Inconsistency Tolerance in SNePS

1
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_at_cse.buffalo.edu
• http//www.cse.buffalo.edu/shapiro/

2
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

3
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

4
SNePS

• A logic- and network-based
• Knowledge representation
• Reasoning
• And acting
• System Shapiro Group 02
• This talk will ignore network and acting aspects.

5
Logic

• Based on R, the logic of relevant implication
• Anderson Belnap 75 Martins Shapiro 88,
  Shapiro 92

6
Supported wffs

• P ltorigin tag, origin setgt

Set of hypotheses From which P has been derived.
hyp hypothesis der derived
Origin set tracks relevance and ATMS assumptions.
7
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

8
Rules of InferenceHypothesis

• Hyp P lthyp,Pgt
• whale(Willy) and free(Willy). wff3
  free(Willy) and whale(Willy) lthyp,wff3gt

9
Rules of InferenceE

• E From A and B ltt,sgt
• infer A ltder,sgt or B ltder,sgt
• wff3 free(Willy) and whale(Willy) lthyp,wff3gt
• free(Willy)? wff2 free(Willy) ltder,wff3gt

10
Rules of InferenceandorE

• The os is the union of os's of parents
• wff3 free(Willy) and whale(Willy) lthyp,wff3gt
• wff6all(x)(andor(0,1)manatee(x), dolphin(x),
  whale(x))
• 
  lthyp,wff6gt dolphin(Willy)?
• wff9 dolphin(Willy) ltder,wff3,wff6gt

At most 1
11
Rules of InferencegtE

• The origin set is the union of os's of parents.
• Since wff10 all(x)(whale(x) gt mammal(x))
  lthyp,wff10gt
• and wff1 whale(Willy)ltder,wff3gt
• I infer wff11 mammal(Willy) ltder,wff3,wff10gt
  

12
Rules of InferencegtI

• origin set is diff of os's of parents.
• wff12 all(x)(orca(x) gt whale(x))
  lthyp,wff12gt
• orca(Keiko) gt mammal(Keiko)?
• Let me assume that wff13 orca(Keiko)
  lthyp,wff13gt
• Since wff12 all(x)(orca(x) gt whale(x))
  lthyp,wff12gtand wff13 orca(Keiko)lthyp,wff13
  gt
• I infer whale(Keiko) ltder,wff12,wff13gt

13
Rules of InferencegtI (contd)

• origin set is diff of os's of parents.
• Since wff10 all(x)(whale(x) gt mammal(x))
  lthyp,wff10gt
• and wff16 whale(Keiko) ltder,wff12,wff13gtI
  infer mammal(Keiko) ltder,wff10,wff12,wff13gt
• Since wff14 mammal(Keiko) ltder,wff10,wff12,wff1
  3gtwas derived assuming
• wff13 orca(Keiko) lthyp,wff13gtI
  infer
• wff15 orca(Keiko) gt mammal(Keiko)
  ltder,wff10,wff12gt

14
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

15
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.

16
Adding Inconsistent Hypotheses

• wff19 all(x)(whale(x) gt fish(x))lthyp,wff19gt
• wff20 all(x)(andor(0,1)mammal(x), fish(x))
• lthyp,wff20gt
• wff21 all(x)(fish(x) ltgt has(x,scales))
• lthyp,wff21gt

17
Finding the Contradiction

• has(Willy, scales)?
• Since wff19 all(x)(whale(x) gt fish(x))
  lthyp,wff19gt
• and wff1 whale(Willy) ltder,wff3gtI infer
  fish(Willy) ltder,wff3,wff19gt
• Since wff21 all(x)(fish(x) ltgt has(x,scales))
• 
  lthyp,wff21gt
• and wff23 fish(Willy) ltder,wff3,wff19gtI
  infer has(Willy,scales) ltder,wff3,wff19,wff21gt
  
• Since wff20
• all(x)(andor(0,1)mammal(x), fish(x))
• lthyp,wff20gt
• and wff11 mammal(Willy) ltder,wff3,wff10gtI
  infer it is not the case that wff23 fish(Willy)

18
Manual Belief Revision

• A contradiction was detected
• within context
  default-defaultct. The contradiction involves
  the newly derived proposition wff24
  fish(Willy) ltder,wff3,wff10,wff20gt and
  the previously existing proposition wff23
  fish(Willy) ltder,wff3,wff19gt You have
  the following options 1. continue anyway,
  knowing that a contradiction is derivable 2.
  re-start the exact same run in a different
  context which is not inconsistent 3.
  drop the run altogether. (please type c, r
  or d)gtlt r

19
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 wff20 all(x)(andor(0,1)mammal(x),fish(x))
  
• 
  lthyp,wff20gt (1 dependent proposition
  (wff24))
• 2 wff19 all(x)(whale(x) gt fish(x))
  lthyp,wff19gt (2 dependent
  propositions (wff23 wff22))
• 3 wff10 all(x)(whale(x) gt
  mammal(x))lthyp,wff10gt (3 dependent
  propositions (wff24 wff15 wff11))
• 4 wff3 free(Willy) and whale(Willy)
  lthyp,wff3gt (8 dependent
  propositions
• (wff24 wff23 wff22 wff11 wff9 wff5
  wff2 wff1))
• User
  deletes 2 wff19.

