Bridges from Classical to Nonmonotonic Logic
David Makinson
King’s College London
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Purpose Message
• Take mystery out of nonmonotonic logic
Not so unfamiliar
Easily accessible given classical logic
• There are natural bridge systems
Monotonic
Supraclassical
Stepping stones
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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
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A Habit to Suspend
• Bridge logics: supraclassical closure opns
• But… how is this possible?
• Not closed under substitution
• Nor are the nonmonotonic ones
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General Picture
vary w ith cu rren t p rem ises
u se ad d it ion a l assu m p tion s
vary w ith cu rren t p rem ises
res tric t se t o f va lu a tion s
vary w ith cu rren t p rem ises
u se ad d it ion a l ru les
c lass ica l con seq u en ce
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First Bridge: Using Additional Assumptions
• Pivotal-assumption consequence
• Fixed set of background assumptions
• Monotonic
• Default-assumption consequence
• Vary background set with current premises
• Nonmonotonic
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Pivotal-Assumption Consequence
• Fix: background set K of formulae
• Define: A |-K x iff KA |- x
• Alias: x CnK(A)
• Class: pivotal-assumption consequence relations: |-K for some set K
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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
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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 KA |- x for every subset K K maxiconsistent with A
• Alias: x CK(A)
• Known as : Poole consequence
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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
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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
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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)
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Default-Valuation Consequence
• Idea– allow set W V to vary with current premises A
– put WA = set of valuations in W minimal among those satisfying
premise set A
– Require the conclusion to be true under all valuations in WA
• Define: A |~W x iff no v WA : v(A) = 1 v(x) = 0
• Alias: x CW(A)
• Known as : preferential consequence (Shoham, KLM….)
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Third Bridge: Using Additional Rules
• Pivotal-rule consequence
• Fixed set of rules
• Monotonic
• Default-rule consequence
• Vary application of rules with current premises
• Nonmonotonic
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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
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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
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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(AnR(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))
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Default-Rule Consequence
• Fix an ordering R of R
• Define CR(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 with An
(if no such rule: An+1 = Cn(An))
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Default-Rule Consequence (ctd)
Facts:
• The sets CR(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) = {CR(A): R an ordering of R}
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Summary TableFixed additional assumptions
• pivotal-assumption: CnK
• paraclassical • OR • compact
Fixed restriction of valuations
• pivotal-valuation: CnW
• paraclassical • OR • compact
Fixed additional rules
• pivotal-rule: CnR
• paraclassical • OR • compact
Vary assumptions with premises
• default-assumption: CK
• consistency constraint• Poole systems • + many variants!
Vary valuation-set with premises
• default-valuation: CW
• minimalization• preferential systems• + many variants!
Vary rules with premises
• default-rule: CR
• consistency constraint• Reiter systems• + many variants!
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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.
