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

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.