20
Willy has no Scales

• Since wff21 all(x)(fish(x) ltgt has(x,scales))
• 
  lthyp,wff21gt
• and it is not the case that wff23 fish(Willy)
• 
  ltder,wff3,wff19gt
• I infer it is not the case that
• wff22 has(Willy,scales) ltder,wff3,wff19,w
  ff21gt
• wff26 has(Willy,scales)ltder,wff3,wff10,wff20,w
  ff21gt

21
Final KB hyps positive ders

• list-asserted-wffs
• wff3 free(Willy) and whale(Willy)
  lthyp,wff3gt
• wff6 all(x)(andor(0,1)manatee(x),dolphin(x),whal
  e(x))
• 
  lthyp,wff6gt
• wff10 all(x)(whale(x) gt mammal(x))
  lthyp,wff10gt
• wff12 all(x)(orca(x) gt whale(x))
  lthyp,wff12gt
• wff20 all(x)(andor(0,1)mammal(x),fish(x))
  lthyp,wff20gt
• wff21 all(x)(fish(x) ltgt has(x,scales))
  lthyp,wff21gt
• wff1 whale(Willy)
  ltder,wff3gt
• wff2 free(Willy)
  ltder,wff3gt
• wff11 mammal(Willy)
  ltder,wff3,wff10gt
• wff15 orca(Keiko) gt mammal(Keiko)
  ltder,wff10,wff12gt

22
Final KB hyps negative ders

• list-asserted-wffs
• wff3 free(Willy) and whale(Willy)
  lthyp,wff3gt
• wff6 all(x)(andor(0,1)manatee(x),dolphin(x),whal
  e(x))
• 
  lthyp,wff6gt
• wff10 all(x)(whale(x) gt mammal(x))
  lthyp,wff10gt
• wff12 all(x)(orca(x) gt whale(x))
  lthyp,wff12gt
• wff20 all(x)(andor(0,1)mammal(x),fish(x))
  lthyp,wff20gt
• wff21 all(x)(fish(x) ltgt has(x,scales))
  lthyp,wff21gt
• wff9 dolphin(Willy)
  ltder,wff3,wff10gt
• wff24 fish(Willy)
  ltder,wff3,wff10,wff20gt
• wff25 (all(x)(whale(x) gt fish(x)))
  ltext,wff3,wff10,wff20gt
• wff26 has(Willy,scales)
  ltder,wff3,wff10,wff20,wff21gt

23
Summary

• Logic is paraconsistent
• Pltt1, h1 higt,
• Pltt2, h(i1) hngt
• hj
• When a contradiction is explicitly found, the
  user is engaged in its resolution.

24
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

25
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.

26
Contradictory Sources

• wff1 all(x)(andor(0,1)mammal(x),fish(x))
  lthyp,wff1gt
• wff2 all(x)(fish(x) ltgt has(x,scales))
  lthyp,wff2gt
• wff3 all(x)(orca(x) gt whale(x)) lthyp,wff3gt
• Source(Melville, all(x)(whale(x) gt fish(x)).).
• wff5 Source(Melville,all(x)(whale(x) gt
  fish(x)))
• 
  lthyp,wff5gt
• Source(Darwin, all(x)(whale(x) gt mammal(x)).).
• wff7 Source(Darwin,all(x)(whale(x) gt
  mammal(x)))
• 
  lthyp,wff7gt Sgreater(Darwin, Melville).
  wff8 Sgreater(Darwin,Melville) lthyp,wff8gt
• wff11 free(Willy) and whale(Willy)
  lthyp,wff11gt

Note Source Sgreater props are regular
object-language props.
27
Finding the Contradiction

