Results 11  20
of
20
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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.
Proof in Flat Specifications
 Algebraic Foundations of Systems Specification, IFIP StateoftheArt Report
"... Introduction This chapter deals with the verification of data types. We put particular emphasis on ffl a uniform syntax for constructorbased specifications of both visible and hidden data types, ffl Gentzen clauses, rules and proofs as a uniform schema for presenting (proofs of) conjectures in a ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Introduction This chapter deals with the verification of data types. We put particular emphasis on ffl a uniform syntax for constructorbased specifications of both visible and hidden data types, ffl Gentzen clauses, rules and proofs as a uniform schema for presenting (proofs of) conjectures in a natural, flexible, structured and implementable way that keeps the gap between informal reasoning and formal deduction as small as possible, ffl a simple model and prooftheoretical basis to which all more or less advanced rules and methods can be reduced for showing their correctness, ffl providing the reader with syntactical criteria for the main conditions on a specification that shall be amenable to efficient proof and prototyping methods. Deductive aspects of specifications are also treated in other chapters of this book. Sections 2.6 through 2.9 provide basic notions and results for equational reasoning, i.e. f
unknown title
"... of Grigore Ro,su is approved, and it is acceptable in quality and form for publication on microfilm: ..."
Abstract
 Add to MetaCart
(Show Context)
of Grigore Ro,su is approved, and it is acceptable in quality and form for publication on microfilm:
Abstract CTCS 2002 Preliminary Version
"... We investigate the reasons for which the existence of certain right adjoints implies the existence of some final coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F ⊣ G be a pair of adjoint functors, and sup ..."
Abstract
 Add to MetaCart
We investigate the reasons for which the existence of certain right adjoints implies the existence of some final coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F ⊣ 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 final coalgebra ˇ 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. 1
Bisimulation and Apartness in Coalgebraic Specification
, 1995
"... . A first basic fact in algebra (or, in algebraic specification) is the existence of free algebras, as suitable sets of terms. In coalgebraic specification, terminal coalgebras are known to exist in Sets (and in other categories) via the standard limit construction. Here we characterize these termin ..."
Abstract
 Add to MetaCart
(Show Context)
. A first basic fact in algebra (or, in algebraic specification) is the existence of free algebras, as suitable sets of terms. In coalgebraic specification, terminal coalgebras are known to exist in Sets (and in other categories) via the standard limit construction. Here we characterize these terminal coalgebras as sets of "trees of observations". It is a standard result that elements of (the carrier of) a coalgebra are bisimilar (i.e. indistinguishable via the coalgebra operations) if and only if they have the same interpretation in the terminal coalgebra. This now becomes: if and only if they have the same tree of observations. Instead of putting emphasis on bisimulationwhich is a rather evasive notionwe consider its negation, which we write as #, and call "apartness ". It behaves like apartness in constructive mathematics. Indeed, the big advantage of apartness over bisimulation is that it can be established in a finite number of steps. It is a positive notion. Finally we show...
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... Circularity and the axioms of set theory In recent years, various kinds of circular phenomena have been studied by researchers from different disciplines. Examples are the study of automata in the science of computation, which generally contain cycles of state transitions; recursive domain equations ..."
Abstract
 Add to MetaCart
(Show Context)
Circularity and the axioms of set theory In recent years, various kinds of circular phenomena have been studied by researchers from different disciplines. Examples are the study of automata in the science of computation, which generally contain cycles of state transitions; recursive domain equations in computer science, which play a role in the semantics of concurrent programming languages with recursion; theories of truth and (self)reference in philosophy; and terminological cycles in artificial intelligence. Let us look at an example, taken from computation science, and treated in Vicious circles in all detail. Streams are infinite sequences of elements over an (alphabet) set A. Let a and b be in A and let s be the stream that informally is defined as consisting of an a followed by a b, again followed by an a and a b, and so on. Because of the fact that when the first two elements of s are removed, the same stream s is obtained again, it seems natural to model it by using the ordinary pairing operator: s = 〈a, 〈b, s〉〉, or, using t to denote the stream obtained by removing the first a from s, by the following system of equations: s = 〈a, t 〉 and
From grammars and automata to algebras and coalgebras
, 2013
"... Abstract. The increasing application of notions and results from category theory, especially from algebra and coalgebra, has revealed that any formal software or hardware model is constructor or destructorbased, a whitebox or a blackbox model. A highlystructured system may involve both construct ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The increasing application of notions and results from category theory, especially from algebra and coalgebra, has revealed that any formal software or hardware model is constructor or destructorbased, a whitebox or a blackbox model. A highlystructured system may involve both constructor and destructorbased components. The two model classes and the respective ways of developing them and reasoning about them are dual to each other. Roughly said, algebras generalize the modeling with contextfree grammars, word languages and structural induction, while coalgebras generalize the modeling with automata, Kripke structures, streams, process trees and all other state or objectoriented formalisms. We summarize the basic concepts of co/algebra and illustrate them at a couple of signatures including those used in language or compiler construction like regular expressions or acceptors. 1