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Incompleteness of Behavioral Logics
, 2000
"... Incompleteness results for behavioral logics are investigated. We show that there is a basic finite behavioral specification for which the behavioral satisfaction problem is not recursively enumerable, which means that there are no automatic methods for proving all true statements; in particular, be ..."
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Cited by 23 (8 self)
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Incompleteness results for behavioral logics are investigated. We show that there is a basic finite behavioral specification for which the behavioral satisfaction problem is not recursively enumerable, which means that there are no automatic methods for proving all true statements; in particular, behavioral logics do not admit complete deduction systems. This holds for all of the behavioral logics of which we are aware. We also prove that the behavioral satisfaction problem is not corecursively enumerable, which means that there is no automatic way to refute false statements in behavioral logics. In fact we show stronger results, that all behavioral logics are # 0 2 hard, and that, for some data algebras, the complexity of behavioral satisfaction is not even arithmetic; matching upper bounds are established for some behavioral logics. In addition, we show for the fixeddata case that if operations mayhave more than one hidden argument, then final models need not exist, so that the coalgebraic flavor of behavioral logic is lost.
Abstract Nonwellfounded trees in categories
"... nonwellfounded sets, as well as nonterminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call Mtypes. We derive existence results for Mtypes in locally cartesian closed pretoposes with a natural numbers object, usi ..."
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nonwellfounded sets, as well as nonterminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call Mtypes. We derive existence results for Mtypes in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with Mtypes under various topostheoretic constructions; namely, slicing, formation of coalgebras (for a cartesian comonad), and sheaves for an internal site. 1
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