inconsistency tolerance in sneps

cse@buff

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

shapiro@cse.buffalo.edu

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

June, 2003 S. C. Shapiro 2

cse@buff

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

cse@buff

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

cse@buff

• A logic- and network-based

• Knowledge representation

• Reasoning

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

This talk will ignore network and acting aspects.

cse@buff

• Based on R, the logic of relevant implication

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

cse@buff

Supported wffs

P{… <origin tag, origin set> …}

hyp hypothesisder derived

Set of hypothesesFrom which Phas been derived.

Origin set tracks relevance and ATMS assumptions.

cse@buff

Rules of Inference:Hypothesis

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

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

cse@buff

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}>}

cse@buff

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

cse@buff

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}>}

cse@buff

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}>}

cse@buff

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}>}

cse@buff

~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.

cse@buff

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}>}

cse@buff

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)

cse@buff

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

cse@buff

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.

cse@buff

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}>}

cse@buff

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}>}

cse@buff

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}>}

cse@buff

Summary

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

~P{<t2, {h(i+1) … hn}>}

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

cse@buff

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.

cse@buff

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.

cse@buff

: 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

cse@buff

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}>}

cse@buff

Summary

• User may select automatic BR.

• Relative credibility is used.

• User is informed of lost beliefs.

cse@buff

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.

cse@buff

Darwin Context

: set-context Darwin ()

: set-default-context Darwin

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

{<hyp,{wff1}>}

cse@buff

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}>}

cse@buff

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}>}

cse@buff

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}>}

cse@buff

Summary

• Contradictory information may be isolated in different contexts.

• Reasoning is performed in a single context.

• Results are available in other contexts.

cse@buff

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]

cse@buff

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.

cse@buff

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))

cse@buff

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)

cse@buff

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)

cse@buff

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))

cse@buff

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))

cse@buff

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}>}

cse@buff

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

cse@buff

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))

cse@buff

Who flies?

wff16: robin(Robin)

wff17: kiwi(Kenneth)

wff18: bird(Betty)

: flies(?x)?

cse@buff

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}>}

cse@buff

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}>}

cse@buff

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.

cse@buff

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.

cse@buff

For more information

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

cse@buff

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.

cse@buff

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

Documents

knowledge representation and reasoning in sneps for...

enhanced assessment of epn station · pdf filetrimble...

inconsistency of faa interpretations - nata |...

pronoun resolution in sneps

reinventing the reinvented shakey in sneps

a sneps approach to the wumpus world agent or cassie meets...

time inconsistency

average inflation targeting: time inconsistency and

the search for deep inconsistency

bayesian inconsistency under misspecification

outliers and inconsistency

parameter inconsistency and network audit program

sneps 2.7.1 user’s manual1 - university at...

en ng product inconsistency

inconsistency robustness for logic programs - arxiv

towards systematic inconsistency identification for

the (mostly harmless) inconsistency of knowledge

inconsistency in medical care

semantic types (sorts) in sneps 3 presentation to...

inconsistency and corporate visual identity