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Mongruences and Cofree Coalgebras
 Algebraic Methods and Software Technology, number 936 in Lect. Notes Comp. Sci
, 1995
"... . A coalgebra is introduced here as a model of a certain signature consisting of a type X with various "destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in objectoriented programming: they provide access to the t ..."
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Cited by 30 (10 self)
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. A coalgebra is introduced here as a model of a certain signature consisting of a type X with various "destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in objectoriented programming: they provide access to the type (or state) X. We show that the category of such coalgebras and structure preserving functions is comonadic over sets. Therefore we introduce the notion of a `mongruence' (predicate) on a coalgebra. It plays the dual role of a congrence (relation) on an algebra. An algebra is a set together with a number of operations on this set which tell how to form (derived) elements in this set, possibly satisfying some equations. A typical example is a monoid, given by a set M with operations 1 ! M , M \Theta M ! M . Here 1 = f;g is a singleton set. In mathematics one usually considers only singletyped algebras, but in computer science one more naturally uses manytyped algebras like 1 ! list(A), A \Theta l...
Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
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Cited by 20 (6 self)
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We investigate the development of theories of types and computability via realizability.
Lax Logical Relations
 In 27th Intl. Colloq. on Automata, Languages and Programming, volume 1853 of LNCS
, 2000
"... Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with th ..."
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Cited by 15 (2 self)
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Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the prelogical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.
A Theory of Program Refinement
, 1998
"... We give a canonical program refinement calculus based on the lambda calculus and classical firstorder predicate logic, and study its proof theory and semantics. The intention is to construct a metalanguage for refinement in which basic principles of program development can be studied. The idea is t ..."
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Cited by 6 (1 self)
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We give a canonical program refinement calculus based on the lambda calculus and classical firstorder predicate logic, and study its proof theory and semantics. The intention is to construct a metalanguage for refinement in which basic principles of program development can be studied. The idea is that it should be possible to induce a refinement calculus in a generic manner from a programming language and a program logic. For concreteness, we adopt the simplytyped lambda calculus augmented with primitive recursion as a paradigmatic typed functional programming language, and use classical firstorder logic as a simple program logic. A key feature is the construction of the refinement calculus in a modular fashion, as the combination of two orthogonal extensions to the underlying programming language (in this case, the simplytyped lambda calculus). The crucial observation is that a refinement calculus is given by extending a programming language to allow indeterminate expressions (or ‘stubs’) involving the construction ‘some program x such that P ’. Factoring this into ‘some x...’
Quotients in Simple Type Theory
 Manuscript, Math. Inst
, 1994
"... Introduction Quotients are used throughout mathematics for constructing new objects from old, by collapsing part of the structure, see for example any textbook on algebra or topology. Here we give a completely general description of such quotients in a type theoretic language. We assume a simple ty ..."
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Cited by 2 (1 self)
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Introduction Quotients are used throughout mathematics for constructing new objects from old, by collapsing part of the structure, see for example any textbook on algebra or topology. Here we give a completely general description of such quotients in a type theoretic language. We assume a simple type theory, together with a predicate logic to reason about types and terms. Then quotients can be described as a left adjoint to a certain equalitypredicate functor. This gives us all the rules we need: formation, introduction, elimination and (fi) and (j)conversions for quotients. These will be described in the next section below. Subsequently, the new syntax is put to use in constructing Z from N, a poset from a preorder, the abelianization of a group, and tensor products\Omega and sums \Phi of abelian groups. All these constructions involve taking a suitable quotient. They will be de
Observational Ultrapowers of Polynomial Coalgebras
, 2001
"... Coalgebras of polynomial functors constructed from set of observable elements have been found useful in modelling various kinds of data types and statetransition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in [6], where it was shown t ..."
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Cited by 1 (1 self)
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Coalgebras of polynomial functors constructed from set of observable elements have been found useful in modelling various kinds of data types and statetransition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in [6], where it was shown that Boolean combinations of equations between terms of observable type form a natural language of observable formulas for specifying properties of polynomial coalgebras, and for giving a HennessyMilner style logical characterisation of observational indistinguishability (bisimilarity) of states. Here we give a structural characterisation of classes of coalgebras definable by observable formulas. This is an analogue for polynomial coalgebras of Birkhoff's celebrated characterisation of equationally definable classes of abstract algebras as being those closed under homomorphic images, subalgebras, and direct products. The coalgebraic characterisation involves a new notion of an observational ultrapower of a coalgebra, obtained from the usual notion of ultrapower by deleting states that assign "nonstandard" values to terms of observable type. A class of polynomial coalgebras is shown to be the class of all models of a set of observable formulas if, and only if, it is closed under images of bisimilarity relations, disjoint unions and observational ultrapowers.
Communication Algebras
, 1996
"... We introduce a semantic framework which generalises algebraic specifications by equipping algebras with descriptions of evaluation strategies. The semantic objects are not merely algebras, but algebras with associated data dependencies. The latter provide separate means for modeling computational co ..."
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Cited by 1 (1 self)
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We introduce a semantic framework which generalises algebraic specifications by equipping algebras with descriptions of evaluation strategies. The semantic objects are not merely algebras, but algebras with associated data dependencies. The latter provide separate means for modeling computational concerns along with the functional specifications usually captured by an algebra. The formalization of evaluation strategies allows the formal manipulation of how the operations of an algebra are to be evaluated. This formalization introduces (1) increased portability among different hardware platforms, as well as (2) allowing a potential increase in execution efficiency, since a chosen evaluation strategy may be tailored to a particular platform. We present the development process where algebraic specifications are equipped with data dependencies, the latter are then refined and, finally, mapped to actual hardware architectures. 1 Introduction The creed of algebraic specification is abstract...
Under consideration for publication in Math. Struct. in Comp. Science Computation Algebras
, 2000
"... We introduce a framework which generalizes algebraic specifications by equipping algebras with descriptions of evaluation strategies. The resulting abstract mathematical description allows one to model the implementation of algebras on various platforms in a way independent of the functionoriented ..."
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We introduce a framework which generalizes algebraic specifications by equipping algebras with descriptions of evaluation strategies. The resulting abstract mathematical description allows one to model the implementation of algebras on various platforms in a way independent of the functionoriented specifications. We are studying algebras with associated data dependencies. The latter provide separate means for modeling computational aspects apart from the functional specifications captured by an algebra. The formalization of evaluation strategies (1) introduces increased portability among different hardware platforms, and (2) allows a potential increase in execution efficiency, since a chosen evaluation strategy may be tailored to a particular platform. We present the development process where algebraic specifications are equipped with data dependencies, the latter are refined and, finally, mapped to actual hardware architectures. 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 ..."
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. 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...
A Modal Proof Theory for Polynomial Coalgebras
, 2004
"... The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe object ..."
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The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for objectoriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets. Acknowledgments I am deeply indebted to my supervisor, Professor Robert Goldblatt, for pointing me in the right direction and keeping my wheels on the tracks. His mathematical advice is the best anyone could hope for. I would like to thank Ranald Clouston for many discussions on logic and life in general. This thesis (and my life in general) are the better for them. I would like to thank all the people at the Centre for Logic, Language and Computation at Victoria who have taught me through my undergraduate years for introducing me to the exciting world of logic. Financially, I have been supported by a scholarship from the Logic and Computation programme of the New Zealand Institute for Mathematics and its Applications. I am grateful for the hospitality of the Institute for