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Bridges from Classical to Nonmonotonic Logic

Bridges from Classical to Nonmonotonic Logic. David Makinson King’s College London. Take mystery out of nonmonotonic logic Not so unfamiliar Easily accessible given classical logic. There are natural bridge systems Monotonic Supraclassical Stepping stones. Purpose Message.

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Bridges from Classical to Nonmonotonic Logic

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  1. Bridges from Classical to Nonmonotonic Logic David Makinson King’s College London

  2. Take mystery out of nonmonotonic logic Not so unfamiliar Easily accessible given classical logic There are natural bridge systems Monotonic Supraclassical Stepping stones Purpose Message

  3. Some Misunderstandings about NMLs

  4. A Habit to Suspend • Bridge logics: supraclassical closure opns • But… how is this possible? • Not closed under substitution • Nor are the nonmonotonic ones

  5. General Picture

  6. Pivotal-assumption consequence Fixed set of background assumptions Monotonic Default-assumption consequence Vary background set with current premises Nonmonotonic First Bridge: Using Additional Assumptions

  7. Pivotal-Assumption Consequence • Fix: background set K of formulae • Define:  A |-Kx iff KA |- x • Alias: xCnK(A) • Class: pivotal-assumption consequence relations: |-K for some set K

  8. Pivotal-Assumption Consequence(ctd) Properties • Paraclassical • Supraclassical (includes classical consequence) • Closure operation (reflexivity + idempotence + monotony) • Disjunction in premises (alias OR) • Compact Representation • Pivotal-assumption consequence iff above three properties

  9. Default-Assumption Consequence • Idea • Allow background assumptions K to vary with current premises A • Diminish K when inconsistent with A • Work with maximal subsets of K that are consistent with A • Define: A |~Kx iff KA |- x for every subset K K maxiconsistent with A • Alias: xCK(A) • Known as : Poole consequence

  10. Pivotal-valuation consequence Fixed subset of the set of all Boolean valuations Monotonic Default-valuation consequence Vary valuation set with current premises Nonmonotonic Second Bridge: Restricting the Valuation Set

  11. Pivotal-Valuation Consequence • Idea: exclude some of the valuations • Fix: subset WV • Define:  A |-Wx iff no v W:v(A) = 1 v(x) = 0 • Class: pivotal-valuation consequence relations: |-W for some set WV

  12. Pivotal-Valuation Consequence(ctd) Properties • Paraclassical • Disjunction in premises • But not compact Fact • {pivotal assumption} = {pivotal valuation}{compact} Representation • Open (when infinite premise sets allowed)

  13. Default-Valuation Consequence • Idea • allow set WV to vary with current premises A • put WA = set of valuations in Wminimal among those satisfying premise set A • Require the conclusion to be true under all valuations in WA • Define: A |~Wx iff no v WA :v(A) = 1 v(x) = 0 • Alias: xCW(A) • Known as : preferential consequence (Shoham, KLM….)

  14. Pivotal-rule consequence Fixed set of rules Monotonic Default-rule consequence Vary application of rules with current premises Nonmonotonic Third Bridge: Using Additional Rules

  15. Pivotal-Rule Consequence • Rule: any ordered pair (a,x) of formulae • Fix: set R of rules • Define:  A |-Rx iff x  every superset of A closed under both Cn and R • Class: pivotal-rule consequence relations: |-R for some set R of rules

  16. Pivotal-Rule Consequence(ctd) Properties • Paraclassical • Compact • But not Disjunction in premises Facts • {pivotal assumption} = {pivotal rule}{OR} = {pivotal rule}{pivotal valuation} Representation • Pivotal-rule consequence iff above two properties

  17. Pivotal-Rule Consequence(ctd) Equivalent definitions of CnR(A) • { X A: X = Cn(X) = R(X)} • {An : n}, where A1 = A and An+1 = Cn(AnR(An)) • {An : n} with A1 = A and An+1 = Cn(An{x}) where (a,x) is first rule in R such that a An but x An (in the case that there is no such rule: An+1 = Cn(An))

  18. Default-Rule Consequence • Fix an ordering R of R • Define CR(A): {An : n} with A1 = A and An+1 = Cn(An{x}) where (a,x) is first rule in R such that: a An , x An , and x is consistent withAn (if no such rule: An+1 = Cn(An))

  19. Default-Rule Consequence(ctd) Facts: • The sets CR(A) for an ordering R of R are precisely the Reiter extensions of A using the normal default rules (a,x) alias (a;x/x) • The ordering makes a difference • Standard inductive definition versus fixpoints Sceptical operation • CR(A) = {CR(A): R an ordering of R}

  20. Summary Table

  21. Further reading • Makinson, David 2003. ‘Bridges between classical and nonmonotonic logic’ Logic Journal of the IGPL 11 (2003) 69-96. Free access: http://www3.oup.co.uk/igpl/Volume_11/Issue_01/ • Makinson, David 1994. ‘General Patterns in Nonmonotonic Reasoning’ pp 35-110 in Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, ed. Gabbay, Hogger and Robinson. Oxford University Press.

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