set is diff of os's of parents.

Since wff10: all(x)(whale(x) => mammal(x)) {<hyp,{wff10}>}and wff16: whale(Keiko) {<der,{wff12,wff13}>}

I infer mammal(Keiko) {<der,{wff10,wff12,wff13}>}

Since wff14: mammal(Keiko) {<der,{wff10,wff12,wff13}>}was derived assuming

wff13: orca(Keiko) {<hyp,{wff13}>}I infer

wff15: orca(Keiko) => mammal(Keiko) {<der,{wff10,wff12}>}


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~I and Belief Revision

• ~I triggered when a contradiction is derived.

• Proposition to be negated must be one of the hypotheses underlying the contradiction.

• Origin set is the rest of the hypotheses.

• SNeBR [Martins & Shapiro ’88] involved in choosing the culprit.


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Adding Inconsistent Hypotheses

wff19: all(x)(whale(x) => fish(x)){<hyp,{wff19}>}

wff20: all(x)(andor(0,1){mammal(x), fish(x)})

{<hyp,{wff20}>}

wff21: all(x)(fish(x) <=> has(x,scales))

{<hyp,{wff21}>}


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Finding the Contradiction: has(Willy, scales)?Since wff19: all(x)(whale(x) => fish(x)) {<hyp,{wff19}>}and wff1: whale(Willy) {<der,{wff3}>}

I infer fish(Willy) {<der,{wff3,wff19}>}

Since wff21: all(x)(fish(x) <=> has(x,scales)) {<hyp,{wff21}>}and wff23: fish(Willy) {<der,{wff3,wff19}>}

I infer has(Willy,scales) {<der,{wff3,wff19,wff21}>}

Since wff20: all(x)(andor(0,1){mammal(x), fish(x)}) {<hyp,{wff20}>}and wff11: mammal(Willy) {<der,{wff3,wff10}>}

I infer it is not the case that wff23: fish(Willy)


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Manual Belief RevisionA contradiction was detected within context default-defaultct.

The contradiction involves the newly derived proposition: wff24: ~fish(Willy) {<der,{wff3,wff10,wff20}>} and the previously existing proposition: wff23: fish(Willy) {<der,{wff3,wff19}>}

You have the following options: 1. [c]ontinue anyway, knowing that a contradiction is derivable; 2. [r]e-start the exact same run in a different context which is not inconsistent; 3. [d]rop the run altogether.

(please type c, r or d)=><= r


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BR AdviceIn order to make the context consistent you must delete

at least one hypothesis from the set listed below.This set of hypotheses is known to be inconsistent: 1 : wff20: all(x)(andor(0,1){mammal(x),fish(x)}) {<hyp,{wff20}>}

(1 dependent proposition: (wff24)) 2 : wff19: all(x)(whale(x) => fish(x)) {<hyp,{wff19}>}

(2 dependent propositions: (wff23 wff22)) 3 : wff10: all(x)(whale(x) => mammal(x)){<hyp,{wff10}>}

(3 dependent propositions: (wff24 wff15 wff11)) 4 : wff3: free(Willy) and whale(Willy) {<hyp,{wff3}>}

(8 dependent propositions: (wff24 wff23 wff22 wff11 wff9 wff5 wff2 wff1))

User deletes #2: wff19.


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Willy has no Scales

Since wff21: all(x)(fish(x) <=> has(x,scales))

{<hyp,{wff21}>}

and it is not the case that wff23: fish(Willy)

{<der,{wff3,wff19}>}

I infer it is not the case that

wff22: has(Willy,scales) {<der,{wff3,wff19,wff21}>}

wff26: ~has(Willy,scales){<der,{wff3,wff10,wff20,wff21}>}


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Final KB: hyps & positive ders: list-asserted-wffs


wff6: all(x)(andor(0,1){manatee(x),dolphin(x),whale(x)})

{<hyp,{wff6}>}

wff10: all(x)(whale(x) => mammal(x)) {<hyp,{wff10}>}


wff20: all(x)(andor(0,1){mammal(x),fish(x)}) {<hyp,{wff20}>}

wff21: all(x)(fish(x) <=> has(x,scales)) {<hyp,{wff21}>}

wff1: whale(Willy) {<der,{wff3}>}

wff2: free(Willy) {<der,{wff3}>}

wff11: mammal(Willy) {<der,{wff3,wff10}>}

wff15: orca(Keiko) => mammal(Keiko) {<der,{wff10,wff12}>}


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Final KB: hyps & negative ders: list-asserted-wffs


wff6: all(x)(andor(0,1){manatee(x),dolphin(x),whale(x)})

