a new computational logic approach to reason with conditionals 2015-49.p… · i indicative...
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I Introduction
I Weak Completion Semantics
I Indicative Conditionals
I Minimal Revision Followed by Abduction
I Subjunctive Conditionals
I Future Work
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 1
A New Computational Logic Approach to Reason withConditionals
Emmanuelle-Anna Dietz and Steffen HolldoblerInternational Center for Computational LogicTechnische Universitat DresdenGermany
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The Firing Squad Example
I If the court orders an execution, then the captain will give the signalupon which rifleman A and B will shoot the prisoner;consequently, the prisoner will be dead Pearl 2000
I We assume that
. the court’s decision is unknown
. both riflemen are accurate, alert and law-abiding
. the prisoner is unlikely to die from any other causes
I Please evaluate the following conditionals (true, false, unknown)
. If the prisoner is alive, then the captain did not signal
. If rifleman A shot, then rifleman B shot as well
. If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 2
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The Firing Squad Example
I If the court orders an execution, then the captain will give the signalupon which rifleman A and B will shoot the prisoner;consequently, the prisoner will be dead Pearl 2000
I We assume that
. the court’s decision is unknown
. both riflemen are accurate, alert and law-abiding
. the prisoner is unlikely to die from any other causes
I Please evaluate the following conditionals (true, false, unknown)
. If the prisoner is alive, then the captain did not signal
. If rifleman A shot, then rifleman B shot as well
. If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 3
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The Firing Squad Example
I If the court orders an execution, then the captain will give the signalupon which rifleman A and B will shoot the prisoner;consequently, the prisoner will be dead Pearl 2000
I We assume that
. the court’s decision is unknown
. both riflemen are accurate, alert and law-abiding
. the prisoner is unlikely to die from any other causes
I Please evaluate the following conditionals (true, false, unknown)
. If the prisoner is alive, then the captain did not signal
. If rifleman A shot, then rifleman B shot as well
. If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 4
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The Firing Squad Example
I If the court orders an execution, then the captain will give the signalupon which rifleman A and B will shoot the prisoner;consequently, the prisoner will be dead Pearl 2000
I We assume that
. the court’s decision is unknown
. both riflemen are accurate, alert and law-abiding
. the prisoner is unlikely to die from any other causes
I Please evaluate the following conditionals (true, false, unknown)
. If the prisoner is alive, then the captain did not signal
. If rifleman A shot, then rifleman B shot as well
. If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 5
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The Firing Squad Example
I If the court orders an execution, then the captain will give the signalupon which rifleman A and B will shoot the prisoner;consequently, the prisoner will be dead Pearl 2000
I We assume that
. the court’s decision is unknown
. both riflemen are accurate, alert and law-abiding
. the prisoner is unlikely to die from any other causes
I Please evaluate the following conditionals (true, false, unknown)
. If the prisoner is alive, then the captain did not signal