• has(Willy, scales)?Since wff4
  all(x)(whale(x) gt fish(x)) lthyp,wff4gtand
  wff9 whale(Willy)
  ltder,wff11gtI infer fish(Willy)
  ltder,wff4,wff11gt
• Since wff2 all(x)(fish(x) ltgt has(x,scales))
  lthyp,wff2gtand wff14 fish(Willy)
  ltder,wff4,wff11gtI infer
  has(Willy,scales)Since wff6 all(x)(whale(x)
  gt mammal(x)) lthyp,wff6gtand wff9
  whale(Willy) ltder,wff11gtI
  infer mammal(Willy)Since wff1
  all(x)(andor(0,1)mammal(x),fish(x))
• 
  lthyp,wff1gtand wff15 mammal(Willy)
  ltder,wff6,wff11gtI infer it is not the case
  that
• wff14 fish(Willy) ltder,wff4,wff11gt
  

28
Automatic BR

• A contradiction was detected within context
  default-defaultct.
• The contradiction involves the newly derived
  proposition wff17 fish(Willy)
  ltder,wff1,wff6,wff11gt
• and the previously existing proposition
  wff14 fish(Willy) ltder,wff4,wff11gt
• 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) gt fish(x)) lthyp,wff4gt
• I no longer believe the following 2
  propositions wff14 fish(Willy)
  ltder,wff4,wff11gt wff13
  has(Willy,scales) ltder,wff2,wff4,wff11gt

29
Summary

• User may select automatic BR.
• Relative credibility is used.
• User is informed of lost beliefs.

30
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

31
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.

32
Darwin Context

• set-context Darwin ()
• set-default-context Darwin
• wff1 all(x)(andor(0,1)mammal(x),fish(x))
• 
  lthyp,wff1gt
• wff2 all(x)(fish(x) ltgt has(x,scales))
  lthyp,wff2gt
• wff3 all(x)(orca(x) gt whale(x))
  lthyp,wff3gt
• wff4 all(x)(whale(x) gt mammal(x))
  lthyp,wff4gt
• wff7 free(Willy) and whale(Willy)
  lthyp,wff7gt

33
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) gt fish(x)).
  wff9 all(x)(whale(x) gt fish(x)) lthyp,wff9gt

34
Melville Willy has scales

• has(Willy, scales)?Since wff9
  all(x)(whale(x) gt fish(x))lthyp,wff9gtand
  wff5 whale(Willy) ltder,wff7gtI infer
  fish(Willy) ltder,wff7,wff9gt
• Since wff2 all(x)(fish(x) ltgt has(x,scales))
• 
  lthyp,wff2gtand wff11 fish(Willy)
  ltder,wff7,wff9gtI infer has(Willy,scales)
  ltder,wff2,wff7,wff9gt
• wff10 has(Willy,scales) ltder,wff2,wff7,wff9
  gt

35
Darwin No scales

• set-default-context Darwin
• has(Willy, scales)?Since wff4
  all(x)(whale(x) gt mammal(x)) lthyp,wff4gtand
  wff5 whale(Willy)
  ltder,wff7gtI infer mammal(Willy)Since
  wff1 all(x)(andor(0,1)mammal(x),fish(x))
• 
  lthyp,wff1gtand wff12 mammal(Willy)
  ltder,wff4,wff7gtI infer it is not the case
  that wff11 fish(Willy)
• Since wff2 all(x)(fish(x) ltgt has(x,scales))
  lthyp,wff2gtand it is not the case that
  wff11 fish(Willy)
• 
  ltder,wff7,wff9gtI infer it is not the case
  that wff10 has(Willy,scales)
• wff15 has(Willy,scales) ltder,wff1,wff2,wff4
  ,wff7gt

36
Summary

• Contradictory information may be isolated in
  different contexts.
• Reasoning is performed in a single context.
• Results are available in other contexts.

37
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

38
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
• dont believe the undercut conclusion.
• Unlike BR, believe the hypotheses, but not a
  conclusion.
• Grosof 97, Bhushan 03

39
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.

40
Animal Modes of Mobility

• wff1 all(x)(orca(x) gt whale(x))
• wff2 all(x)(whale(x) gt mammal(x))
• wff3 all(x)(deer(x) gt mammal(x))
• wff4 all(x)(tuna(x) gt fish(x))
• wff5 all(x)(canary(x) gt bird(x))
• wff6 all(x)(penguin(x) gt bird(x))
• wff7 all(x)(andor(0,1)swims(x),flies(x),runs(x)
  )
• wff8 all(x)(mammal(x) gt runs(x))
• wff9 all(x)(fish(x) gt swims(x))
• wff10 all(x)(bird(x) gt flies(x))
• wff11 all(x)(whale(x) gt swims(x))
• wff12 all(x)(penguin(x) gt swims(x))

