Multiple Kernel Contraction

. This paper focuses on the extension of AGM that allows change for a belief base by a set of sentences instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Contraction and a construction based on the AGM Partial Meet Contraction . We propose for their model another way to construct functions: Multiple Kernel Contraction , that is a modiﬁcation of Kernel Contraction , proposed by Hansson [Han94] to construct classical AGM contractions and belief base contractions. This construction works out the unsolved problem pointed out by Hansson in [Han99, pp. 369].


Introduction
In logic of theory change, the standard model is AGM, proposed by Alchourrón, Gärdenfors and Makinson [AGM85]. During the '90 the AGM model was extended in several ways, among them: models of belief change in which one or more of the postulates of the original AGM model are not satisfied; models that allow for sequences of operations (iterated or global functions); models that extend the language or the representation of the belief state; models that modify the representation of a belief state by introducing belief bases, etc. When a new model is proposed, not only are the intuitions of it important, but also its axiomatic (which determines the behaviour of the functions) and the ways to construct functions. This paper focuses on the extension of AGM that allows change for a belief base by a set of sentences instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Contraction and † Personal note from Fermé: This paper pays a double tribute to Sven Ove Hansson. First, it combines his work on two topics: kernel contraction and multiple contraction. Second, the research reported here is the result of work done by Saez and Sanz in connection with their degree theses, and I have personally advised them following the example of Sven Ove, who was my own PhD advisor.
Special Issue On Belief Revision Dedicated to Sven Ove Hansson on the Occasion of his 50th Birthday Edited by Erik J. Olsson a construction based on the AGM Partial Meet Contraction. We propose for their model another way to construct functions: Multiple Kernel Contraction, that is a modification of Kernel Contraction, proposed by Hansson [Han94] to construct classical AGM contractions and belief base contractions.
In section Background we introduce all the background needed to develop our method: the AGM model, belief base functions, kernel contraction and multiple contraction. In section Multiple Kernel Contraction we present our proposal and its axiomatic characterization. Proofs are deferred to an Appendix.

Background
Formal preliminaries: We will assume a language L that is closed under truth-functional operations and a consequence operator Cn for L. Cn satisfies the standard Tarskian properties, namely inclusion (A ⊆ Cn(A)), monotony (if A ⊆ B, then Cn(A) ⊆ Cn(B)), and iteration (Cn(A) = Cn(Cn(A))). It is supraclassical and compact, and satisfies deduction (if β ∈ Cn(A ∪ {α}), then (α → β) ∈ Cn(A)). A α will be used as an alternative notation for α ∈ Cn(A), α for α ∈ Cn(∅) and Cn(α) for Cn ({α}). Upper-case letters denote subsets of L. Lower-case Greek letters denote elements of L.
is an arbitrary tautology and ⊥ an arbitrary contradiction.

Belief Sets
In the AGM model of theory change [AGM85] belief states are represented by belief sets, i.e., set of sentences closed under Cn. Changes in beliefs are represented by operations on such sets. Among these operations, the contraction of a belief set A by a sentence α should be a new belief set A ⊆ A that does not contain α. The most widely known method of contracting a sentence from a belief set is partial meet contraction, introduced by Alchourrón, Gärdenfors and Makinson [AGM85] in their seminal paper. For any set A of propositions, A⊥α be the set of all maximal subsets of A that do not imply α. Let γ be any function such that for any proposition α,γ(A⊥α) is a nonempty subset of A if the latter is non-empty, and γ(A⊥α) = {A} in the limiting case that A⊥α is empty. The partial meet contraction on A that is generated by γ is the operation ∼ γ such that for all sentence α: One of the major achievements of AGM theory is the characterization of partial meet contraction for belief sets in terms of a set of postulates: Theorem 1 ( [AGM85]). Let A be a belief set. An operator − on A is a partial meet contraction function if and only if − satisfies: Closure A−α is a belief set whenever A is a belief set.
Other methods to construct contraction functions are Safe Contraction [AM85], its generalization Kernel Contraction [Han94]; and contraction functions based on Epistemic Entrenchment [GM88].

Belief Bases
In recent years, alternative models have been presented in which the belief states are represented by belief bases, sets of sentences that are not necessarily closed under logical consequence. Hansson characterizes partial meet contraction for belief bases in terms of a set of postulates: Theorem 2 ( [Han92]). Let A be a belief set. An operator − on A is a partial meet contraction function for A if and only if − satisfies success, inclusion,

Uniformity: If it holds for all subsets
Uniformity is a strong version of extensionality, that is extended to sentences that have the same "behaviour" in a belief base and relevance is the postulate that represents the rationality criteria of minimal change.

