Inter-Definability of Horn Contraction and Horn Revision

Abstract

There have been a number of publications in recent years on generalising the AGM paradigm to the Horn fragment of propositional logic. Most of them focused on adapting AGM contraction and revision to the Horn setting. It remains an open question whether the adapted Horn contraction and Horn revision are inter-definable as in the AGM case through the Levi and Harper identities. In this paper, we give a positive answer by providing methods for generating contraction and revision from their dual operations. Noticeably, we cannot apply the Levi and Harper identities directly in such methods as the Horn fragment does not fully support negation. To overcome this difficulty, a Horn approximation technique called Horn strengthening is used. We show that Horn contraction generated from Horn revision is always plausible whereas Horn revision generated from Horn contraction is, in general, implausible and, to regain plausibility, the generating contraction has to be properly restricted.

Appendix A: Proofs of Results

Proof Proof for Lemma 1

• 1. By model theory H ₂⊆H ₁ implies |H ₁|⊆|H ₂|. Again by model theory |H ₁|⊆|H ₂| implies C n(H ₂)⊆C n(H ₁). Since H ₁ = C n _H (H ₁) and H ₂ = C n _H (H ₂), C n(H ₂)⊆C n(H ₁) implies H ₂⊆H ₁.

• 2. Immediate from Observation 1.28 in [18].

• 3. It can be derived from H ₁ = C n _H (H ₁) and H ₂ = C n _H (H ₂) that $H_{1}\cap H_{2}=\mathcal {H}(Cn(H_{1})\cap Cn(H_{2}))$. By model theory we have |C n(H ₁)∩C n(H ₂)| = |H ₁|∪|H ₂|. Then by the Horn closure property we have $|\mathcal {H}(Cn(H_{1})\cap Cn(H_{2}))|=Cl_{\cap }(|H_{1}|\cup |H_{2}|)$, thus |H ₁∩H ₂| = C l _∩(|H ₁|∪|H ₂|).

• 4.5. Immediate from the definition of C l _∩.

□

Proof Proof for Lemma 2

We first show that if ψ is a Horn formula such that ψ⊩ϕ, then there is $\chi \in \mathcal {H}\mathcal {S}(\phi )$ such that ψ⊩χ. If $\psi \in \mathcal {H}\mathcal {S}(\phi )$, then the result trivially holds. Suppose $\psi \not \in \mathcal {H}\mathcal {S}(\phi )$, then by the definition of Horn strengthening there is $\chi _{1}\in \mathcal {L}_{\text {\texttt {H}}}$ such that |ψ|⊂|χ ₁|⊆|ϕ|. Again, if $\chi _{1}\not \in \mathcal {H}\mathcal {S}(\phi )$, then there must be $\chi _{2}\in \mathcal {L}_{\text {\texttt {H}}}$ such that |χ ₁|⊂|χ ₂|⊆|ϕ|. Since |ϕ| is finite, eventually we will find a χ _n which is a Horn strengthening of ϕ. Since |ψ|⊂|χ _n |, ψ⊩χ _n .

Now we show |ϕ| = |χ ₁|∪⋯∪|χ _n |. |χ ₁|∪⋯∪|χ _n |⊆|ϕ| follows directly from the definition of Horn strengthening. For the other inclusion, assume there is μ ∈ |ϕ| such that μ∉|χ ₁|∪⋯∪|χ _n |. Let $\psi \in \mathcal {L}$ be such that |ψ| = {μ}. Since C l _∩({μ}) = {μ}, ψ is a Horn formula and |ψ|⊆|ϕ|. Then by the above result there is $\chi \in \mathcal {H}\mathcal {S}(\phi )$ such that |ψ|⊆|χ|. A contradiction ensues. □

Proof for Theorem 2

Suppose $\dot {-}$ is a MHC function for H with the determining pre-order ≼. $(H\dot {-} 1)$, $(H\dot {-} 2)$, $(H\dot {-} 4)$, and $(H\dot {-} 6)$ follow immediately from the construction of MHC. Let’s show the proof for the rest of the postulates.

$(H\dot {-} 3)$: Suppose ϕ∉H. Then we have |¬ϕ|∩|H|≠∅. Thus by the faithfulness of ≼ we have m i n(|¬ϕ|, ≼)⊆|H|. Thus $|H\dot {-}\phi |=Cl_{\cap }(min(|\neg \phi |, \preceq )\cup |H|)= Cl_{\cap }(|H|)=|H|$. Then $H\dot {-}\phi =H$ follows from $(H\dot {-} 1)$.

$(H\dot {-} de)$: We first show that $\dot {-}$ satisfies the following postulate

$$(H\dot{-} mc) \text{ If } \psi\in H\setminus H\dot{-}\phi, \text{ then }|H\dot{-}\phi|\not\subseteq |\phi\vee\psi|. $$

Suppose $\psi \in H\setminus H\dot {-}\phi$, it suffices to show there is $\mu \in |H\dot {-}\phi |\cap |\neg \phi |$ such that μ∉|ψ|. Assume to the contrary that $|H\dot {-}\phi |\cap |\neg \phi |\subseteq |\psi |$. Then we have, by the construction of MHC, that C l _∩(m i n(|¬ϕ|, ≼)∪|H|)∩|¬ϕ|⊆|ψ|. It then follows from m i n(|¬ϕ|, ≼)⊆|¬ϕ| that m i n(|¬ϕ|, ≼)⊆|ψ|. Since ψ ∈ H, we have |H|⊆|ψ|. Thus m i n(|¬ϕ|, ≼)∪|H|⊆|ψ| which implies C l _∩(m i n(|¬ϕ|, ≼)∪|H|)⊆|ψ|. However, it follows from $\psi \not \in H\dot {-}\phi$, by the construction of MHC that C l _∩(m i n(|¬ϕ|, ≼)∪|H|)⫅̸|ψ|. So we have a contradiction and this completes the proof of $(H\dot {-} mc)$.

