Abstract
Approximation fixpoint theory (AFT) is an algebraic framework for the study of fixpoints of operators on bilattices, which has been applied to the study of the semantics for a number of nonmonotonic formalisms. A central notion of AFT is that of stable revision based on an underlying approximating operator (called approximator), where the negative information used in fixpoint computation is by default. This raises a problem in systems that combine different formalisms, where both default negation and established negation may be present in reasoning. In this paper we extend AFT to allow more flexible approximators. The main idea is to formulate and propose ternary approximators, of which traditional binary approximators are a special case. The extra parameter allows separation of two kinds of negative information, by entailment and by default, respectively. The new approach is motivated by the need to integrate different knowledge representation and reasoning (KRR) systems, in particular to support combined reasoning by nonmonotonic rules with ontologies. However, this small change by allowing flexible approximators raises a mathematical question - whether the resulting AFT is a sound fixpoint theory. The main result of this paper is a proof that answers this question positively.
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Notes
- 1.
By the Knaster-Tarski fixpoint theory, the least fixpoint can be computed iteratively from the least element of the underlying lattice - in this case, u is the least element in the lattice domain represented by the interval \([u,\top ]\).
- 2.
Again, because \(\bot \) is the least element of the domain \([\bot ,\top ]\).
- 3.
This example specifies a system in which states are represented by a pair of factors - high and low. Here, all states are stable except the one in which both factors are high. This state may be transmitted to an “inconsistent state” with the first factor high and the second low. This state is the only inconsistent one, and it itself is stable.
- 4.
Note that we never need a parameter to carry (already computed) true atoms, as in computing \(\textit{lfp}(\mathcal{A}^1(\cdot , v))\) we do not make default assumptions, and the monotonicity of the operator \(\mathcal{A}^1\) guarantees that any previously computed true atoms are derived again.
- 5.
For example, if v is a set of possibly true atoms, smaller v means more atoms that are false.
- 6.
Default negation here is evaluated independently of L, which has been called local closed world reasoning [11].
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Liu, F., Bi, Y., Chowdhury, M.S., You, JH., Feng, Z. (2016). Flexible Approximators for Approximating Fixpoint Theory. In: Khoury, R., Drummond, C. (eds) Advances in Artificial Intelligence. Canadian AI 2016. Lecture Notes in Computer Science(), vol 9673. Springer, Cham. https://doi.org/10.1007/978-3-319-34111-8_28
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DOI: https://doi.org/10.1007/978-3-319-34111-8_28
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