In [Han94] Hansson introduced Kernel Contraction, a generalization of Safe
Contraction. It is based on a selection among the sentences of a set A that contribute effectively to imply α; and how to use this selection in contracting by α. Formally: Definition 1 ( [Han94]). Let A be a set in L and α a sentence. Then A⊥ ⊥α is the set such that B ∈ A⊥ ⊥α if and only if: A⊥ ⊥α is called the kernel set of A with respect to α and its elements are the α-kernels of A.
Definition 2 ( [Han94]). Let A be a set of sentences. Let A⊥ ⊥α be the kernel set of A respect to α. An incision function σ for A is a function such that for all sentences α: Han94]). Let A be a set of sentences and σ an incision function for A. The kernel contraction − σ for A is defined as follows:

An operator − for a set A is a kernel contraction if and only if there is an incision function
Hansson also provided an axiomatic characterization for kernel contraction.

Theorem 3 ([Han94]). The operator − for a set of sentences A is a kernel contraction if and only if it satisfies success, inclusion, uniformity and
Core-retainment If β ∈ A and β / ∈ A−α then there is some set A such that Since core-retainment is weaker than relevance, it follows for belief bases that all partial meet contraction are kernel contraction 1 .

Multiple Contraction
Another extension of the AGM contraction consists in extending the model to contracting by sets of sentences [Fuh88,Nie91,FH94]. This extension allows two variants: package contraction where all the sentences must be removed from the belief base and choice contraction, where it is sufficient to remove at least one of the sentences. Fuhrmann and Hansson [FH94] propose the partial meet multiple contraction for choice and package. For package contraction we will redefine the remainder operator "⊥" for sets by sets, whereas for choice contraction we will introduce the operator " ".
Definition 4 (Package remainders [FH94]). X ∈ A⊥B if and only if: Definition 5 ( [FH94]). X ∈ A B if and only if: A B is the set of all maximal subsets of A that do not imply the whole elements of B.
It is important to note that B ∩ Cn(X) = ∅ means that X does not imply any element of B (closely related with package contraction) whereas B ⊆ Cn(X) means that there X does not imply all elements of B (closely related with choice contraction).
Partial meet package and choice contraction are defined based based on a selection from A ⊥ B and A B respectively: Definition 6 ( [FH94]). γ is a package selection function for A if and only if for all sets B: 1. If A⊥B is non-empty, then γ(A⊥B) is a non-empty subset of A⊥B. Theorem 4 ( [FH94]). An operator ÷ for a set A is an operator of partial meet package contraction if and only if it satisfies the following conditions:

Multiple Kernel Contraction
The idea of Multiple Kernel Contraction is to define a constructive method to contract a belief base by a set of sentences. Multiple partial meet contraction is based on the maximal subsets of A that do not imply B (in its both means of implication, package and choice). Multiple Kernel Contraction is based on a selection among the elements of A that effectively contribute to imply B. The method to construct multiple kernel contraction is rather similar to kernel contraction for single sentences.
We must to define at first the kernel set of A with respect to B. We distinguish between package and choice.
Definition 8 (Package Kernel Set). Let A, B be sets in L. Then A⊥ ⊥ P B is the set such that X ∈ A⊥ ⊥ P B if and only if: It is important to note that A⊥ ⊥ P B is different from α∈B A⊥ ⊥α. For example let β 1 , β 2 ∈ B, X ∈ A⊥ ⊥β 1 , Y ∈ A⊥ ⊥β 2 and X ⊂ Y . then Y ∈ α∈B A⊥ ⊥α, however by Definition 8, Y / ∈ A⊥ ⊥ P B. The relationship among the elements from A⊥ ⊥ P B and from α∈B A⊥ ⊥α, α ∈ B can be seen in the following observation: For Choice, the kernel set is defined as follows: Definition 9 (Choice Kernel Set). Let A, B be sets in L. Then A⊥ ⊥ C B is the set such that X ∈ A⊥ ⊥ P B if and only if: The following observation, inspired by the case of single sentences [Han94], will be useful in the characterization of multiple kernel contraction: Observation 2. The following conditions are equivalent: As in Kernel Contraction we must define an incision function to determine which sentences would be removed.

Definition 10. A function σ is a incision function for A if and only if it satisfies for all B:
The incision function gives rise to the multiple kernel contraction function.
Definition 11. Let σ be an incision function for A. The Multiple Kernel Contraction ≈ σ for A is defined as follows: • An operator ÷ for A is a multiple package contraction if and only if there is some package incision function σ for A such that A ÷ B = A ≈ σ B for all the sets B.
• An operator ÷ for A is a multiple choice contraction if and only if there is some choice incision function σ for A such that A ÷ B = A ≈ σ B for all the sets B.