Now suppose $\psi \in H\setminus H\dot {-}\phi$. Then it follows from $(H\dot {-} mc)$ that $|H\dot {-}\phi |\not \subseteq |\phi \vee \psi |$ . Thus there is $\mu \in |H\dot {-}\phi |$ such that μ∉|ϕ∨ψ|. Since |χ|⊆|ϕ∨ψ| for all $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$, μ∉|χ| for all $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$ . Then it follows from $\mu \in |H\dot {-}\phi |$ that $H\dot {-}\phi \not \vdash \chi$ for all $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$.

$(H\dot {-} 7)$: By the definition of MHC, we have

$$|H\dot{-}\phi\wedge\psi|=Cl_{\cap}(|H|\cup min(|\neg\phi|\cup|\neg\psi|,\preceq))$$

and

$$Cl_{\cap}(|H\dot{-}\phi|\cup|H\dot{-}\psi|)=Cl_{\cap}(|H|\cup min(|\neg\phi|,\preceq)\cup min(|\neg\psi|,\preceq)).$$

It is easy to see that |H| ∪ m i n(|¬ϕ| ∪ |¬ψ|, ≼)⊆|H| ∪ m i n(|¬ϕ|, ≼)∪m i n(|¬ψ|, ≼). Thus $|H\dot {-}\phi \wedge \psi |\subseteq Cl_{\cap }(|H\dot {-}\phi |\cup |H\dot {-}\psi |)$ which implies $(H\dot {-}\phi )\cap (H\dot {-}\psi ) \subseteq H\dot {-}\phi \wedge \psi$.

$(H\dot {-} 8)$: If ⊩ϕ, ⊩ψ, ϕ∉H or ψ∉H, then $(H\dot {-} 8)$ is trivially satisfied. So suppose ϕ, ψ ∈ H, ⊯ϕ, and ⊯ψ. Let $\phi \not \in H\dot {-}\phi \wedge \psi$, we need to show $H\dot {-} \phi \wedge \psi \subseteq H\dot {-}\phi$. By the construction of MHC, it suffices to show C l _∩(M i n(|¬ϕ|, ≼)∪|H|)⊆C l _∩(m i n(|¬ϕ| ∪ |¬ψ|, ≼)∪|H|). Since $\phi \not \in H\dot {-}\phi \wedge \psi$, we have C l _∩(m i n(|¬ϕ| ∪ |¬ψ|, ≼)∪|H|)⫅̸|ϕ| which implies m i n(|¬ϕ| ∪ |¬ψ|, ≼)∪|H|⫅̸|ϕ|. Since ϕ ∈ H, we have |H|⊆|ϕ|. Thus there is μ ∈ m i n(|¬ϕ| ∪ |¬ψ|, ≼) such that μ ∈ |¬ϕ|. Then we have for all ν ∈ |¬ϕ| ∪ |¬ψ|, μ≼ν. Let ω ∈ m i n(|¬ϕ|, ≼). It then follows from μ ∈ |¬ϕ|, that ω≼μ. By the transitivity of ≼, ω≼ν follows from ω≼μ and μ≼ν. As ω ∈ |¬ϕ| ∪ |¬ψ|, we have ω ∈ m i n(|¬ϕ| ∪ |¬ψ|, ≼). Thus m i n(|¬ϕ|, ≼)⊆m i n(|¬ϕ| ∪ |¬ψ|, ≼) which implies C l _∩(m i n(|¬ϕ|, ≼)∪|H|)⊆C l _∩(m i n(|¬ϕ| ∪ |¬ψ|, ≼)∪|H|). □

Proof for Theorem 3

For the first part, suppose − is a model-based contraction function for K with a determining pre-order ≼. Define ≼ _H as follows:

• 1. If μ, ν∉|K|∖|H|, then μ≼ _H ν iff μ≼ν,

• 2. If μ, ν ∈ |K|∖|H|, then μ≃ _H ν, μ≺ _H ω ₁, and ω ₂≺ _H μ for all ω ₁∉|K| and ω ₂∈|H|.

Clearly ≼ _H is a faithful pre-order for H. Let $\dot {-}$ be a MHC for H that is determined by ≼ _H . It remains to show $H\dot {-}\phi =\mathcal {H}(K-\phi )$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$.

