alo cse f buf Inconsistency Tolerance in SNe

alo @ cse f buf Inconsistency Tolerance in SNe. PS 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

alo f buf @ cse • • • Acknowledgements João Martins Frances L. Johnson Bharat Bhushan The SNe. PS Research Group NSF, Instituto Nacional de Investigação Cientifica, Rome Air Development Center, AFOSR, U. S. Army CECOM 2 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 3 June, 2003 S. C. Shapiro

alo f buf @ cse • • • SNe. PS A logic- and network-based Knowledge representation Reasoning And acting System [Shapiro & Group ’ 02] This talk will ignore network and acting aspects. 4 June, 2003 S. C. Shapiro

alo @ cse f buf Logic • Based on R, the logic of relevant implication [Anderson & Belnap ’ 75; Martins & Shapiro ’ 88, Shapiro ’ 92] 5 June, 2003 S. C. Shapiro

alo @ cse f buf Supported wffs P{… <origin tag, origin set> hypothesis derived …} Set of hypotheses From which P has been derived. Origin set tracks relevance and ATMS assumptions. 6 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 7 June, 2003 S. C. Shapiro

alo f buf @ cse Rules of Inference: Hypothesis Hyp: P {<hyp, {P}>} : whale(Willy) and free(Willy). wff 3: free(Willy) and whale(Willy) {<hyp, {wff 3}>} 8 June, 2003 S. C. Shapiro

alo f buf @ cse Rules of Inference: &E &E: From A and B {<t, s>} infer A {<der, s>} or B {<der, s>} wff 3: free(Willy) and whale(Willy) {<hyp, {wff 3}>} : free(Willy)? wff 2: free(Willy) {<der, {wff 3}>} 9 June, 2003 S. C. Shapiro

alo f buf @ cse Rules of Inference: andor. E The os is the union of os's of parents wff 3: free(Willy) and whale(Willy) {<hyp, {wff 3}>} wff 6: all(x)(andor(0, 1){manatee(x), dolphin(x), whale(x)}) {<hyp, {wff 6}>} At most 1 : dolphin(Willy)? wff 9: ~dolphin(Willy) {<der, {wff 3, wff 6}>} 10 June, 2003 S. C. Shapiro

alo f buf @ cse Rules of Inference: =>E The origin set is the union of os's of parents. Since wff 10: all(x)(whale(x) => mammal(x)) {<hyp, {wff 10}>} and wff 1: whale(Willy){<der, {wff 3}>} I infer wff 11: mammal(Willy) {<der, {wff 3, wff 10}>} 11 June, 2003 S. C. Shapiro

alo f buf Rules of Inference: =>I origin set is diff of os's of parents. @ cse wff 12: all(x)(orca(x) => whale(x)) {<hyp, {wff 12}>} : orca(Keiko) => mammal(Keiko)? Let me assume that wff 13: orca(Keiko) {<hyp, {wff 13}>} Since wff 12: all(x)(orca(x) => whale(x)) {<hyp, {wff 12}>} and wff 13: orca(Keiko){<hyp, {wff 13}>} I infer whale(Keiko) {<der, {wff 12, wff 13}>} 12 June, 2003 S. C. Shapiro

alo f buf @ cse Rules of Inference: =>I (cont’d) origin set is diff of os's of parents. Since wff 10: all(x)(whale(x) => mammal(x)) {<hyp, {wff 10}>} and wff 16: whale(Keiko) {<der, {wff 12, wff 13}>} I infer mammal(Keiko) {<der, {wff 10, wff 12, wff 13}>} Since wff 14: mammal(Keiko) {<der, {wff 10, wff 12, wff 13}>} was derived assuming wff 13: orca(Keiko) {<hyp, {wff 13}>} I infer wff 15: orca(Keiko) => mammal(Keiko) {<der, {wff 10, wff 12}>} 13 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 14 June, 2003 S. C. Shapiro

alo @ cse f buf ~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. • SNe. BR [Martins & Shapiro ’ 88] involved in choosing the culprit. 15 June, 2003 S. C. Shapiro

alo f buf @ cse Adding Inconsistent Hypotheses wff 19: all(x)(whale(x) => fish(x)){<hyp, {wff 19}>} wff 20: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 20}>} wff 21: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 21}>} 16 June, 2003 S. C. Shapiro