. If rifleman A shot, then rifleman B shot as well
. If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 6
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The Firing Squad Example
I If the court orders an execution, then the captain will give the signalupon which rifleman A and B will shoot the prisoner;consequently, the prisoner will be dead Pearl 2000
I We assume that
. the court’s decision is unknown
. both riflemen are accurate, alert and law-abiding
. the prisoner is unlikely to die from any other causes
I Please evaluate the following conditionals (true, false, unknown)
. If the prisoner is alive, then the captain did not signal
. If rifleman A shot, then rifleman B shot as well
. If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 7
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Introduction
I The weak completion semantics (WCS)
. is a new cognitive theory
. is based on ideas presented in Stenning, van Lambalgen 2008
. is mathematically sound H., Kencana Ramli 2009
. has been successfully applied to model–among others–the suppression task, the selection task, and the belief bias effectDietz, H., Ragni 2012, Dietz, H., Ragni 2013, Pereira, Dietz, H. 2014
I Now, we want to apply WCS to reason about conditionals
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 8
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Introduction
I The weak completion semantics (WCS)
. is a new cognitive theory
. is based on ideas presented in Stenning, van Lambalgen 2008
. is mathematically sound H., Kencana Ramli 2009
. has been successfully applied to model–among others–the suppression task, the selection task, and the belief bias effectDietz, H., Ragni 2012, Dietz, H., Ragni 2013, Pereira, Dietz, H. 2014
I Now, we want to apply WCS to reason about conditionals
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 9
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Logic Programs and their Weak Completion
I Let P be a ground logic program
I Let S be a finite and consistent set of ground literals
I def (S,P) = {A← body ∈ P | A ∈ S ∨ ¬A ∈ S}
I Weak completion wcP = completion of P \ {A↔ ⊥ | def (A,P) = ∅}
wc{a ← b, a ← c, c ← ⊥} = {a ↔ b ∨ c, c ↔ ⊥}wc{c ← >, c ← ⊥} = {c ↔ >∨⊥}
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Logic Programs and their Weak Completion
I Let P be a ground logic program
I Let S be a finite and consistent set of ground literals
I def (S,P) = {A← body ∈ P | A ∈ S ∨ ¬A ∈ S}
I Weak completion wcP = completion of P \ {A↔ ⊥ | def (A,P) = ∅}
wc{a ← b, a ← c, c ← ⊥} = {a ↔ b ∨ c, c ↔ ⊥}wc{c ← >, c ← ⊥} = {c ↔ >∨⊥}
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Logic Programs and their Weak Completion
I Let P be a ground logic program
I Let S be a finite and consistent set of ground literals
I def (S,P) = {A← body ∈ P | A ∈ S ∨ ¬A ∈ S}
I Weak completion wcP = completion of P \ {A↔ ⊥ | def (A,P) = ∅}
wc{a ← b, a ← c, c ← ⊥} = {a ↔ b ∨ c, c ↔ ⊥}wc{c ← >, c ← ⊥} = {c ↔ >∨⊥}
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Logic Programs and their Weak Completion
I Let P be a ground logic program
I Let S be a finite and consistent set of ground literals
I def (S,P) = {A← body ∈ P | A ∈ S ∨ ¬A ∈ S}
I Weak completion wcP = completion of P \ {A↔ ⊥ | def (A,P) = ∅}
wc{a ← b, a ← c, c ← ⊥} = {a ↔ b ∨ c, c ↔ ⊥}
wc{c ← >, c ← ⊥} = {c ↔ >∨⊥}
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Logic Programs and their Weak Completion
I Let P be a ground logic program
I Let S be a finite and consistent set of ground literals
I def (S,P) = {A← body ∈ P | A ∈ S ∨ ¬A ∈ S}
I Weak completion wcP = completion of P \ {A↔ ⊥ | def (A,P) = ∅}
wc{a ← b, a ← c, c ← ⊥} = {a ↔ b ∨ c, c ↔ ⊥}wc{c ← >, c ← ⊥} = {c ↔ >∨⊥}