⊇: Suppose $\psi \in \mathcal {L}_{\text {\texttt {H}}}$ and ψ ∈ K − ϕ. We need to show $\psi \in H\dot {-}\phi$. Since ψ ∈ K − ϕ, we have by $(K\dot {-} 2)$ that ψ ∈ K. Since H contains all the Horn formulas of K, we have ψ ∈ H. If ϕ∉K, then since H is a subset of K we have ϕ∉H which implies by $(H\dot {-} 3)$ that $H=H\dot {-}\phi$. Thus $\psi \in H\dot {-}\phi$ for the case of ϕ∉K. Now suppose ϕ ∈ K. Then we have |¬ϕ|∩|K| = ∅. Since ψ ∈ K − ϕ, we have by the construction of model-based contraction that m i n(|¬ϕ|, ≼)⊆|ψ|. By the definition of ≼ _H (part 1), we have m i n(|¬ϕ|, ≼ _H ) = m i n(|¬ϕ|, ≼). Thus m i n(|¬ϕ|, ≼ _H )⊆|ψ|. Since ψ ∈ H implies |H|⊆|ψ|, we have m i n(|¬ϕ|, ≼ _H )∪|H|⊆|ψ| which implies C l _∩(m i n(|¬ϕ|, ≼ _H )∪|H|)⊆|ψ|. Finally, it follows from the construction of MHC, $|H\dot {-}\phi |\subseteq |\psi |$.

⊆: Suppose $\psi \in H\dot {-}\phi$. We need to show ψ ∈ K − ϕ. Since $\psi \in H\dot {-}\phi$ we have by $(H\dot {-} 2)$ that ψ ∈ H. Since H is a subset of K, we have ψ ∈ K. If ϕ∉H, then since H contains all the Horn formulas of K, we have ψ∉K which implies by $(K\dot {-} 3)$ that K = K − ϕ. Thus ψ ∈ K − ϕ for the case of ϕ∉H. Now suppose ϕ ∈ H. Then ϕ ∈ K which implies |¬ϕ|∩|K| = ∅. Since $\psi \in H\dot {-}\phi$, we have by the construction of MHC, C l _∩(m i n(|¬ϕ|, ≼ _H )∪|H|)⊆|ψ| which implies m i n(|¬ϕ|, ≼ _H )⊆|ψ|. By the definition of ≼ _H (part 1), we have m i n(|¬ϕ|, ≼ _H ) = m i n(|¬ϕ|, ≼). Thus m i n(|¬ϕ|, ≼)⊆|ψ|. Since ψ ∈ K implies |K|⊆|ψ|, we have m i n(|¬ϕ|, ≼)∪⊆|ψ| which implies by the construction of model-based contraction that |K − ϕ|⊆|ψ|.

The second part can be proved in a similar manner. This time we need to generate a pre-order for K from the one for H. □

Proof for Theorem 4

It suffices to show (1) if $\dot {-}$ is a MHC function for H, then there is a TRPMHC function − for H such that $H\dot {-}\phi =H-\phi$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$ and (2) if $\dot {-}$ is a TRPMHC function for H, then there is a MHC function − for H such that $H\dot {-}\phi =H-\phi$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$.

For the first part, suppose $\dot {-}$ is a MHC function for H and is determined by the pre-order ≼. We first derive a relation ≤ over all weak remainder sets of H as follows:

For each pair X, Y of weak remainder sets of H, let Y≤X iff X, Y are such that |X| = C l _∩ (|H| ∪ {μ}), |Y| = C l _∩ (|H| ∪ {ν}), and μ ≼ ν.

Now we show ≤ is transitive. Suppose X, Y, Z are weak remainder sets of H, X≤Y and Y≤Z, it suffices to show X≤Z. By the derivation of ≤, there are interpretations μ, ν, and ω such that |X| = C l _∩(|H| ∪ {μ}), |Y| = C l _∩(|H| ∪ {ν}), and |Z| = C l _∩(|H| ∪ {ω}). Moreover, it follows from X≤Y that ν≼μ and it follows from Y≤Z that ω≼ν. Since ≼ is transitive, it follows from ν≼μ and ω≼ν that ω≼μ. Again by derivation of ≤, it follows from ω≼μ, that X≤Z.

Since ≤ is a transitive relation over all weak remainder sets of H, it can generate a TRPMHC function for H. Suppose the generated function is −, it remains to show $H\dot {-}\phi =H-\phi$ for all $\phi \in \mathcal {L}_{H}$. If ϕ∉H or ⊩ϕ, then we can easily obtain $H\dot {-}\phi =H-\phi =H$. So suppose ϕ ∈ H and ⊯ϕ. Let m i n(|¬ϕ|, ≼) = {μ ₁,…,μ _n }. Then we have by the definition of MHC that $H\dot {-}\phi =\mathcal {T}_{H}(|H|\cup \{\mu _{1},\ldots ,\mu _{n}\})$. By the definition of weak remainder set, there are X ₁,…,X _n ∈H ↓ _w ϕ such that |X _i | = C l _∩(|H| ∪ {μ _i }) for 1≤i≤n. Since m i n(|¬ϕ|, ≼) = {μ ₁,…,μ _n }, we have, by the derivation of ≤ that {X ∈ H ↓ _w ϕ | Y≤X for all Y ∈ H ↓ _w ϕ} = {X ₁,…,X _n }. Then we have by the definition of TRPMHC that $H-\phi =X_{1}\cap \cdots \cap X_{n}=\mathcal {T}_{H}(|X_{1}\cap \cdots \cap X_{n}|)= \mathcal {T}_{H}(Cl_{\cap }(|H|\cup \{\mu _{1}\}\cup \cdots \cup |H|\cup \{\mu _{n}\}))= \mathcal {T}_{H}(Cl_{\cap }(\{\mu _{1},\ldots ,\mu _{n}\}\cup |H|))= H\dot {-}\phi$.