{<hyp,{wff6}>}



wff20: all(x)(andor(0,1){mammal(x),fish(x)}) {<hyp,{wff20}>}


wff9: ~dolphin(Willy) {<der,{wff3,wff10}>}

wff24: ~fish(Willy) {<der,{wff3,wff10,wff20}>}

wff25: ~(all(x)(whale(x) => fish(x))) {<ext,{wff3,wff10,wff20}>}

wff26: ~has(Willy,scales) {<der,{wff3,wff10,wff20,wff21}>}


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Summary

• Logic is paraconsistent:P{<t1, {h1 … hi}>},

~P{<t2, {h(i+1) … hn}>}

~hj

• When a contradiction is explicitly found, the user is engaged in its resolution.


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Credibility Ordering and Automatic Belief Revision*

• Hypotheses may be given sources.• Sources may be given relative credibility.• Hypotheses inherit relative credibility from

sources.• Hypotheses may be given relative

credibility directly. (Not shown.)• SNeBR may use relative credibility to

choose a culprit by itself. [Shapiro & Johnson ’00]

*Not yet in released version.


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Contradictory Sourceswff1: all(x)(andor(0,1){mammal(x),fish(x)}) {<hyp,{wff1}>}



: Source(Melville, all(x)(whale(x) => fish(x)).).

wff5: Source(Melville,all(x)(whale(x) => fish(x)))

{<hyp,{wff5}>}

: Source(Darwin, all(x)(whale(x) => mammal(x)).).

wff7: Source(Darwin,all(x)(whale(x) => mammal(x)))

{<hyp,{wff7}>}

: Sgreater(Darwin, Melville). wff8: Sgreater(Darwin,Melville) {<hyp,{wff8}>}


Note: Source & Sgreater props are regular object-language props.


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: has(Willy, scales)?

Since wff4: all(x)(whale(x) => fish(x)) {<hyp,{wff4}>}and wff9: whale(Willy) {<der,{wff11}>}I infer fish(Willy) {<der,{wff4,wff11}>}

Since wff2: all(x)(fish(x) <=> has(x,scales)) {<hyp,{wff2}>}and wff14: fish(Willy) {<der,{wff4,wff11}>}I infer has(Willy,scales)

Since wff6: all(x)(whale(x) => mammal(x)) {<hyp,{wff6}>}and wff9: whale(Willy) {<der,{wff11}>}I infer mammal(Willy)

Since wff1: all(x)(andor(0,1){mammal(x),fish(x)}) {<hyp,{wff1}>}

and wff15: mammal(Willy) {<der,{wff6,wff11}>}I infer it is not the case that

wff14: fish(Willy) {<der,{wff4,wff11}>}

Finding the Contradiction


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Automatic BRA contradiction was detected within context default-defaultct.The contradiction involves the newly derived proposition:

wff17: ~fish(Willy) {<der,{wff1,wff6,wff11}>}

and the previously existing proposition: wff14: fish(Willy) {<der,{wff4,wff11}>}

The least believed hypothesis: (wff4) The most common hypothesis: (nil) The hypothesis supporting the fewest wffs: (wff1)

I removed the following belief: wff4: all(x)(whale(x) => fish(x)) {<hyp,{wff4}>}

I no longer believe the following 2 propositions: wff14: fish(Willy) {<der,{wff4,wff11}>}



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Summary

• User may select automatic BR.

• Relative credibility is used.

• User is informed of lost beliefs.


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Reasoning in Different Contexts

• A context is a set of hypotheses and all propositions derived from them.

• Reasoning is performed within a context.• A conclusion is available in every context that

is a superset of its origin set. [Martins & Shapiro ’83]

• Contradictions across contexts are not noticed.


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Darwin Context

: set-context Darwin ()

: set-default-context Darwin

wff1: all(x)(andor(0,1){mammal(x),fish(x)})

{<hyp,{wff1}>}






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Melville Context

: describe-context((assertions (wff8 wff7 wff4 wff3 wff2 wff1)) (restriction nil) (named (science)))

: set-context Melville (wff8 wff7 wff3 wff2 wff1)((assertions (wff8 wff7 wff3 wff2 wff1)) (restriction nil) (named (melville)))

: set-default-context Melville((assertions (wff8 wff7 wff3 wff2 wff1)) (restriction nil) (named (melville)))

: all(x)(whale(x) => fish(x)). wff9: all(x)(whale(x) => fish(x)) {<hyp,{wff9}>}


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Melville: Willy has scales: has(Willy, scales)?