alo f buf @ cse Finding the Contradiction : has(Willy, scales)? Since wff 19: all(x)(whale(x) => fish(x)) {<hyp, {wff 19}>} and wff 1: whale(Willy) {<der, {wff 3}>} I infer fish(Willy) {<der, {wff 3, wff 19}>} Since wff 21: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 21}>} and wff 23: fish(Willy) {<der, {wff 3, wff 19}>} I infer has(Willy, scales) {<der, {wff 3, wff 19, wff 21}>} Since wff 20: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 20}>} and wff 11: mammal(Willy) {<der, {wff 3, wff 10}>} I infer it is not the case that wff 23: fish(Willy) 17 June, 2003 S. C. Shapiro

alo f buf @ cse Manual Belief Revision A contradiction was detected within context default-defaultct. The contradiction involves the newly derived proposition: wff 24: ~fish(Willy) {<der, {wff 3, wff 10, wff 20}>} and the previously existing proposition: wff 23: fish(Willy) {<der, {wff 3, wff 19}>} 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 June, 2003 S. C. Shapiro 18

alo f buf @ cse BR Advice In 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 : wff 20: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 20}>} (1 dependent proposition: (wff 24)) 2 : wff 19: all(x)(whale(x) => fish(x)) {<hyp, {wff 19}>} (2 dependent propositions: (wff 23 wff 22)) 3 : wff 10: all(x)(whale(x) => mammal(x)){<hyp, {wff 10}>} (3 dependent propositions: (wff 24 wff 15 wff 11)) 4 : wff 3: free(Willy) and whale(Willy) {<hyp, {wff 3}>} (8 dependent propositions: (wff 24 wff 23 wff 22 wff 11 wff 9 wff 5 wff 2 wff 1)) User deletes #2: wff 19. 19 June, 2003 S. C. Shapiro

alo f buf @ cse Willy has no Scales Since wff 21: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 21}>} and it is not the case that wff 23: fish(Willy) {<der, {wff 3, wff 19}>} I infer it is not the case that wff 22: has(Willy, scales) {<der, {wff 3, wff 19, wff 21}>} wff 26: ~has(Willy, scales){<der, {wff 3, wff 10, wff 21}>} 20 June, 2003 S. C. Shapiro

alo f buf @ cse Final KB: hyps & positive ders : list-asserted-wffs wff 3: free(Willy) and whale(Willy) {<hyp, {wff 3}>} wff 6: all(x)(andor(0, 1){manatee(x), dolphin(x), whale(x)}) {<hyp, {wff 6}>} wff 10: all(x)(whale(x) => mammal(x)) {<hyp, {wff 10}>} wff 12: all(x)(orca(x) => whale(x)) {<hyp, {wff 12}>} wff 20: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 20}>} wff 21: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 21}>} wff 1: wff 2: wff 11: wff 15: whale(Willy) free(Willy) mammal(Willy) orca(Keiko) => mammal(Keiko) {<der, {wff 3}>} {<der, {wff 3, wff 10}>} {<der, {wff 10, wff 12}>} 21 June, 2003 S. C. Shapiro

alo f buf @ cse Final KB: hyps & negative ders : list-asserted-wffs wff 3: free(Willy) and whale(Willy) {<hyp, {wff 3}>} wff 6: all(x)(andor(0, 1){manatee(x), dolphin(x), whale(x)}) {<hyp, {wff 6}>} wff 10: all(x)(whale(x) => mammal(x)) {<hyp, {wff 10}>} wff 12: all(x)(orca(x) => whale(x)) {<hyp, {wff 12}>} wff 20: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 20}>} wff 21: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 21}>} wff 9: wff 24: wff 25: wff 26: ~dolphin(Willy) {<der, {wff 3, wff 10}>} ~fish(Willy) {<der, {wff 3, wff 10, wff 20}>} ~(all(x)(whale(x) => fish(x))) {<ext, {wff 3, wff 10, wff 20}>} ~has(Willy, scales) {<der, {wff 3, wff 10, wff 21}>} 22 June, 2003 S. C. Shapiro

alo @ cse f buf Summary • Logic is paraconsistent: P{<t 1, {h 1 … hi}>}, ~P{<t 2, {h(i+1) … hn}>} ~hj • When a contradiction is explicitly found, the user is engaged in its resolution. 23 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 24 June, 2003 S. C. Shapiro