The second part can be proved in a similar manner. This time we need to derive a pre-order over Ω for H from a transitive relation over all weak remainder sets of H. □

Proof for Theorem 5

Let $\dot {-}$ be a SHCMHC function for H. We need to show there is a EHC function − for H such that $H\dot {-}\phi =H-\phi$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$.

Let ≼ be a pre-order. We use m i n(|ϕ|, ≼)≼m i n(|ψ|, ≼) to denote that for all μ ∈ m i n(|ψ|, ≼) there is ν ∈ m i n(|ϕ|, ≼) such that ν≼μ and m i n(|ϕ|, ≼)≺m i n(|ψ|, ≼) to denote that for all μ ∈ m i n(|ψ|, ≼) there is ν ∈ m i n(|ϕ|, ≼) such that ν≺μ.

We first show that if a MHC function $\dot {-}$ for H is determined by the pre-order ≼, then $\dot {-}$ satisfies the following condition:

$$\psi\in H\dot{-}\phi iff \psi\in H~\text{and either} \vdash\phi~\text{or}~min(|\neg\phi|,\preceq)\prec min(|\neg\phi\wedge\neg\psi|,\preceq). $$

For one direction, suppose $\psi \in H\dot {-}\phi$ and ⊯ϕ. We need to show ψ ∈ H and m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼). Since $\psi \in H\dot {-}\phi$, ψ ∈ H follows from $(H\dot {-} 2)$. It remains to show m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼). If |¬ϕ∧¬ψ| = ∅, then the result holds trivially. So suppose |¬ϕ∧¬ψ|≠∅. By the definition of MHC, $\psi \in H\dot {-}\phi$ implies C l _∩(|H| ∪ m i n(|¬ϕ|, ≼))⊆|ψ|. Thus m i n(|¬ϕ|, ≼)∩|¬ψ| = ∅. Let μ ∈ m i n(|¬ϕ∧¬ψ|, ≼). Then we have for all ν ∈ m i n(|¬ϕ|, ≼), ν≺μ for otherwise m i n(|¬ϕ|, ≼)∩|¬ψ|≠∅. Thus m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼).

For the other direction, suppose ψ ∈ H, m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼), and ⊯ϕ. We need to show $\psi \in H\dot {-}\phi$. Since ψ ∈ H, we have |H|⊆|ψ|. Since m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼), we have m i n(|¬ϕ|, ≼)∩|¬ψ| = ∅ which implies m i n(|¬ϕ|, ≼)⊆|ψ|. Thus C l _∩(|H| ∪ m i n(|¬ϕ|, ≼))⊆|ψ| which implies by the definition of MHC that $\psi \in H\dot {-} \phi$.

Now we show that if ≼ is strict Horn compliant, then m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼) iff there is $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$ such that m i n(|¬ϕ|, ≼)≺m i n(|¬χ|, ≼).

For one direction, suppose m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼). We have to show there is $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$ such that m i n(|¬ϕ|, ≼)≺m i n(|¬χ|, ≼). If ϕ∨ψ is a Horn formula, then its only Horn strengthening is itself thus the result holds trivially. So suppose ϕ∨ψ is non-Horn. Then there exist two largest (by set inclusion) model sets X, Y⊆|ϕ∨ψ| such that X∩Y = ∅, for each x ∈ X there is y ∈ Y such that x∩y = w and w ∈ |¬ϕ∧¬ψ|, and for each y ∈ Y there is x ∈ X such that x∩y = w and w ∈ |¬ϕ∧¬ψ|. Since ≼ is strict Horn compliant, for each pair of x and y we have either w≼x or w≼y. Assume, without loss of generality, that w≼y for all such pairs. Let |χ| = |ϕ∨ψ|∖Y then, by the definition of Horn strengthening, we have $\chi \in \mathcal {H}\mathcal {S}(\phi )$. Since |¬χ| = |¬ϕ∧¬ψ| ∪ Y, then by the derivation of χ we have m i n(|¬ϕ∧¬ψ|, ≼)≼m i n(|¬χ|, ≼). Then by the transitivity of ≼ we have m i n(|¬ϕ|, ≼)≺m i n(|¬χ|, ≼).

For the other direction, suppose there is $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$ such that m i n(|¬ϕ|, ≼)≺m i n(|¬χ|, ≼). We have to show m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼). Since |χ|⊆|ϕ∨ψ| we have |¬ϕ∧¬ψ|⊆|¬χ| which implies m i n(|¬χ|, ≼)≼m i n(|¬ϕ∧¬ψ|, ≼). Then by the transitivity of ≼ we have m i n(|¬ϕ|, ≼)≺m i n(|¬ϕ∧¬ψ|, ≼).

It follows from the above results that if $\dot {-}$ is a SHCMHC function for H that is determined by the pre-order ≼, then it satisfies the following condition:

$(SHC\dot {-})$: $\psi \in H\dot {-}\phi$ iff ψ ∈ H and either ⊩ϕ or there is $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$ such that m i n(|¬ϕ|, ≼)≺m i n(|¬χ|, ≼).