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Weak Completion Semantics
I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsIn: Hill, Warren (eds), Logic Programming, LNCS 5649, 464-478: 2009
I We consider the three-valued Łukasiewicz logic Łukasiewicz 1920
. U← U = > (compared to U← U = U under Kripke-Kleene logic)
I Each weakly completed program admits a least modelMP under Ł-logic
I MP is the least fixed point of ΦP(I) = 〈J>, J⊥〉, where
J> = {A | A← body ∈ P and I(body) = >}J⊥ = {A | def (A,P) 6= ∅ and I(body) = ⊥ for all A← body ∈ def (A,P)}
(ΦP is due to Stenning, vanLambalgen 2008)
I P |=wcs F iff MP(F ) = >
{a ← b, a ← c, c ← ⊥} 6|=wcs a ∨ ¬a
{c ← >, c ← ⊥} |=wcs c
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Weak Completion Semantics
I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsIn: Hill, Warren (eds), Logic Programming, LNCS 5649, 464-478: 2009
I We consider the three-valued Łukasiewicz logic Łukasiewicz 1920
. U← U = > (compared to U← U = U under Kripke-Kleene logic)
I Each weakly completed program admits a least modelMP under Ł-logic
I MP is the least fixed point of ΦP(I) = 〈J>, J⊥〉, where
J> = {A | A← body ∈ P and I(body) = >}J⊥ = {A | def (A,P) 6= ∅ and I(body) = ⊥ for all A← body ∈ def (A,P)}
(ΦP is due to Stenning, vanLambalgen 2008)
I P |=wcs F iff MP(F ) = >
{a ← b, a ← c, c ← ⊥} 6|=wcs a ∨ ¬a
{c ← >, c ← ⊥} |=wcs c
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Weak Completion Semantics
I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsIn: Hill, Warren (eds), Logic Programming, LNCS 5649, 464-478: 2009
I We consider the three-valued Łukasiewicz logic Łukasiewicz 1920
. U← U = > (compared to U← U = U under Kripke-Kleene logic)
I Each weakly completed program admits a least modelMP under Ł-logic
I MP is the least fixed point of ΦP(I) = 〈J>, J⊥〉, where
J> = {A | A← body ∈ P and I(body) = >}J⊥ = {A | def (A,P) 6= ∅ and I(body) = ⊥ for all A← body ∈ def (A,P)}
(ΦP is due to Stenning, vanLambalgen 2008)
I P |=wcs F iff MP(F ) = >
{a ← b, a ← c, c ← ⊥} 6|=wcs a ∨ ¬a
{c ← >, c ← ⊥} |=wcs c
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Weak Completion Semantics
I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsIn: Hill, Warren (eds), Logic Programming, LNCS 5649, 464-478: 2009
I We consider the three-valued Łukasiewicz logic Łukasiewicz 1920
. U← U = > (compared to U← U = U under Kripke-Kleene logic)
I Each weakly completed program admits a least modelMP under Ł-logic
I MP is the least fixed point of ΦP(I) = 〈J>, J⊥〉, where
J> = {A | A← body ∈ P and I(body) = >}J⊥ = {A | def (A,P) 6= ∅ and I(body) = ⊥ for all A← body ∈ def (A,P)}
(ΦP is due to Stenning, vanLambalgen 2008)
I P |=wcs F iff MP(F ) = >
{a ← b, a ← c, c ← ⊥} 6|=wcs a ∨ ¬a
{c ← >, c ← ⊥} |=wcs c
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Weak Completion Semantics
I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsIn: Hill, Warren (eds), Logic Programming, LNCS 5649, 464-478: 2009
I We consider the three-valued Łukasiewicz logic Łukasiewicz 1920
. U← U = > (compared to U← U = U under Kripke-Kleene logic)
I Each weakly completed program admits a least modelMP under Ł-logic
I MP is the least fixed point of ΦP(I) = 〈J>, J⊥〉, where
J> = {A | A← body ∈ P and I(body) = >}J⊥ = {A | def (A,P) 6= ∅ and I(body) = ⊥ for all A← body ∈ def (A,P)}
(ΦP is due to Stenning, vanLambalgen 2008)
I P |=wcs F iff MP(F ) = >
{a ← b, a ← c, c ← ⊥} 6|=wcs a ∨ ¬a
{c ← >, c ← ⊥} |=wcs c
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Weak Completion Semantics
I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsIn: Hill, Warren (eds), Logic Programming, LNCS 5649, 464-478: 2009
