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Semantical Principles in the Modal Logic of Coalgebraic
"... Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natur ..."
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Cited by 31 (7 self)
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Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.
Representations of First Order Function Types as Terminal Coalgebras
 In Typed Lambda Calculi and Applications, TLCA 2001, number 2044 in Lecture Notes in Computer Science
, 2001
"... terminal coalgebras ..."
On the construction of free algebras for equational systems
 IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE
, 2009
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applica ..."
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Cited by 5 (4 self)
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.
Iteration monads
 Proceedings CMCS'01. Electronic Notes in Theoretical Computer Science 44
, 2001
"... ..."
Induction, Coinduction, and Adjoints
, 2002
"... We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and supp ..."
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Cited by 3 (2 self)
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We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(Y ) = X + F (Y ) exists; then a right adjoint G(X) to F (X) exists if and only if a nal coalgebra G(X) of the functor K(Y ) = X G(Y ) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y ) = X + F (Y ) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.
Term Equational Systems and Logics (Extended Abstract)
"... We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an intern ..."
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Cited by 2 (0 self)
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We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an application, we synthesise a sound and complete nominal equational logic, called Synthetic Nominal Equational Logic, based on the category of Nominal Sets.
A note on expressive coalgebraic logics for finitary set functors
 J. Log. Comput
"... This paper has two purposes. The first is to present a final coalgebra construction for finitary endofunctors on Set that uses a certain subset L ∗ of the limit L of the first ω terms in the final sequence. L ∗ is the set of points in L which arise from all coalgebras using their canonical morphisms ..."
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This paper has two purposes. The first is to present a final coalgebra construction for finitary endofunctors on Set that uses a certain subset L ∗ of the limit L of the first ω terms in the final sequence. L ∗ is the set of points in L which arise from all coalgebras using their canonical morphisms into L, and it was used earlier for different purposes in Kurz and Pattinson [5]. Viglizzo in [11] showed that the same set L ∗ carried a final coalgebra structure for functors in a certain inductively defined family. Our first goal is to generalize this to all finitary endofunctors; the result is implicit in Worrell [12]. The second goal is to use the final coalgebra construction to propose coalgebraic logics similar to those in [6] but for all finitary endofunctors F on Set. This time one can dispense with all conditions on F, construct a logical language LF directly from it, and prove that two points in a coalgebra satisfy the same sentences of LF iff they are identified by the final coalgebra morphism. The language LF is very spare, having no boolean connectives. This work on LF is thus a reworking of coalgebraic logic for finitary functors on sets. 1