Now we derive a relation ≤ over $\mathcal {L}$ from the pre-order ≼ as follows:

For $\phi ,\psi \in \mathcal {L}$, ϕ≤ψ iff m i n(|¬ϕ|, ≼)≼m i n(|¬ψ|, ≼)

It has been shown that the relation ≤ derived from ≼ as above is an epistemic entrenchment [26]. Thus ≤ can be used to determine an EHC function via $(HC\dot {-})$. Let − be the EHC function for H that is determined by ≤. It remains to show $H\dot {-}\phi =H-\phi$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$.

For one direction, suppose $\psi \in H\dot {-}\phi$. We need to show ψ ∈ H − ϕ. By $(SHC\dot {-})$, $\psi \in H\dot {-}\phi$ implies ψ ∈ H and either ⊩ϕ or there is $\chi \in \mathcal {H}\mathcal {S}(\phi \vee \psi )$ such that m i n(|¬ϕ|, ≼)≺m i n(|¬χ|, ≼). By the derivation of ≤, the last part implies ϕ<χ. Thus it follows from $(HC\dot {-})$ that ψ ∈ H − ϕ. The other direction can be proved in a similar manner. □

Proof for Theorem 6

Suppose |H|∩|ϕ|≠∅. By the construction of SHCIR, $|H*\phi |=Cl_{\cap }(|H\dot {-}\chi _{1}|\cup \cdots \cup |H\dot {-}\chi _{n}|)\cap |\phi |$ where $\dot {-}$ is the SHCMHC function that generates ∗ and $\mathcal {H}\mathcal {S}(\neg \phi )=\{\chi _{1},\ldots ,\chi _{n}\}$. By the definition of Horn strengthenings we have |χ _i |⊆|¬ϕ| which implies |ϕ|⊆|¬χ _i | for 1≤i≤n. Thus |H|∩|¬χ _i |≠∅ which implies H⊯χ _i . It then follows from $(H\dot {-} 3)$ that $H\dot {-}\chi _{i}=H$. Thus |H ∗ ϕ| = |H|∩|ϕ|. Then by the faithfulness of ≼, |H|∩|ϕ| = m i n(|ϕ|, ≼).

Now suppose |H|∩|ϕ| = ∅. For one direction, suppose μ ∈ |H ∗ ϕ|. We need to show μ ∈ m i n(|ϕ|, ≼). By the definition of SHCIR, |H ∗ ϕ| = C l _∩(C l _∩(H∪m i n(|¬χ ₁|,≼))∪⋯∪C l _∩(H∪m i n(|¬χ _n |,≼)))∩|ϕ|. Since μ ∈ |H ∗ ϕ|, by Lemma 5 there is ν ∈ C l _∩(H∪m i n(|¬χ ₁|,≼))∪⋯∪C l _∩(H∪m i n(|¬χ _n |,≼)) such that μ≼ν. Thus there is χ _i such that ν ∈ C l _∩(H∪m i n(|¬χ _i |,≼)). Again, by Lemma 5, there is σ ∈ H∪m i n(|¬χ _i |,≼) such that ν≼σ. By the faithfulness of ≼ we cannot have σ ∈ |H|, thus it must be that σ ∈ m i n(|¬χ _i |,≼). Since |ϕ|⊆|¬χ _i |, σ ∈ m i n(|¬χ _i |,≼) implies σ≼x for all x ∈ |ϕ|. By the transitivity of ≼, it follows from μ≼ν and ν≼σ that μ≼σ. Thus we have μ≼x for all x ∈ |ϕ| which implies μ ∈ m i n(|ϕ|, ≼).

For the other direction, suppose μ ∈ m i n(|ϕ|, ≼). We need to show μ ∈ |H ∗ ϕ|. Assume to the contrary that μ∉|H ∗ ϕ|. Recall that |H ∗ ϕ| = C l _∩(C l _∩(H∪m i n(|¬χ ₁|,≼))∪⋯∪C l _∩(H∪m i n(|¬χ _n |,≼)))∩|ϕ|. We first show that the assumption leads to |H ∗ ϕ|∩|ϕ| = ∅.

Assume there is ω ∈ |ϕ| such that ω ∈ |H ∗ ϕ| which implies ω ∈ C l _∩(C l _∩(H∪m i n(|¬χ ₁|,≼))∪⋯∪C l _∩(H∪m i n(|¬χ _n |,≼))). Again, by Lemma 5, there is σ ∈ m i n(|¬χ _i |,≼) such that ω≼σ for some $\chi _{i}\in \mathcal {H}\mathcal {S}(\neg \phi )$. Since μ ∈ m i n(|ϕ|, ≼) and ω ∈ |ϕ|, we have μ≼ω. It then follows from ω≼σ and μ≼ω that μ≼ω which implies μ ∈ |ϕ|⊆|¬χ _i |, a contradiction. Thus we have |H ∗ ϕ|∩|ϕ| = ∅.