I We consider the three-valued Łukasiewicz logic Łukasiewicz 1920
. U← U = > (compared to U← U = U under Kripke-Kleene logic)
I Each weakly completed program admits a least modelMP under Ł-logic
I MP is the least fixed point of ΦP(I) = 〈J>, J⊥〉, where
J> = {A | A← body ∈ P and I(body) = >}J⊥ = {A | def (A,P) 6= ∅ and I(body) = ⊥ for all A← body ∈ def (A,P)}
(ΦP is due to Stenning, vanLambalgen 2008)
I P |=wcs F iff MP(F ) = >
{a ← b, a ← c, c ← ⊥} 6|=wcs a ∨ ¬a
{c ← >, c ← ⊥} |=wcs c
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Abduction
I Consider the abductive framework 〈P,AP ,IC, |=wcs〉, where
. AP =⋃
{A|def (A,P)=∅}{A← >, A← ⊥} is the set of abducibles
. IC is a finite set of integrity constraints
I An observationO is a set of ground literals
. O is explainable in 〈P,AP ,IC, |=wcs〉iff there exists a minimal E ⊆ AP called explanation such thatMP∪E satisfies IC and P ∪ E |=wcs O
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Abduction
I Consider the abductive framework 〈P,AP ,IC, |=wcs〉, where
. AP =⋃
{A|def (A,P)=∅}{A← >, A← ⊥} is the set of abducibles
. IC is a finite set of integrity constraints
I An observationO is a set of ground literals
. O is explainable in 〈P,AP ,IC, |=wcs〉iff there exists a minimal E ⊆ AP called explanation such thatMP∪E satisfies IC and P ∪ E |=wcs O
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Revision
I rev(P,S) = (P \ def (S,P)) ∪ {A← > | A ∈ S} ∪ {A← ⊥ | ¬A ∈ S}
I Properties
. rev is non-monotonic in general
. rev is monotonic, i.e.,MP ⊆Mrev(P,S), ifMP(L) = U for all L ∈ S
. Mrev(P,S)(S) = >
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Revision
I rev(P,S) = (P \ def (S,P)) ∪ {A← > | A ∈ S} ∪ {A← ⊥ | ¬A ∈ S}
I Properties
. rev is non-monotonic in general
. rev is monotonic, i.e.,MP ⊆Mrev(P,S), ifMP(L) = U for all L ∈ S
. Mrev(P,S)(S) = >
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Conditionals
I Conditionals are statements of the form if condition then consequence
I Indicative conditionals are conditionals
. whose condition may or may not be true
. whose consequence may or may not be true
. but the consequence is asserted to be true if the condition is true
I Subjunctive conditionals are conditionals
. whose condition is false
. whose consequence may or may not be true
. but in the counterfactual circumstance of the condition being true,the consequence is asserted to be true as well
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Conditionals
I Conditionals are statements of the form if condition then consequence
I Indicative conditionals are conditionals
. whose condition may or may not be true
. whose consequence may or may not be true
. but the consequence is asserted to be true if the condition is true
I Subjunctive conditionals are conditionals
. whose condition is false
. whose consequence may or may not be true
. but in the counterfactual circumstance of the condition being true,the consequence is asserted to be true as well
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Conditionals
I Conditionals are statements of the form if condition then consequence
I Indicative conditionals are conditionals
. whose condition may or may not be true
. whose consequence may or may not be true
. but the consequence is asserted to be true if the condition is true
I Subjunctive conditionals are conditionals
. whose condition is false
. whose consequence may or may not be true
. but in the counterfactual circumstance of the condition being true,the consequence is asserted to be true as well
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Indicative Conditionals
I In the sequel, let cond(C,D) be an indicative conditional, where
