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Observational Equality, Now!
 A SUBMISSION TO PLPV 2007
, 2007
"... This paper has something new and positive to say about propositional equality in programming and proof systems based on the CurryHoward correspondence between propositions and types. We have found a way to present a propositional equality type • which is substitutive, allowing us to reason by repla ..."
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Cited by 23 (8 self)
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This paper has something new and positive to say about propositional equality in programming and proof systems based on the CurryHoward correspondence between propositions and types. We have found a way to present a propositional equality type • which is substitutive, allowing us to reason by replacing equal for equal in propositions; • which reflects the observable behaviour of values rather than their construction: in particular, we have extensionality— functions are equal if they take equal inputs to equal outputs; • which retains strong normalisation, decidable typechecking and canonicity—the property that closed normal forms inhabiting datatypes have canonical constructors; • which allows inductive data structures to be expressed in terms of a standard characterisation of wellfounded trees; • which is presented syntactically—you can implement it directly, and we are doing so—this approach stands at the core of Epigram 2; • which you can play with now: we have simulated our system by a shallow embedding in Agda 2, shipping as part of the standard examples package for that system [20]. Until now, it has always been necessary to sacrifice some of these aspects. The closest attempt in the literature is Altenkirch’s construction of a setoidmodel for a system with canonicity and extensionality on top of an intensional type theory with proofirrelevant propositions [4]. Our new proposal simplifies Altenkirch’s construction by adopting McBride’s heterogeneous approach to equality [18].
The Interpretation of Inuitionistic . . .
, 2008
"... We give an intuitionistic view of Seely’s interpretation of MartinLöf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use MartinLöf type theory itself as metalanguage, and Ecategories, the appropriate notion of categories when working in this metalanguage. As a ..."
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We give an intuitionistic view of Seely’s interpretation of MartinLöf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use MartinLöf type theory itself as metalanguage, and Ecategories, the appropriate notion of categories when working in this metalanguage. As an Ecategorical substitute for the formal system of MartinLöf type theory we use Ecategories with families (Ecwfs). These come in two flavours: groupoidstyle Ecwfs and proofirrelevant Ecwfs. We then analyze Seely’s interpretation as consisting of three parts. The first part is purely categorical: the interpretation of groupoidstyle Ecwfs in Elocally cartesian closed categories. (The key part of this interpretation has been typechecked in the Coq system.) The second is a coherence problem which relates groupoidstyle Ecwfs with proofirrelevant ones. The third is a purely syntactic problem: that proofirrelevant Ecwfs are equivalent to traditional lambda calculus based formulations of MartinLöf type theory.
Unified Categorical Models for Three Typical Resource Allocation Problems ⋆
"... This paper concerns formalization of resource allocation using category theory, rather than a new algorithm to solve these problems. The unified and efficient categorical models for three specific resource allocation problemsincluding dinning philosophers problem, drinking philosophers problem and ..."
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This paper concerns formalization of resource allocation using category theory, rather than a new algorithm to solve these problems. The unified and efficient categorical models for three specific resource allocation problemsincluding dinning philosophers problem, drinking philosophers problem and committee coordination problem is originally presented based on ChandyMisra’s acyclic precedence graph strategy and our previous experience in defining the categorical semantics for distributed dinning philosophers problem. Four categories (including Dinners Category, Drinkers Category, Committees Category and Functors Category) defined in our paper not only facilitate to formalize usual concepts (such as task, resource, precedence) of resource allocation problems, but also give good directions to reason the relationships between these three typical problems. Finally, we formally proof some properties of symmetry, safety (nondeadlock), liveness (nonstarvation) and concurrency, which all satisfied in our models.