Since wff9: all(x)(whale(x) => fish(x)){<hyp,{wff9}>}and wff5: whale(Willy) {<der,{wff7}>}I infer fish(Willy) {<der,{wff7,wff9}>}

Since wff2: all(x)(fish(x) <=> has(x,scales))

{<hyp,{wff2}>}and wff11: fish(Willy) {<der,{wff7,wff9}>}I infer has(Willy,scales) {<der,{wff2,wff7,wff9}>}



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Darwin: No scales: set-default-context Darwin: has(Willy, scales)?

Since wff4: all(x)(whale(x) => mammal(x)) {<hyp,{wff4}>}and wff5: whale(Willy) {<der,{wff7}>}I infer mammal(Willy)

Since wff1: all(x)(andor(0,1){mammal(x),fish(x)}) {<hyp,{wff1}>}

and wff12: mammal(Willy) {<der,{wff4,wff7}>}I infer it is not the case that wff11: fish(Willy)

Since wff2: all(x)(fish(x) <=> has(x,scales)) {<hyp,{wff2}>}and it is not the case that wff11: fish(Willy)

{<der,{wff7,wff9}>}I infer it is not the case that wff10: has(Willy,scales)

wff15: ~has(Willy,scales) {<der,{wff1,wff2,wff4,wff7}>}


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Summary

• Contradictory information may be isolated in different contexts.

• Reasoning is performed in a single context.

• Results are available in other contexts.


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Default Reasoning by Preferential Ordering

• No special syntax for default rules.

• If P and ~P are derived– but argument for one is undercut by an

argument for the other– don’t believe the undercut conclusion.

• Unlike BR, believe the hypotheses, but not a conclusion.

[Grosof ’97, Bhushan ’03]


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Preclusion Rules in SNePS*

• P undercuts ~P if– Precludes(P, ~P) or

– Every origin set of ~P has some hyp h such that there is some hyp q in an origin set of P such that Precludes(q, h).

• Precludes(P, Q) is a proposition like any other.

*Not yet in released version.


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Animal Modes of Mobility

wff1: all(x)(orca(x) => whale(x))

wff2: all(x)(whale(x) => mammal(x))

wff3: all(x)(deer(x) => mammal(x))

wff4: all(x)(tuna(x) => fish(x))

wff5: all(x)(canary(x) => bird(x))

wff6: all(x)(penguin(x) => bird(x))

wff7: all(x)(andor(0,1){swims(x),flies(x),runs(x)})

wff8: all(x)(mammal(x) => runs(x))

wff9: all(x)(fish(x) => swims(x))

wff10: all(x)(bird(x) => flies(x))

wff11: all(x)(whale(x) => swims(x))

wff12: all(x)(penguin(x) => swims(x))


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Using Preclusion for Exceptions

wff13: Precludes(all(x)(whale(x) => swims(x)),

all(x)(mammal(x) => runs(x)))

wff14: Precludes(all(x)(penguin(x) => swims(x)),

all(x)(bird(x) => flies(x)))

wff15: orca(Willy)

wff16: tuna(Charlie)

wff17: deer(Bambi)

wff18: canary(Tweety)

wff19: penguin(Opus)


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Who Swims?(Contradictory Conclusions)

: swims(?x)?

I infer swims(Opus)

I infer swims(Charlie)

I infer swims(Willy)

I infer flies(Tweety)

I infer it is not the case that swims(Tweety)

I infer flies(Opus)

I infer it is not the case that wff20: swims(Opus)

I infer runs(Willy)

I infer it is not the case that wff24: swims(Willy)

I infer runs(Bambi)

I infer it is not the case that swims(Bambi)


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Using Preclusionto Arbitrate Contradictions (1)

Since wff13: Precludes(all(x)(whale(x) => swims(x)), all(x)(mammal(x) => runs(x)))

and wff11: all(x)(whale(x) => swims(x)) {<hyp,{wff11}>} holds within the BS defined by context default-defaultct

Therefore wff34: ~swims(Willy)containing in its support wff8: all(x)(mammal(x) => runs(x))

is precluded by wff24: swims(Willy)that contains in its support wff11:all(x)(whale(x) => swims(x))


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Using Preclusionto Arbitrate Contradictions (2)

Since wff14: Precludes(all(x)(penguin(x) => swims(x)),

all(x)(bird(x) => flies(x)))

and wff12: all(x)(penguin(x) => swims(x))

holds within the BS defined by context default-defaultct

Therefore wff31: ~swims(Opus)

containing in its support

wff10:all(x)(bird(x) => flies(x))

is precluded by wff20: swims(Opus)

that contains in its support

wff12: all(x)(penguin(x) => swims(x))