. Condition C and consequenceDare finite and consistent sets of ground literals
I Conditionals are evaluated wrt a given P and IC
. We assume thatMP satisfies IC
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Indicative Conditionals
I In the sequel, let cond(C,D) be an indicative conditional, where
. Condition C and consequenceDare finite and consistent sets of ground literals
I Conditionals are evaluated wrt a given P and IC
. We assume thatMP satisfies IC
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Evaluating Indicative Conditionals – Our Approach
I Given P , IC, and cond(C,D)
. IfMP(C) = > and MP(D) = > then cond(C,D) is true
. IfMP(C) = > and MP(D) = ⊥ then cond(C,D) is false
. IfMP(C) = > and MP(D) = U then cond(C,D) is unknown
. IfMP(C) = ⊥ then cond(C,D) is vacuous
. IfMP(C) = U then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and
E ⊆ Arev(P,S) is an explanation for C \ S such that
P′ |=wcs C and MP′ satisfies IC
Minimal revision followed by abduction
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Modeling the Firing Squad Example
I Ps ← era ← srb ← sd ← rad ← rba ← ¬d
I MP〈∅, ∅〉
I AP{e ← >, e ← ⊥}
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
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Modeling the Firing Squad Example
I Ps ← era ← srb ← sd ← rad ← rba ← ¬d
I MP〈∅, ∅〉
I AP{e ← >, e ← ⊥}
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
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Modeling the Firing Squad Example
I Ps ← era ← srb ← sd ← rad ← rba ← ¬d
I MP〈∅, ∅〉
I AP{e ← >, e ← ⊥}
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
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Modeling the Firing Squad Example
I Ps ← era ← srb ← sd ← rad ← rba ← ¬d
I MP〈∅, ∅〉
I AP{e ← >, e ← ⊥}
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
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Modeling the Firing Squad Example
I Ps ← era ← srb ← sd ← rad ← rba ← ¬d
I MP〈∅, ∅〉
I AP{e ← >, e ← ⊥}
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
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Modeling the Firing Squad Example
I Ps ← era ← srb ← sd ← rad ← rba ← ¬d
I MP〈∅, ∅〉
I AP{e ← >, e ← ⊥}
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
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The Firing Squad Examples
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
I If the prisoner is alive, then the captain did not signal
cond(a,¬s) : P ⇒ P ∪ E⊥ ⇒ true
I If rifleman A shot, then rifleman B shot as well
cond(ra, rb) : P ⇒ P ∪ E> ⇒ true
I If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
cond({¬s, ra},¬e) : P ⇒ rev(P, ra) ∪ E⊥ ⇒ true
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The Firing Squad Examples
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
I If the prisoner is alive, then the captain did not signal
cond(a,¬s) : P ⇒ P ∪ E⊥ ⇒ true
I If rifleman A shot, then rifleman B shot as well
cond(ra, rb) : P ⇒ P ∪ E> ⇒ true
I If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
cond({¬s, ra},¬e) : P ⇒ rev(P, ra) ∪ E⊥ ⇒ true
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The Firing Squad Examples
I Observations
. E> = {e ← >} explains {s, ra, rb, d,¬a}
. E⊥ = {e ← ⊥} explains {¬s,¬ra,¬rb,¬d, a}
. {¬s, ra} cannot be explained
I If the prisoner is alive, then the captain did not signal
cond(a,¬s) : P ⇒ P ∪ E⊥ ⇒ true
I If rifleman A shot, then rifleman B shot as well
cond(ra, rb) : P ⇒ P ∪ E> ⇒ true
I If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
cond({¬s, ra},¬e) : P ⇒ rev(P, ra) ∪ E⊥ ⇒ true
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The Last Firing Squad Example Revisited
I If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