Now Let ω ∈ m i n(|¬χ _i |,≼) for some $\chi _{i}\in \mathcal {H}\mathcal {S}(\neg \phi )$. Then there is $\chi _{j}\in \mathcal {H}\mathcal {S}(\neg \phi )$ such that ω∉|¬χ _j | for otherwise it follows from Lemma 2 that ω ∈ |ϕ| which contradicts |H ∗ ϕ|∩|ϕ| = ∅. Let ν ∈ m i n(|¬χ _j |,≼). Since ω ∈ |χ _j |, ν∉|χ _j |, and ν ∈ |¬ϕ|, we have by the definition of Horn strengthening that ω∩ν ∈ |ϕ|. Since ω, ν ∈ C l _∩(C l _∩(H∪m i n(|¬χ ₁|,≼))∪⋯∪C l _∩(H∪m i n(|¬χ _n |,≼))), we have ω∩ν ∈ C l _∩(C l _∩(H∪m i n(|¬χ ₁|,≼))∪⋯∪C l _∩(H∪m i n(|¬χ _n |,≼))) which implies |H ∗ ϕ|∩|ϕ|≠∅, a contradiction. □

Proof for Theorem 8

Let $\phi \in \mathcal {L}_{\text {\texttt {H}}}$ be such that $\mathcal {H}\mathcal {S}(\neg \phi )=\{\chi _{1},\ldots ,\chi _{n}\}$. By Lemma 2 we have |¬ϕ| = |χ ₁|∪⋯∪|χ _n | which implies, by De Morgan’s laws, |ϕ| = |¬χ ₁|∩⋯∩|¬χ _n |. Assume to the contrary that m i n(|¬χ|, ≼)∩|ϕ| = ∅ for all $\chi \in \mathcal {H}\mathcal {S}(\neg \phi )$. Let μ ∈ m i n(|¬χ _i |,≼) for some $\chi _{i}\in \mathcal {H}\mathcal {S}(\neg \phi )$, then there is $\chi _{j}\in \mathcal {H}\mathcal {S}(\neg \phi )$ such that μ∉|¬χ _j | for otherwise it follows from |ϕ| = |¬χ ₁|∩⋯∩|¬χ _n | that μ ∈ |ϕ| which contradicts the fact that m i n(|¬χ|, ≼)∩|ϕ| = ∅ for all $\chi \in \mathcal {H}\mathcal {S}(\neg \phi )$. Let ν ∈ m i n(|¬χ _j |,≼). Since μ ∈ |χ _j |, ν∉|χ _j | and ν ∈ |¬ϕ|, we have by the definition of Horn strengthening that μ∩ν ∈ |ϕ|. It then follows from |ϕ| = |¬χ ₁|∩⋯∩|¬χ _n |, that μ∩ν ∈ |¬χ _i | and μ∩ν ∈ |¬χ _j |. Due to the strict Horn compliance of ≼, we have either μ∩ν≼μ or μ∩ν≼ν which implies either μ∩ν ∈ m i n(|¬χ _i ,≼)| or μ∩ν ∈ m i n(|¬χ _j ,≼)|, a contradiction. Thus there is $\chi \in \mathcal {H}\mathcal {S}(\neg \phi )$ such that m i n(|¬χ|, ≼)∩|ϕ|≠∅.

Let’s assume, without loss of generality, that $\chi _{1}\in \mathcal {H}\mathcal {S}(\neg \phi )$ is such that m i n(|¬χ ₁|,≼)∩|ϕ|≠∅. By Theorem 6 we have |H ∗ ϕ| = m i n(|ϕ|, ≼). Thus it suffices to show $|H\dot {-}\chi _{1}+\phi | = min(|\phi |,\preceq )$. For one direction, we have, by the construction of MHC and IR, that $|H\dot {-}\chi _{1}+\phi |=Cl_{\cap } (|H|\cup min(|\neg \chi _{1}|,\preceq ))\cap |\phi | \subseteq Cl_{\cap }(Cl_{\cap }(H\cup min(|\neg \chi _{1}|,\preceq ))\cup \cdots \cup Cl_{\cap }(H\cup min(|\neg \chi _{n}|,\preceq )))\cap |\phi |=|H*\phi |= min(|\phi |,\preceq )$. For the other direction, suppose μ ∈ m i n(|ϕ|, ≼). Let ν ∈ m i n(|¬χ ₁|,≼)∩|ϕ|. Then we have μ≼ν. Since |ϕ|⊆|¬χ ₁|, we have μ ∈ |¬χ ₁|. Thus it follows from ν ∈ m i n(|¬χ ₁|,≼) and μ≼ν that μ ∈ m i n(|¬χ ₁|,≼). Thus m i n(|ϕ|, ≼)⊆m i n(|¬χ ₁|,≼)∩|ϕ|⊆C l _∩(|H| ∪ m i n _≼|¬χ ₁|)∩|ϕ| = |H − χ ₁ + ϕ|. □

Proof Proof for Lemma 6

Suppose μ, ν ∈ |ϕ| and μ≼ν. Since $\mathcal {H}\mathcal {S}(\neg \phi )=\{\chi _{1},\chi _{2},\ldots ,\chi _{n}\}$, we have by the definition of Horn strengthenings that |χ _i |⊆|¬ϕ| which implies |ϕ|⊆|¬χ _i | for 1≤i≤n. We first show $\mu \preceq ^{-_{1}}_{\chi _{1}} \nu$. There are three cases:

• Case 1,  μ, ν∉C l _∩(|H| ∪ m i n(|¬χ ₁|,≼)): $\mu \preceq ^{-_{1}}_{\chi _{1}} \nu$ follows immediately from HC2.