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The Swimmersand Non-Swimmers

wff38: ~swims(Bambi) {<der,{wff3,wff7,wff8,wff17}>}

wff28: ~swims(Tweety) {<der,{wff5,wff7,wff10,wff18}>}

wff24: swims(Willy) {<der,{wff1,wff11,wff15}>}

wff22: swims(Charlie) {<der,{wff4,wff9,wff16}>}

wff20: swims(Opus) {<der,{wff12,wff19}>}


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Two-Level Preclusion

wff1: all(x)(robin(x) => bird(x))wff2: all(x)(kiwi(x) => bird(x))

wff3: all(x)(bird(x) => flies(x))wff4: all(x)(bird(x) => (~flies(x)))

wff5: all(x)(robin(x) => flies(x))wff6: all(x)(kiwi(x) => (~flies(x)))

Example from Delgrande & Schaub ‘00


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Preferenceswff7: Precludes(all(x)(robin(x) => flies(x)), all(x)(bird(x) => (~flies(x))))wff8: Precludes(all(x)(kiwi(x) => (~flies(x))), all(x)(bird(x) => flies(x)))

wff12: (~location(New Zealand)) => Precludes(all(x)(bird(x) => flies(x)), all(x)(bird(x) => (~flies(x))))wff14: location(New Zealand) => Precludes(all(x)(bird(x) => (~flies(x))), all(x)(bird(x) => flies(x)))

wff10: ~location(New Zealand)wff15: Precludes(location(New Zealand), ~location(New Zealand))


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Who flies?

wff16: robin(Robin)

wff17: kiwi(Kenneth)

wff18: bird(Betty)

: flies(?x)?


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Outside New Zealand

wff24: ~flies(Kenneth){<der,{wff6,wff17}>,

<der,{wff2,wff4,wff17}>,

<der,{wff2,wff4,wff6,wff17}>}

wff21: flies(Robin) {<der,{wff5,wff16}>,

<der,{wff1,wff3,wff16}>}

wff19: flies(Betty) {<der,{wff3,wff18}>}


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Inside New Zealand: location("New Zealand").

wff9: location(New Zealand)

: flies(?x)?

wff24: ~flies(Kenneth) {<der,{wff6,wff17}>,

<der,{wff2,wff4,wff17}>,

<der,{wff2,wff4,wff6,wff17}>}

wff21: flies(Robin) {<der,{wff5,wff16}>,

<der,{wff1,wff3,wff16}>}

wff20: ~flies(Betty) {<der,{wff4,wff18}>}


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Summary• Contradictions may be handled by DR

instead of by BR.

• Hypotheses retained; conclusion removed.

• DR uses preferential ordering among contradictory conclusions or among supporting hypotheses.

• Precludes forms object-language proposition that may be reasoned with or reasoned about.


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SummaryInconsistency Tolerance in SNePS

• Inconsistency across contexts is harmless.• Inconsistency about unrelated topic is harmless.• Explicit contradiction may be resolved by user.• Explicit contradiction may be resolved by

system using relative credibility of propositions or sources.

• Explicit contradiction may be resolved by system using preferential ordering of conclusions or hypotheses.


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For more information

http://www.cse.buffalo.edu/sneps/


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References IA. R. Anderson, A. R. and N. D. Belnap, Jr. (1975) Entailment Volume I

(Princeton: Princeton University Press).

B. Bhushan (2003) Preferential Ordering of Beliefs for Default Reasoning, M.S. Thesis, Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY.

J. P. Delgrande and T. Schaub (2000) The role of default logic in knowledge representation. In J. Minker, ed. Logic-Based Artificial Intelligence (Boston: Kluwer Academic Publishers) 107-126.

B. N. Grosof (1997) Courteous Logic Programs: Prioritized Conflict Handling for Rules, IBM Research Report RC 20836, revised.


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References IIJ. P. Martins and S. C. Shapiro (1983) Reasoning in multiple belief spaces,

Proc. Eighth IJCAI (Los Altos, CA: Morgan Kaufmann) 370-373. J. P. Martins and S. C. Shapiro (1988) A model for belief revision, Artificial

Intelligence 35, 25-79.

S. C. Shapiro (1992) Relevance logic in computer science. In A. R. Anderson, N. D. Belnap, Jr., M. Dunn, et al. Entailment Volume II (Princeton: Princeton University Press) 553-563.

S. C. Shapiro and The SNePS Implementation Group (2002)

SNePS 2.6 User's Manual, Department of Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY.

S. C. Shapiro and F. L. Johnson (2000) Automatic belief revision in SNePS. In C.

Baral & M. Truszczyński, eds., Proc. 8th International Workshop on Non-Monotonic Reasoning.

inconsistency tolerance in sneps

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