alo @ cse f buf 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. ) • SNe. BR may use relative credibility to choose a culprit by itself. [Shapiro & Johnson ’ 00] *Not yet in released version. 25 June, 2003 S. C. Shapiro

alo f buf @ cse Contradictory Sources wff 1: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 1}>} wff 2: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 2}>} wff 3: all(x)(orca(x) => whale(x)) {<hyp, {wff 3}>} : Source(Melville, all(x)(whale(x) => fish(x)). ). wff 5: Source(Melville, all(x)(whale(x) => fish(x))) {<hyp, {wff 5}>} : Source(Darwin, all(x)(whale(x) => mammal(x)). ). wff 7: Source(Darwin, all(x)(whale(x) => mammal(x))) {<hyp, {wff 7}>} : Sgreater(Darwin, Melville). wff 8: Sgreater(Darwin, Melville) {<hyp, {wff 8}>} wff 11: free(Willy) and whale(Willy) {<hyp, {wff 11}>} Note: Source & Sgreater props are regular object-language props. June, 2003 S. C. Shapiro 26

alo f buf @ cse Finding the Contradiction : has(Willy, scales)? Since wff 4: all(x)(whale(x) => fish(x)) {<hyp, {wff 4}>} and wff 9: whale(Willy) {<der, {wff 11}>} I infer fish(Willy) {<der, {wff 4, wff 11}>} Since wff 2: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 2}>} and wff 14: fish(Willy) {<der, {wff 4, wff 11}>} I infer has(Willy, scales) Since wff 6: all(x)(whale(x) => mammal(x)) {<hyp, {wff 6}>} and wff 9: whale(Willy) {<der, {wff 11}>} I infer mammal(Willy) Since wff 1: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 1}>} and wff 15: mammal(Willy) {<der, {wff 6, wff 11}>} I infer it is not the case that 27 wff 14: fish(Willy) {<der, {wff 4, wff 11}>} June, 2003 S. C. Shapiro

alo f buf @ cse Automatic BR A contradiction was detected within context default-defaultct. The contradiction involves the newly derived proposition: wff 17: ~fish(Willy) {<der, {wff 1, wff 6, wff 11}>} and the previously existing proposition: wff 14: fish(Willy) {<der, {wff 4, wff 11}>} The least believed hypothesis: (wff 4) The most common hypothesis: (nil) The hypothesis supporting the fewest wffs: (wff 1) I removed the following belief: wff 4: all(x)(whale(x) => fish(x)) {<hyp, {wff 4}>} I no longer believe the following 2 propositions: wff 14: fish(Willy) {<der, {wff 4, wff 11}>} wff 13: has(Willy, scales) {<der, {wff 2, wff 4, wff 11}>} 28 June, 2003 S. C. Shapiro

alo @ cse f buf Summary • User may select automatic BR. • Relative credibility is used. • User is informed of lost beliefs. 29 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 30 June, 2003 S. C. Shapiro

alo @ cse f buf 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 noticed. 31 June, 2003 S. C. Shapiro

alo f buf @ cse Darwin Context : set-context Darwin () : set-default-context Darwin wff 1: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 1}>} wff 2: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 2}>} wff 3: all(x)(orca(x) => whale(x)) {<hyp, {wff 3}>} wff 4: all(x)(whale(x) => mammal(x)) {<hyp, {wff 4}>} wff 7: free(Willy) and whale(Willy) {<hyp, {wff 7}>} 32 June, 2003 S. C. Shapiro

alo f buf @ cse Melville Context : describe-context ((assertions (wff 8 wff 7 wff 4 wff 3 wff 2 wff 1)) (restriction nil) (named (science))) : set-context Melville (wff 8 wff 7 wff 3 wff 2 wff 1) ((assertions (wff 8 wff 7 wff 3 wff 2 wff 1)) (restriction nil) (named (melville))) : set-default-context Melville ((assertions (wff 8 wff 7 wff 3 wff 2 wff 1)) (restriction nil) (named (melville))) : all(x)(whale(x) => fish(x)). wff 9: all(x)(whale(x) => fish(x)) {<hyp, {wff 9}>} 33 June, 2003 S. C. Shapiro

