Results 1 
5 of
5
Free modal algebras: a coalgebraic perspective
"... Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1
Completeness for flat modal fixpoint logics
 Annals of Pure and Applied Logic, 162(1):55 – 82
, 2010
"... This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for each γ ∈ Γ. The term ♯γ(ϕ1,..., ϕn) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x, ϕ1,..., ϕn). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L♯(Γ) on Kripke frames. We prove two results that solve this problem. First, let K♯(Γ) be the logic obtained from the basic polymodal K by adding a KozenPark style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ♯γ. Provided that each indexing formula γ satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K + ♯ (Γ) of K♯(Γ). This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ♯γ, of size bounded by the length of γ. Thus the axiom system K + (Γ) is finite whenever Γ is finite.
Completeness for the coalgebraic cover modality
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2012
"... We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics of ∇T is defined in terms of a categorically defined relation lifting operation T. As the main contributions of our paper we introduce a derivation system M, and prove that M provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, and we employ Pattinson’s stratification method, showing that our derivation system can be stratified in ω many layers, corresponding to the modal depth of the formulas involved. In the proof of our main result we identify some new concepts and obtain some auxiliary results of independent interest. We survey properties of the notion T of relation lifting, induced by an arbitrary but fixed set functor T. We introduce a category Pres of Boolean algebra presentations, and establish an adjunction between Pres and the category BA of Boolean algebras. Given the fact that our derivation system M involves only formulas of depth one, it can be encoded as a functor
GENERALIZED POWERLOCALES VIA RELATION LIFTING
"... Abstract. This paper introduces an endofunctor VT ..."
MSc in Logic
, 2011
"... In this thesis we study relation liftings in the context of coalgebraic modal logic. ..."
Abstract
 Add to MetaCart
In this thesis we study relation liftings in the context of coalgebraic modal logic.