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Datatypes in Memory
"... Abstract. Besides functional correctness, specifications must describe other properties of permissible implementations. We want to use simple algebraic techniques to specify resource usage alongside functional behaviour. In this paper we examine the space behaviour of datatypes, which depends on the ..."
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Abstract. Besides functional correctness, specifications must describe other properties of permissible implementations. We want to use simple algebraic techniques to specify resource usage alongside functional behaviour. In this paper we examine the space behaviour of datatypes, which depends on the representation of values in memory. In particular, it varies according to how much values are allowed to overlap, and how much they must be kept apart to ensure correctness for destructive spacereusing operations. We introduce a mechanism for specifying datatypes represented in a memory, with operations that may be destructive to varying degrees. We start from an abstract model notion for datainmemory and then show how to specify the observable behaviour of models. The method is demonstrated by specifications of listsinmemory and pointers; with a suitable definition of implementation, we show that listsinmemory may be implemented by pointers. We then present a method for proving implementations correct and show that it is sound and, under certain assumptions, complete. 1
Dynamic Coalgebraic Modalities
"... With this work we aim to place dynamic modal logics such as Propositional Dynamic Logic (PDL) [1] and Game Logic (GL) [4] in a uniform coalgebraic framework. In our view, a dynamic system S consists of the following ingredients: 1. A set S which represents the global states of S. 2. An algebra L of ..."
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With this work we aim to place dynamic modal logics such as Propositional Dynamic Logic (PDL) [1] and Game Logic (GL) [4] in a uniform coalgebraic framework. In our view, a dynamic system S consists of the following ingredients: 1. A set S which represents the global states of S. 2. An algebra L of labels (denoting actions, programs, games,...). 3. An interpretation of labels as Gcoalgebras on the state space S. 4. A collection of labelled modalities [α], for α ∈ L, where intuitively [α]ϕ reads: “after α, ϕ holds”. Formally, the interpretation of labels is a map σ: L → (GS) S which describes how actions change the global system state. The algebraic structure on L describes how one can compose actions into more complex ones. The same type of algebraic structure should be carried by (GS) S, and we say that σ is standard, if σ is an algebra homomorphism, which means that the semantics of actions is compositional. By considering the exponential adjoint ̂σ: S → (GS) L we obtain a behavioural description of the system in the form of a G Lcoalgebra. These two (equivalent) views of a dynamic system form the basis of our modelling. In short, σ describes structure and dynamics, and ̂σ describes behaviour and induces modalities. σ: L → (GS) S (algebraic view: structure, dynamics)
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