alo f buf @ cse Melville: Willy has scales : has(Willy, scales)? Since wff 9: all(x)(whale(x) => fish(x)){<hyp, {wff 9}>} and wff 5: whale(Willy) {<der, {wff 7}>} I infer fish(Willy) {<der, {wff 7, wff 9}>} Since wff 2: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 2}>} and wff 11: fish(Willy) {<der, {wff 7, wff 9}>} I infer has(Willy, scales) {<der, {wff 2, wff 7, wff 9}>} wff 10: has(Willy, scales) {<der, {wff 2, wff 7, wff 9}>} 34 June, 2003 S. C. Shapiro

alo f buf @ cse Darwin: No scales : set-default-context Darwin : has(Willy, scales)? Since wff 4: all(x)(whale(x) => mammal(x)) {<hyp, {wff 4}>} and wff 5: whale(Willy) {<der, {wff 7}>} I infer mammal(Willy) Since wff 1: all(x)(andor(0, 1){mammal(x), fish(x)}) {<hyp, {wff 1}>} and wff 12: mammal(Willy) {<der, {wff 4, wff 7}>} I infer it is not the case that wff 11: fish(Willy) Since wff 2: all(x)(fish(x) <=> has(x, scales)) {<hyp, {wff 2}>} and it is not the case that wff 11: fish(Willy) {<der, {wff 7, wff 9}>} I infer it is not the case that wff 10: has(Willy, scales) wff 15: June, 2003 ~has(Willy, scales) {<der, {wff 1, wff 2, wff 4, wff 7}>} 35 S. C. Shapiro

alo @ cse f buf Summary • Contradictory information may be isolated in different contexts. • Reasoning is performed in a single context. • Results are available in other contexts. 36 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 37 June, 2003 S. C. Shapiro

alo @ cse f buf 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] 38 June, 2003 S. C. Shapiro

alo @ cse f buf Preclusion Rules in SNe. PS* • 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 . 39 June, 2003 S. C. Shapiro

alo f buf @ cse Animal Modes of Mobility wff 1: wff 2: wff 3: wff 4: wff 5: wff 6: all(x)(orca(x) => whale(x)) all(x)(whale(x) => mammal(x)) all(x)(deer(x) => mammal(x)) all(x)(tuna(x) => fish(x)) all(x)(canary(x) => bird(x)) all(x)(penguin(x) => bird(x)) wff 7: all(x)(andor(0, 1){swims(x), flies(x), runs(x)}) wff 8: all(x)(mammal(x) => runs(x)) wff 9: all(x)(fish(x) => swims(x)) wff 10: all(x)(bird(x) => flies(x)) wff 11: all(x)(whale(x) => swims(x)) wff 12: all(x)(penguin(x) => swims(x)) 40 June, 2003 S. C. Shapiro

alo f buf @ cse Using Preclusion for Exceptions wff 13: Precludes(all(x)(whale(x) => swims(x)), all(x)(mammal(x) => runs(x))) wff 14: Precludes(all(x)(penguin(x) => swims(x)), all(x)(bird(x) => flies(x))) wff 15: wff 16: wff 17: wff 18: wff 19: orca(Willy) tuna(Charlie) deer(Bambi) canary(Tweety) penguin(Opus) 41 June, 2003 S. C. Shapiro

alo f buf @ cse : I I I Who Swims? (Contradictory Conclusions) swims(? x)? infer swims(Opus) infer swims(Charlie) infer swims(Willy) infer flies(Tweety) infer it is not the case infer flies(Opus) infer it is not the case infer runs(Willy) infer it is not the case infer runs(Bambi) infer it is not the case that swims(Tweety) that wff 20: swims(Opus) that wff 24: swims(Willy) that swims(Bambi) 42 June, 2003 S. C. Shapiro

alo f buf @ cse Using Preclusion to Arbitrate Contradictions (1) Since wff 13: Precludes(all(x)(whale(x) => swims(x)), all(x)(mammal(x) => runs(x))) and wff 11: all(x)(whale(x) => swims(x)) {<hyp, {wff 11}>} holds within the BS defined by context default-defaultct Therefore wff 34: ~swims(Willy) containing in its support wff 8: all(x)(mammal(x) => runs(x)) is precluded by wff 24: swims(Willy) that contains in its support wff 11: all(x)(whale(x) => swims(x)) 43 June, 2003 S. C. Shapiro