• Case 2,  μ ∈ C l _∩(|H| ∪ m i n(|¬χ ₁|,≼)): It follows from HC1 that μ is minimal in $\preceq ^{-_{1}}_{\chi _{1}}$. Thus $\mu \preceq ^{-_{1}}_{\chi _{1}}\nu$.

• Case 3,  ν ∈ C l _∩(|H| ∪ m i n(|¬χ ₁|,≼)) and μ∉C l _∩(|H| ∪ m i n(|¬χ ₁|,≼)): If ν ∈ |H| ∪ m i n(|¬χ ₁|,≼), then μ≼ν implies μ ∈ |H| ∪ m i n(|¬χ ₁|,≼), a contradiction. If ν∉|H| ∪ m i n(|¬χ ₁|,≼), then by Lemma 5, there is δ ∈ |H| ∪ m i n(|¬χ ₁|,≼) such that ν≼δ. It then follows from μ≼ν and the transitivity of ≼ that μ≼δ which implies μ ∈ |H| ∪ m i n(|¬χ ₁|,≼), a contradiction.

$\mu \preceq ^{-_{i}}_{\chi _{i}}\nu$ for 2≤i≤n can be proved inductively in the same manner as for $\preceq ^{-_{1}}_{\chi _{1}}$. The proof for the opposite direction is similar. □

Proof for Theorem 9

Suppose $\mathcal {H}\mathcal {S}(\neg \phi )=\{\chi _{1},\chi _{2},\ldots ,\chi _{n}\}$. Then by the definition of Horn strengthenings we have |χ _i |⊆|¬ϕ| which implies |ϕ|⊆|¬χ _i | for 1≤i≤n. It follows from the definition of SHCSR that $|H*\phi |= |((\cdots ((H-_{1}\chi _{1})-_{2}\chi _{2})\cdots )-_{n}\chi _{n})+\phi |= Cl_{\cap }(|H|\cup min(|\neg \chi _{1}|,\preceq ) \cup min(|\neg \chi _{2}|,\preceq ^{-_{1}}_{\chi _{1}})\cup \cdots \cup min(|\neg \chi _{n}|,\preceq ^{-_{n-1}}_{\chi _{n-1}}))\cap |\phi |$ where −₁ is the SHCMHC function that generates ∗. Note that HC1 and HC2 are used for deriving all the posterior pre-orders $\preceq _{\chi _{1}}^{-_{1}},\ldots ,\preceq _{\chi _{n-1}}^{-_{n-1}}$ which in turn determine the MHC functions −₂,…,− _n . Let $\preceq ^{-_{0}}_{\chi _{0}}=\preceq$.

For one direction, suppose ω ∈ |H ∗ ϕ|, we need to show ω ∈ m i n(|ϕ|, ≼) There are three cases:

• Case 1,  ω ∈ |H|: It follows from the faithfulness of ≼ that ω ∈ m i n(|ϕ|, ≼).

• Case 1,  there is χ _i such that $\omega \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$: Since ω ∈ |ϕ|⊆|¬χ _i |, $\omega \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$ implies $\omega \in min(|\phi |,\preceq ^{-_{i-1}}_{\chi _{i-1}})$. Since $min(|\phi |,\preceq )=min(|\phi |,\preceq ^{-_{i-1}}_{\chi _{i}-1})$ follows from Lemma 6, we have ω ∈ m i n(|ϕ|, ≼).

• Case 1,  ω is induced by models in $|H|\cup min(|\neg \chi _{1}|,\preceq ) \cup min(|\neg \chi _{2}|,\preceq ^{-_{1}}_{\chi _{1}})\cup \cdots \cup min(|\neg \chi _{n}|,\preceq ^{-_{n-1}}_{\chi _{n-1}})$ : Since ≼ satisfies SHC, it follows from Lemma 5 that there is $\mu \in |H|\cup min(|\neg \chi _{1}|,\preceq ) \cup min(|\neg \chi _{2}|,\preceq ^{-_{1}}_{\chi _{1}})\cup \cdots \cup min(|\neg \chi _{n}|,\preceq ^{-_{n-1}}_{\chi _{n-1}})$ such that ω≼μ. If μ ∈ |H|, then it follows from ω∉|H| and the faithfulness of ≼ that μ≺ω which contradicts ω≼μ. So there is χ _i such that $\mu \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$. Due to HC1 and HC2, the rankings for models of ϕ are not downgraded throughout the contraction sequence. It then follows from ω ∈ |ϕ| and ω≼μ that $\omega \preceq ^{-_{i-1}}_{\chi _{i-1}} \mu$. Thus $\omega \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$ which implies, as in Case 2, ω ∈ m i n(|ϕ|, ≼).

For the other direction, suppose ω ∈ m i n(|ϕ|, ≼), we need to show ω ∈ |H ∗ ϕ|. Assume ω∉|H ∗ ϕ|. We first show that the assumption implies |H ∗ ϕ|∩|ϕ| = ∅. Assume there is μ ∈ |ϕ| such that μ ∈ |H ∗ ϕ|. There are three cases.