Axiomatic Characterization of Multiple Kernel Contraction
Theorem 6. An operator ÷ for a set A is an operator of kernel package contraction if and only if it satisfies the following conditions:

C-uniformity If for every subset A of A B ⊆ Cn(A ) if and only if
Inclusion and success are the two basic postulates for a contraction function. Uniformity is a strong version of extensionality, that is extended to sentences that have the same "behavior" in a belief base, and core-retainment is the postulate that represent the rationality criteria of minimal change and is a postulate closely relate to the idea of "kernel-set", i.e., C ∪ {β} is a Bkernel element of A.

Conclusions and Remarks
We have defined multiple kernel contraction. This work can be considered as an extension of previous models in different ways: • We extend the model of kernel contraction to multiple contraction.
• We define a new constructive method for multiple contraction.
• Since, as in the singleton case, all the multiple partial meet contractions are multiple kernel contractions, our model is a generalization of multiple partial meet contraction.
The connection with the supplementary postulates and its (probable) relationship with multiple safe contraction still awaits research.

Appendix: Proofs
Proof of Observation 1.
Let α ∈ B, B ∩ Cn(∅) = ∅ and A α. Let X ∈ A⊥ ⊥α: then it follows that X ⊆ A and (since α ∈ B) B ∩ Cn(X) = ∅. If X ∈ A⊥ ⊥ P B, let Y = X and we done. Let X / ∈ A⊥ ⊥ P B, then (due the compactness of the underlying logic) there exists a subset of X, X such that B∩Cn(X ) = ∅ and for all X such that X ⊂ X , it follows B∩Cn(X ) = ∅. Hence X ∈ A⊥ ⊥ P B.
Proof of Observation 2.
(1. ⇒ 2.) Let A⊥ ⊥ P B = A⊥ ⊥ P C. We can, without loss of generality, assume that there exists X, such that X ∈ A⊥ ⊥ P B and X / (2. ⇒ 1.) Let X ⊆ A such that B ∩ Cn(X) = ∅ and C ∩ Cn(X) = ∅. Then (due the compactness of the underlying logic) there exists X ⊆ X such that X ∈ A⊥ ⊥ P B . Since X ⊆ X and C ∩ Cn(X) = ∅ it follows by definition 8 that X / ∈ A⊥ ⊥ P C.
(1. ⇒ 3.) Let A⊥ ⊥ C B = A⊥ ⊥ C C. We can, without loss of generality, assume that there exists X, such that X ∈ A⊥ ⊥ C B and X / ∈ A⊥ ⊥ C C. If C ⊆ Cn(X), we done, since B ⊆ Cn(X). If C ⊆ Cn(X), then there exists by definition 9 (since X / ∈ A⊥ ⊥ C C) Y ⊂ X, such that C ⊆ Cn(Y ). But B ⊆ Cn(Y ) and we done.
(3. ⇒ 1.) Let X ⊆ A such that B ⊆ Cn(X) and C ⊆ Cn(X). Then (by definition 9 and due the compactness of the underlying logic ) there Proof of Theorem 6. Construction to Postulates: Let ≈ σ be a Multiple Package Kernel Contraction operator for A. P-success: Due to observation 1 there exists X ⊆ X and X ∈ A⊥ ⊥ P B. By definition 10 there exists β ∈ X and β ∈ σ(A⊥ ⊥ P B).
P-inclusion: It follows trivially from definition 11. P-core-retainment: Let β ∈ A and β / ∈ A ≈ σ B. Then β ∈ σ(A⊥ ⊥ P B). P-uniformity: Suppose that every subset A of A implies some element of B if and only if A implies some element of C. By observation 2, A⊥ ⊥ P B = A⊥ ⊥ P C, from which it follows that σ(A⊥ ⊥ P B) = σ(A⊥ ⊥ P C). Hence by definition 11 A ≈ σ B = A ≈ σ C.
Postulates to Construction: Let ÷ be an operator for A that satisfies P-success, P-inclusion, P-core-retainment, and P-uniformity. We will prove that ÷ is a kernel package contraction.
Let σ such that for all B : σ(A⊥ ⊥ P B) = A \ (A ÷ B). We must prove that σ is a package incision function for A, proving that: (a) σ is a function, (b) σ satisfies conditions (1) and (2) from definition 10, and (c) that A ÷ B = A \ σ(A⊥ ⊥ P B).