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

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

3 Some Misunderstandings about NMLs Weaker or stronger? Non-classical?One or many? Fewer Horn properties Include classical logic Unlike usual non-CLs A way of using CL Which is correct? Essential multiplicity A few basic kinds

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

7 Pivotal-Assumption Consequence Fix: background set K of formulae Define: A |- K x iff K  A |- x Alias: x  Cn K (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 |~ K x iff K  A |- x for every subset K  K maxiconsistent with A Alias : x  C K (A) Known as : Poole consequence

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

11 Pivotal-Valuation Consequence Idea: exclude some of the valuations Fix: subset W  V Define: A |- W x 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 W A = set of valuations in W minimal among those satisfying premise set A –Require the conclusion to be true under all valuations in W A Define: A |~ W x iff no v  W A : v(A) = 1 v(x) = 0 Alias : x  C W (A) Known as : preferential consequence (Shoham, KLM….)

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

15 Pivotal-Rule Consequence Rule: any ordered pair (a,x) of formulae Fix: set R of rules Define: A |- R x 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 Cn R (A)  { X  A: X = Cn(X) = R(X)}  {A n : n   }, where A 1 = A and A n+1 = Cn(A n  R(A n ))  {A n : n   } with A 1 = A and A n+1 = Cn(A n  {x}) where (a,x) is first rule in  R  such that a  A n but x  A n (in the case that there is no such rule: A n+1 = Cn(A n ))

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

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 C R (A) =  {C  R  (A):  R  an ordering of R}

20 Summary Table Fixed additional assumptions pivotal-assumption: Cn K paraclassical  OR  compact  Fixed restriction of valuations pivotal-valuation: Cn W paraclassical  OR  compact  Fixed additional rules pivotal-rule: Cn R paraclassical  OR  compact  Vary assumptions with premises default-assumption: C K consistency constraint Poole systems + many variants! Vary valuation-set with premises default-valuation: C W minimalization preferential systems + many variants! Vary rules with premises default-rule: C R consistency constraint Reiter systems + many variants!

21 Further reading Makinson, David ‘Bridges between classical and nonmonotonic logic’ Logic Journal of the IGPL 11 (2003) Free access: Makinson, David ‘General Patterns in Nonmonotonic Reasoning’ pp in Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, ed. Gabbay, Hogger and Robinson. Oxford University Press.