• Case 1,  μ ∈ |H|: It follows from the faithfulness of ≼ and ω≼μ that ω ∈ |H|, a contradiction.

• Case 2,  there is χ _i such that $\mu \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$: It follows from Lemma 6 and ω≼μ that $\omega \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$, a contradiction.

• Case 3,  μ is induced by models in $|H|\cup min(|\neg \chi _{1}|,\preceq ) \cup min(|\neg \chi _{2}|,\preceq ^{-_{1}}_{\chi _{1}})\cup \cdots \cup min(|\neg \chi _{n}|,\preceq ^{-_{n-1}}_{\chi _{n-1}})$ : Since ≼ satisfies SHC, it follows from Lemma 5 that there is $\nu \in |H|\cup min(|\neg \chi _{1}|,\preceq ) \cup min(|\neg \chi _{2}|,\preceq ^{-_{1}}_{\chi _{1}})\cup \cdots \cup min(|\neg \chi _{n}|,\preceq ^{-_{n-1}}_{\chi _{n-1}})$ such that μ≼ν. It then follows from ω≼μ and the transitivity of ≼ that ω≼ν. Then, with the same reasoning as in the Case 3 above, we can derive a contradiction.

Since all cases lead to a contradiction, we have |H ∗ ϕ|∩|ϕ| = ∅. Let $\mu \in min(|\neg \chi _{i}|,\preceq ^{-_{i-1}}_{\chi _{i-1}})$ . Then there is χ _j such that μ∉|¬χ _j | for otherwise μ ∈ |ϕ| which leads to |H ∗ ϕ|∩|ϕ|≠∅. Let $\nu \in min(|\neg \chi _{j}|,\preceq ^{-_{j-1}}_{\chi _{j-1}})$. Since μ ∈ |χ _j |, ν∉|χ _j |, and ν ∈ |¬ϕ|, we have by the definition of Horn strengthening that μ∩ν ∈ |ϕ|. Since μ, ν ∈ |H ∗ ϕ| = C l _∩(|H ∗ ϕ|), we have μ∩ν ∈ |H ∗ ϕ| which implies |H ∗ ϕ|∩|ϕ|≠∅, a contradiction. □

Proof Proof for Lemma 7

For one direction suppose ω ∈ m i n(|ϕ|, ≼), we need to show ω ∈ m i n(|χ ₁|,≼)∪⋯∪m i n(|χ _n |,≼). By Lemma 2 there is $\chi _{i}\in \mathcal {H}\mathcal {S}(\phi )$ such that ω ∈ |χ _i |. By the definition of Horn strengthening, |χ|⊆|ϕ| for all Horn strengthenings χ of ϕ. Thus there is no χ such that μ ∈ |χ| and μ≺ω which implies $\chi _{i}\in min(\mathcal {H}\mathcal {S}(\phi ),\preceq )$ and we are done.

For the other direction, suppose ω ∈ m i n(|χ ₁|,≼)∪⋯∪m i n(|χ _n |,≼), we need to show ω ∈ m i n(|ϕ|, ≼). Without loss of generality, let ω ∈ m i n(|χ ₁|,≼). Since |χ ₁|⊆|ϕ|, we have ω ∈ |ϕ|. By the first part of the proof, we have m i n(|χ|, ≼)⊆m i n(|χ ₁|,≼)∪⋯∪m i n(|χ _n |,≼). Thus it follows from the definition of most preferred formulas and ω ∈ m i n(|χ ₁|,≼) that μ⊀ω for all μ ∈ m i n(|χ|, ≼) which implies ω ∈ m i n(|ϕ|, ≼). □

Proof for Theorem 11

For one direction, suppose $\dot {-}$ is a MHC function for H that is determined by the pre-order ≼, we need to show $\dot {-}$ is an IC function for H. Suppose ∗ is the MHR function for H that is determined by ≼. Let − be the IC function generated by ∗. It suffices to show $|H\dot {-}\phi |=|H-\phi |$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$. Let $min(\mathcal {H}\mathcal {S}(\neg \phi ),\preceq )=\{\chi _{1},\ldots ,\chi _{n}\}$. By the definition of MHC and IC functions, we have $|H\dot {-}\phi |=Cl_{\cap }(|H|\cup min(|\neg \phi |,\preceq ))$ and |H − ϕ| = C l _∩(|H| ∪ m i n(|χ ₁|,≼),…,m i n(|χ _n |,≼)). Thus it suffices to show |H| ∪ m i n(|¬ϕ|, ≼) = |H| ∪ m i n(|χ ₁|,≼),…,m i n(|χ _n |,≼) which follows immediately from Lemma 7.

For the other direction, suppose $\dot {-}$ is an IC function for H that is generated from the MHR function ∗ for H, we need to show $\dot {-}$ is a MHC function for H. Suppose the determining pre-order for ∗ is ≼. Now suppose − is a MHC function for H that is determined by ≼. It suffices to show $|H\dot {-}\phi |=|H-\phi |$ for all $\phi \in \mathcal {L}_{\text {\texttt {H}}}$. As in the first part of the proof, this comes down to showing |H| ∪ m i n(|¬ϕ|, ≼) = |H| ∪ m i n(|χ ₁|,≼),…,m i n(|χ _n |,≼) which follows immediately from Lemma 7. □