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Final coalgebras and the HennessyMilner property
 Annals of Pure and Applied Logic
"... The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the HennessyMilner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisi ..."
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The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the HennessyMilner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisimilarity relation is transitive. There are cases of functors that do have logics with the HennessyMilner property, but the only such logics have a proper class of formulas. The main theorem gives a representation of states of the final coalgebra as certain satisfiable sets of formulas. The key technical fact used is that any function between coalgebras that is truthpreserving and has a simple codomain must be a coalgebraic morphism.
A Logical Treatment of Constructive Duality
, 1997
"... We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a selfdual logic; constructible duality. We develop a selfdual model by considering an interval of worlds in an intuitionistic Kripke mo ..."
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We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a selfdual logic; constructible duality. We develop a selfdual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards, that is forward in the dual model. We define a symmetrization of the BethFitting construction which transforms an interval Kripke model in a selfdual algebra for the logic of constructible duality and back. We then show that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a pseudoBoolean algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a pseudoBoolea...
Subtyping, Declaratively An Exercise in Mixed Induction and Coinduction
"... Abstract. It is natural to present subtyping for recursive types coinductively. However, Gapeyev, Levin and Pierce have noted that there is a problem with coinductive definitions of nontrivial transitive inference systems: they cannot be “declarative”—as opposed to “algorithmic ” or syntaxdirected ..."
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Abstract. It is natural to present subtyping for recursive types coinductively. However, Gapeyev, Levin and Pierce have noted that there is a problem with coinductive definitions of nontrivial transitive inference systems: they cannot be “declarative”—as opposed to “algorithmic ” or syntaxdirected—because coinductive inference systems with an explicit rule of transitivity are trivial. We propose a solution to this problem. By using mixed induction and coinduction we define an inference system for subtyping which combines the advantages of coinduction with the convenience of an explicit rule of transitivity. The definition uses coinduction for the structural rules, and induction for the rule of transitivity. We also discuss under what conditions this technique can be used when defining other inference systems. The developments presented in the paper have been mechanised using Agda, a dependently typed programming language and proof assistant. 1
Christ Jesus, our Lord and Savior. PREDICATE ANSWER SET PROGRAMMING WITH COINDUCTION
, 2009
"... to ..."
ControlFlow Analysis of Function Calls and Returns by Abstract Interpretation
, 2012
"... Abstract interpretation techniques are used to derive a controlflow analysis for a simple higherorder functional language. The analysis approximates the interprocedural controlflow of both function calls and returns in the presence environment, the analysis computes for each expression an abstrac ..."
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Abstract interpretation techniques are used to derive a controlflow analysis for a simple higherorder functional language. The analysis approximates the interprocedural controlflow of both function calls and returns in the presence environment, the analysis computes for each expression an abstract callstack, effectively approximating where function calls return. The analysis is systematically machine of Flanagan et al. using a series of Galois connections. We prove that the analysis is equivalent to an analysis obtained by first transforming the program into continuationpassing style and then performing control flow analysis of the transfored program. We then show how the analysis induces an equivalent constraintbased formulation, thereby providing a rational reconstruction of a constraintbased CFA from abstract interpretation principles. 1.