A Type-Directed Negation Elimination

Abstract

In the modal µ-calculus, a formula is well-formed if each recursive variable occurs underneath an even number of negations. By means of De Morgan's laws, it is easy to transform any well-formed formula ϕ into an equivalent formula without negations -the negation normal form of ϕ. Moreover, if ϕ is of size n, the negation normal form of ϕ is of the same size O(n). The full modal µ-calculus and the negation normal form fragment are thus equally expressive and concise.In this paper we extend this result to the higher-order modal fixed point logic (HFL), an extension of the modal µ-calculus with higher-order recursive predicate transformers. We present a procedure that converts a formula of size n into an equivalent formula without negations of size O(n 2 ) in the worst case and O(n) when the number of variables of the formula is fixed.

Higher-Order Program Verification via HFL Model Checking

“…, ϕ k ) on integers (here, the arguments of integer operations or predicates will be restricted to integer expressions by the type system introduced below). Following [19], we have omitted negations, as any formula can be transformed to an equivalent negation-free formula [30]. We explain the meaning of each formula informally; the formal semantics is given in Sect.…”

On the relationship between higher-order recursion schemes and higher-order fixpoint logic

“…In its original formulation [34], HFL includes negations. In our setting, these are disallowed for simplicity, which is not a restriction since any closed HFL formula can be transformed to an equivalent negation-free formula [21].…”

Alternation Is Strict For Higher-Order Modal Fixpoint Logic

“…where P ∈ P, x ∈ V and X ∈ Y and τ is a simple type. Note that negation is not present explicitly in the logic since it can be eliminated [11].…”

“…where P ∈ P, x ∈ V and X ∈ Y and τ is a simple type. Note that negation is not present explicitly in the logic since it can be eliminated [11]. The binder λ (x v : τ).ϕ binds x in ϕ, the binder σ (X : τ).ϕ with σ ∈ {µ, ν} binds X in ϕ.…”