41
Using Preclusion for Exceptions

• wff13 Precludes(all(x)(whale(x) gt swims(x)),
• all(x)(mammal(x) gt runs(x)))
• wff14 Precludes(all(x)(penguin(x) gt swims(x)),
• all(x)(bird(x) gt flies(x)))
• wff15 orca(Willy)
• wff16 tuna(Charlie)
• wff17 deer(Bambi)
• wff18 canary(Tweety)
• wff19 penguin(Opus)

42
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)

43
Using Preclusionto Arbitrate Contradictions (1)

• Since wff13 Precludes(all(x)(whale(x) gt
  swims(x)),
• all(x)(mammal(x) gt
  runs(x)))
• and wff11 all(x)(whale(x) gt swims(x))
  lthyp,wff11gt
• holds within the BS defined by context
  default-defaultct
• Therefore wff34 swims(Willy)
• containing in its support
• wff8 all(x)(mammal(x) gt runs(x))
• is precluded by wff24 swims(Willy)
• that contains in its support
• wff11all(x)(whale(x) gt swims(x))

44
Using Preclusionto Arbitrate Contradictions (2)

• Since wff14 Precludes(all(x)(penguin(x) gt
  swims(x)),
• all(x)(bird(x) gt
  flies(x)))
• and wff12 all(x)(penguin(x) gt swims(x))
• holds within the BS defined by context
  default-defaultct
• Therefore wff31 swims(Opus)
• containing in its support
• wff10all(x)(bird(x) gt flies(x))
• is precluded by wff20 swims(Opus)
• that contains in its support
• wff12 all(x)(penguin(x) gt
  swims(x))

45
The Swimmersand Non-Swimmers

• wff38 swims(Bambi) ltder,wff3,wff7,wff8,wff17
  gt
• wff28 swims(Tweety) ltder,wff5,wff7,wff10,wff18
  gt
• wff24 swims(Willy) ltder,wff1,wff11,wff15gt
• wff22 swims(Charlie) ltder,wff4,wff9,wff16gt
• wff20 swims(Opus) ltder,wff12,wff19gt

46
Two-Level Preclusion

• wff1 all(x)(robin(x) gt bird(x))
• wff2 all(x)(kiwi(x) gt bird(x))
• wff3 all(x)(bird(x) gt flies(x))
• wff4 all(x)(bird(x) gt (flies(x)))
• wff5 all(x)(robin(x) gt flies(x))
• wff6 all(x)(kiwi(x) gt (flies(x)))
• Example from Delgrande Schaub 00

47
Preferences

• wff7 Precludes(all(x)(robin(x) gt flies(x)),
• all(x)(bird(x) gt (flies(x))))
• wff8 Precludes(all(x)(kiwi(x) gt (flies(x))),
• all(x)(bird(x) gt flies(x)))
• wff12 (location(New Zealand))
• gt Precludes(all(x)(bird(x) gt flies(x)),
• all(x)(bird(x) gt
  (flies(x))))
• wff14 location(New Zealand)
• gt Precludes(all(x)(bird(x) gt
  (flies(x))),
• all(x)(bird(x) gt flies(x)))
• wff10 location(New Zealand)
• wff15 Precludes(location(New Zealand),
• location(New Zealand))

48
Who flies?

• wff16 robin(Robin)
• wff17 kiwi(Kenneth)
• wff18 bird(Betty)
• flies(?x)?

49
Outside New Zealand

• wff24 flies(Kenneth)ltder,wff6,wff17gt,
• ltder,wff2,wff4,wff17gt,
• ltder,wff2,wff4,wff6,wff17
  gt
• wff21 flies(Robin) ltder,wff5,wff16gt,
• ltder,wff1,wff3,wff16gt
• wff19 flies(Betty) ltder,wff3,wff18gt

50
Inside New Zealand

• location("New Zealand").
• wff9 location(New Zealand)
• flies(?x)?
• wff24 flies(Kenneth) ltder,wff6,wff17gt,
• ltder,wff2,wff4,wff17gt,
• ltder,wff2,wff4,wff6,wff17
  gt
• wff21 flies(Robin) ltder,wff5,wff16gt,
• ltder,wff1,wff3,wff16gt
• wff20 flies(Betty) ltder,wff4,wff18gt

51
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.

52
Outline

• Introduction
• Some Rules of Inference
• I and Belief Revision
• Credibility Ordering and Automatic BR
• Reasoning in Different Contexts
• Default Reasoning by Preferential Ordering
• Summary

53
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.

54
For more information

• http//www.cse.buffalo.edu/sneps/

55
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, 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.
•  

56
 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 SNePS Implementation Group
  (2002) SNePS 2.6 User's Manual, Department of
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  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.
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  Workshop on Non-Monotonic Reasoning.