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Rw)) = Rw
N. Olivetti, G. Pozzato (2005)
KLMLean 1.0: a Theorem Prover for Logics of Default Reasoning
(s, w)) |= {(¬)A i |∼ B i }. Furthermore, we show that < ′ satisfies the smoothness condition on L-formulas
′ (s, w) = V (w)
Laura Giordano, Valentina Gliozzi, G. Pozzato (2007)
KLMLean 2.0: A Theorem Prover for KLM Logics of Nonmonotonic Reasoning
s, w)) |= ¬(A |∼ B). Then, there is a (s ′ , w ′ ) ∈ M in ′ < (LA) s.t
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Ian Parker (2020)
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Belief functions and default reasoning
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Bernhard Beckert, J. Posegga (1996)
Logic Programming as a Basis for Lean Automated DeductionJ. Log. Program., 28
We can reason similarly to what done in Facts A.9 and A.10 above to prove the following Fact: Fact A.14. For every conditional formula (¬)A |∼ B we have that
Laura Giordano, Valentina Gliozzi, N. Olivetti, C. Schwind (2003)
Tableau Calculi for Preference-Based Conditional Logics
We have hence proven that (i), (ii), and (iii) have a closed tableau without (Weak-Cut), then we can conclude by applying (|∼ + ) to them to obtain a closed tableau for Γ
A. Artosi, Guido Governatori, A. Rotolo (2002)
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B. Frieden (2001)
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D. Makinson (2003)
Bridges between Classical and Nonmonotonic LogicLog. J. IGPL, 11
B, then there exists Γ k in B such that z < x ∈ Γ k
By absurd, suppose there was a w ′ s
Given a consistent finite set of formulas Γ, there is a consistent, finite, and saturated set Γ ′ ⊇ Γ
D. Gabbay (1989)
Theoretical Foundations for Non-Monotonic Reasoning in Expert Systems
We conclude that (iii)Γ, LC i , ¬LC i , LD i has a closed tableau without (Weak-Cut), obtained by applying (|∼ + ) to (I')
Laura Giordano, Valentina Gliozzi, N. Olivetti, G. Pozzato (2006)
Analytic Tableau Calculi for KLM Rational Logic R
(ANY) > is used to represent the case in which the leftmost conclusion of (Weak-Cut) (1) is obtained by an application of (L − ), whereas the inner conclusion (2) is obtained by an application of
G. Pozzato (2008)
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Sita Gupta, Deepak Mohta, Vinod Todwal, Sushma Singh (2001)
Knowledge RepresentationJournal of Experimental & Theoretical Artificial Intelligence, 13
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Rw)) |≡ (¬)A |∼ B. We conclude that (M ′ , (w, Rw)) |≡ {(¬)A i |∼ B i }. Furthermore
N. Dershowitz, Z. Manna (1979)
Proving termination with multiset orderings
For every negated conditional formula ¬(A |∼ B) we have that (M, s) |≡ ¬(A |∼ B) iff (M ′ , (s, w)) |= ¬(A |∼ B)
M. Fitting (1998)
leanTAP RevisitedJ. Log. Comput., 8
N. Friedman, Joseph Halpern (1996)
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Observe that for each s ∈ S there is at least one corresponding (s, w) ∈ W ′ , since
Let the set of conditionals {(¬)A i |∼ B i } be satisfied in a possible world w in the CL-preferential model
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J. Leeuwen (2002)
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s, w)) |= LA. By absurd, suppose there exists a (s ′ , w ′ ) s.t. (M ′ , (s ′ , w ′ )) |= LA, and (s ′ , w ′ ) < (s, w
Laura Giordano, Valentina Gliozzi, N. Olivetti, G. Pozzato (2005)
Analytic Tableaux for KLM Preferential and Cumulative Logics
Let (M, s) |≡ A. By definition, for all w ∈ l(s), (M, w) |= A. By induction on the complexity of A, we can easily show that (M ′ , (s, w)) |= A. Since R(s, w) = {(s, w ′ ) | w ′ ∈ l(s)}
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As far as the smoothness condition, see Fact A.15 below. Furthermore, for all (w, Rw) ∈ S, l(w, Rw) = ∅, since R is serial
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(|∼ + ) >. The list is exhaustive. -( †) < (ANY)(ANY)(L − ) >: the tableau for (3) is started with an application of (L − ) to a formula
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Laura Giordano, Valentina Gliozzi, N. Olivetti, G. Pozzato (2007)
Tableau Calculi for KLM Logics: extended version
E. Weydert (2003)
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N. Friedman, Joseph Halpern, D. Koller (2000)
First-order conditional logic for default reasoning revisitedACM Trans. Comput. Log., 1
-(s, w) < ′ (s ′ , w ′ ) if s < s ′
D. Makinson (2005)
Bridges from classical to nonmonotonic logic, 5
Sarit Kraus, D. Lehmann, M. Magidor (1990)
Nonmonotonic Reasoning, Preferential Models and Cumulative LogicsArXiv, cs.AI/0202021
R. Goré (1999)
Tableau Methods for Modal and Temporal Logics
O. Arieli, A. Avron (2000)
General Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible RelationsLog. J. IGPL, 8
H. Kyburg, P. Gärdenfors (1988)
Knowledge in Flux
We present tableau calculi for the logics of nonmonotonic reasoning defined by Kraus, Lehmann and Magidor (KLM). We give a tableau proof procedure for all KLM logics, namely preferential, loop-cumulative, cumulative, and rational logics. Our calculi are obtained by introducing suitable modalities to interpret conditional assertions. We provide a decision procedure for the logics considered and we study their complexity.
ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery
Published: Apr 1, 2009
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