CONTENTS

232 MIHAIELA LUPEA (1)
form d = α:β
γ , where: α is called prerequisite, β is called justification and γ is called
consequent. A default d = α:β
γ can be applied and thus derive γ if α is believed and
”it is consistent to assumed β”(meaning that ¬β is not believed).
From all versions of default logics, constrained default logic has the most desirable
formal properties: supraclassicality, semi-monotonicity, commitment to assumptions,
due to the global consistency condition for the applied defaults.
Definition 2.2.[4] Let D be a set of defaults. The set of residues and the set
of justifications of D with respect to a set C of formulas are defined as follows:
ResC D =
α α:β
γ | γ ∈ D, C ∪ {β, γ} ⇒ false ,
Justif C D =
β| α:β
γ ∈ D, C ∪ {β, γ} ⇒ false .
The residues corresponding to the applied defaults are monotonic rules and are
used to reduce the nonmonotonic reasoning process modelled by constrained default
logic into a monotonic one according to the following theorem based on a theorem
from [4].
Theorem 2.1. Let ∆ = (D, W ) be a default theory. (E, C) is a constrained
extension of ∆ if E = T hres (W, ResC D ), C = T h(T hres (W, ResC D ) ∪ Justif C D ), where
T h(·) is the classical consequence operator and T hres (·, R) is the consequence operator
of the predicate formal system enhanced with the set R of residues.
E is the actual extension embedded in the reasoning context C. We remark that
the nonmonotonic reasoning process modelled by constrained default logic is guided
by a maximal consistent reasoning context.
3. Sequent and anti-sequent calculi for residues
The two complementary systems: sequent and anti-sequent calculi [1] for classical
logics are enhanced with specific rules for residues [1, 5] preserving the soundness and
completness of the new axiomatic systems. We use the same metasymbols ⇒ and ⇒
to express the inference relations in classical sequent/anti-sequent calculi and in the
corresponding systems, enhanced with residues.
Sequent rules for residues: Anti-sequent rules for residues:
(Re1) Γ⇒Ψ
Γ, α
γ ⇒Ψ Γ⇒α Γ,γ⇒Ψ
(Re2) Γ, α
γ ⇒Ψ Γ⇒α Γ⇒Ψ
(Re3) Γ, α
γ ⇒Ψ
(Re4) Γ,γ⇒Ψ
Γ, α
γ ⇒Ψ
4. Axiomatization of skeptical reasoning in constrained logic
Definition 4.1. Let ∆ = (D, W ) be a default theory. A skeptical constrained
default sequent has the syntax: Constr; (W, D); Res ↦−→ U. The set U of formulas is
called succedent. The antecedent contains Constr(a set of constraints expressed using
the modalities: M and L), the default theory (W, D) and Res (the set of residues
corresponding to the applied defaults).