P ⇒ rev(P, ra) ∪ E⊥
·d
·ra
·rb
·s
·e
·a
◦d
◦ra
·rb
·s
·e
•a
◦d
◦ra
•rb
•s
•e
•a
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The Last Firing Squad Example Revisited
I If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
P ⇒ rev(P, ra) ∪ E⊥
·d
·ra
·rb
·s
·e
·a
◦d
◦ra
·rb
·s
·e
•a
◦d
◦ra
•rb
•s
•e
•a
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The Last Firing Squad Example Revisited
I If the captain gave no signal and rifleman A decides to shoot,then the court did not order an execution
P ⇒ rev(P, ra) ∪ E⊥
·d
·ra
·rb
·s
·e
·a
◦d
◦ra
·rb
·s
·e
•a
◦d
◦ra
•rb
•s
•e
•a
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An Alternative Approach
I Katrin Schulz: Minimal Models vs. Logic Programming: The Case of CounterfactualConditionals. Journal of Applied Non-Classical Logics, 24, 153-168: 2014
I Lemma For all n ∈ N, we find Φrev(P,S) ↑ n ⊆ ΨP,S ↑ n ⊆ Φrev(P,S) ↑ (n + 1)
I Theorem lfp Φrev(P,S) = lfp ΨP,S
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Subjunctive Conditionals–The Forest Fire Example
I If there had not been so many dry leaves on the forest floor,then the forest fire would not have occurred Byrne 2007
P = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← >}MP = 〈{dl, l, ff}, {ab}〉
I Subjunctive conditional cond(¬dl,¬ff )
rev(P,¬dl) = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← ⊥}
Φrev(P,¬dl)↑ 0 〈∅, ∅〉↑ 1 〈{l}, {dl}〉↑ 2 〈{l, ab}, {dl}〉↑ 3 〈{l, ab}, {dl, ff}〉
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Subjunctive Conditionals–The Forest Fire Example
I If there had not been so many dry leaves on the forest floor,then the forest fire would not have occurred Byrne 2007
P = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← >}MP = 〈{dl, l, ff}, {ab}〉
I Subjunctive conditional cond(¬dl,¬ff )
rev(P,¬dl) = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← ⊥}
Φrev(P,¬dl)↑ 0 〈∅, ∅〉↑ 1 〈{l}, {dl}〉↑ 2 〈{l, ab}, {dl}〉↑ 3 〈{l, ab}, {dl, ff}〉
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Subjunctive Conditionals–The Forest Fire Example
I If there had not been so many dry leaves on the forest floor,then the forest fire would not have occurred Byrne 2007
P = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← >}MP = 〈{dl, l, ff}, {ab}〉
I Subjunctive conditional cond(¬dl,¬ff )
rev(P,¬dl) = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← ⊥}
Φrev(P,¬dl)↑ 0 〈∅, ∅〉↑ 1 〈{l}, {dl}〉↑ 2 〈{l, ab}, {dl}〉↑ 3 〈{l, ab}, {dl, ff}〉
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Subjunctive Conditionals–The Forest Fire Example
I If there had not been so many dry leaves on the forest floor,then the forest fire would not have occurred Byrne 2007
P = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← >}MP = 〈{dl, l, ff}, {ab}〉
I Subjunctive conditional cond(¬dl,¬ff )
rev(P,¬dl) = {ff ← l ∧ ¬ab, l ← >, ab ← ¬dl, dl ← ⊥}
Φrev(P,¬dl)↑ 0 〈∅, ∅〉↑ 1 〈{l}, {dl}〉↑ 2 〈{l, ab}, {dl}〉↑ 3 〈{l, ab}, {dl, ff}〉
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The Extended Forest Fire Example
I If there had not been so many dry leaves on the forest floor,then the forest fire would not have occurred Pereira, Dietz, H. 2014
P = {ff ← l ∧ ¬ab1, ff ← a ∧ ¬ab2,
l ← >, ab1 ← ¬dl, dl ← >, ab2 ← ⊥}MP = 〈{dl, l, ff}, {ab1, ab2}〉
rev(P,¬dl) = {ff ← l ∧ ¬ab1, ff ← a ∧ ¬ab2,
l ← >, ab1 ← ¬dl, dl ← ⊥, ab2 ← ⊥}Mrev(P,¬dl) = 〈{l, ab1}, {dl, ab2}〉
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 54
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Future Work
I Evaluating subjunctive conditionals
I Applying abduction before evaluating the consequent of a conditional
I Experiments
Emmanuelle-Anna Dietz and Steffen HolldoblerA New Computational Logic Approach to Reason with Conditionals 55