alo f buf @ cse Using Preclusion to Arbitrate Contradictions (2) Since wff 14: Precludes(all(x)(penguin(x) => swims(x)), all(x)(bird(x) => flies(x))) and wff 12: all(x)(penguin(x) => swims(x)) holds within the BS defined by context default-defaultct Therefore wff 31: ~swims(Opus) containing in its support wff 10: all(x)(bird(x) => flies(x)) is precluded by wff 20: swims(Opus) that contains in its support wff 12: all(x)(penguin(x) => swims(x)) 44 June, 2003 S. C. Shapiro

alo f buf @ cse wff 38: wff 24: wff 22: wff 20: The Swimmers and Non-Swimmers ~swims(Bambi) ~swims(Tweety) swims(Willy) swims(Charlie) swims(Opus) {<der, {wff 3, wff 7, wff 8, wff 17}>} {<der, {wff 5, wff 7, wff 10, wff 18}>} {<der, {wff 1, wff 15}>} {<der, {wff 4, wff 9, wff 16}>} {<der, {wff 12, wff 19}>} 45 June, 2003 S. C. Shapiro

alo f buf @ cse Two-Level Preclusion wff 1: wff 2: all(x)(robin(x) => bird(x)) all(x)(kiwi(x) => bird(x)) wff 3: wff 4: all(x)(bird(x) => flies(x)) all(x)(bird(x) => (~flies(x))) wff 5: wff 6: all(x)(robin(x) => flies(x)) all(x)(kiwi(x) => (~flies(x))) Example from Delgrande & Schaub ‘ 00 46 June, 2003 S. C. Shapiro

alo f buf @ cse Preferences wff 7: Precludes(all(x)(robin(x) => flies(x)), all(x)(bird(x) => (~flies(x)))) wff 8: Precludes(all(x)(kiwi(x) => (~flies(x))), all(x)(bird(x) => flies(x))) wff 12: (~location(New Zealand)) => Precludes(all(x)(bird(x) wff 14: location(New Zealand) => Precludes(all(x)(bird(x) => flies(x)), => (~flies(x)))) => (~flies(x))), => flies(x))) wff 10: ~location(New Zealand) wff 15: Precludes(location(New Zealand), ~location(New Zealand)) 47 June, 2003 S. C. Shapiro

alo @ cse f buf wff 16: wff 17: wff 18: Who flies? robin(Robin) kiwi(Kenneth) bird(Betty) : flies(? x)? 48 June, 2003 S. C. Shapiro

alo @ cse f buf Outside New Zealand wff 24: ~flies(Kenneth){<der, {wff 6, wff 17}>, <der, {wff 2, wff 4, wff 6, wff 17}>} wff 21: flies(Robin) {<der, {wff 5, wff 16}>, <der, {wff 1, wff 3, wff 16}>} wff 19: flies(Betty) {<der, {wff 3, wff 18}>} 49 June, 2003 S. C. Shapiro

alo @ cse f buf Inside New Zealand : location("New Zealand"). wff 9: location(New Zealand) : flies(? x)? wff 24: ~flies(Kenneth) {<der, {wff 6, wff 17}>, <der, {wff 2, wff 4, wff 6, wff 17}>} wff 21: flies(Robin) {<der, {wff 5, wff 16}>, <der, {wff 1, wff 3, wff 16}>} wff 20: ~flies(Betty) {<der, {wff 4, wff 18}>} 50 June, 2003 S. C. Shapiro

alo @ cse f buf 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. 51 June, 2003 S. C. Shapiro

alo f buf @ cse Outline Ø Introduction Ø Some Rules of Inference Ø ~I and Belief Revision Ø Credibility Ordering and Automatic BR Ø Reasoning in Different Contexts Ø Default Reasoning by Preferential Ordering Ø Summary 52 June, 2003 S. C. Shapiro

alo f buf Summary Inconsistency Tolerance in SNe. PS @ cse • • 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. 53 June, 2003 S. C. Shapiro

alo @ cse f buf For more information http: //www. cse. buffalo. edu/sneps/ 54 June, 2003 S. C. Shapiro

alo f buf @ cse References I A. 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, 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. 55 June, 2003 S. C. Shapiro

alo @ cse f buf References II J. 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 SNe. PS Implementation Group (2002) SNe. PS 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 SNe. PS. In C. Baral & M. Truszczyński, eds. , Proc. 8 th International Workshop on Non. Monotonic Reasoning. 56 June, 2003 S. C. Shapiro