Impredicative types
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being … Zobacz więcej The terms "predicative" and "impredicative" were introduced by Russell (1907), though the meaning has changed a little since then. Solomon Feferman provides a historical review of … Zobacz więcej • Gödel, Escher, Bach • Impredicative polymorphism • Logicism • Richard's paradox Zobacz więcej Witryna11 paź 2024 · Known impredicative encodings of various inductive types in System F, such as the type of natural numbers do not satisfy the relevant -computation rules. The aim of this work is to refine the System F encodings by moving to a system of HoTT with an impredicative universe, so that the relevant -rules are satisfied (along with all the …
Impredicative types
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Witrynasynthetic guarded domain theory, BI-hyperdoctrine, impredicative polymorphism, recursive types 1 Introduction State of the art program logics such as Iris [30] and the Veri ed Software Toolchain [1] typically combine two design decisions: the basic sorts of the logic are generated by the raw, untyped terms of the programming WitrynaType theories tend towards predicativity mainly socio-technical reasons. First, the informal concept of impredicativity can be formalized in (at least) two different ways. …
WitrynaIt is well known that impredicative type systems do not have set theoretical semantics. This paper takes a look at semantics of inductive types in impredicative type systems. A generalized inductive type is interpreted as an omega set generated by effectivizing a certain rule set. The result provides a semantic justification of inductive types in the … Witryna10 maj 2024 · Inductive types can be better but weak inductive types are cool too. I know you can use impredicative universes as in System F to encode inductive types. Or you can just accept a universe bump. And apparently if you internalize a small amount of parametricity you can construct appropriate induction principles.
Witryna10 maj 2024 · Inductive types can be better but weak inductive types are cool too. I know you can use impredicative universes as in System F to encode inductive … Witryna23 maj 2024 · Idea Relation to Predicativity Definition Methods Upper bounds Lower bounds Table of Ordinal Analyses Ordinal notations Systems of first-order arithmetic (FOA) Systems of second-order arithmetic (SOA) Kripke Platek Set theories Martin-Löf type theories Constructive Set theories Explicit mathematics systems References Idea
WitrynaWe make no mention of any “calculus” and pay no attention to impredicative type systems. While these applications are interesting and important, they are not the main point of parametricity in our view. Rather, parametricity re-invokes the same intuitions that led to the notion of natural transformations and the definition of 3
In a predicative type system (also known as a prenex polymorphic system), type variables may not be instantiated with polymorphic types. Predicative type theories include Martin-Löf type theory and NuPRL. This is very similar to what is called "ML-style" or "Let-polymorphism" (technically ML's Let-polymorphism has a few other syntactic restrictions). This restriction makes the distinction between polymorphic and non-polymorphic types very important; thus in predicative systems po… citrix files for outlook not showing upWitryna25 cze 2016 · The first approach, where you take the maximum among the type of A and B is the so called predicative approach, giving rise to a strictly stratified type theory. … dickinsonmouthWitrynaBesides higher-rank types, HMF also supports the other ingredient for first-class polymorphism, namely impredicative instantiation, where type variables can be instantiated with polymorphic types (instead of just monomorphic types). We believe that this is a cru-cial property that enables the use of normal polymorphic abstrac- citrix file for outlookWitryna40. Coq has a type Prop of proof irrelevant propositions which are discarded during extraction. What are the reason for having this if we use Coq only for proofs. Prop is impredicative, so Prop : Prop, however, Coq automatically infers universe indexes and we can use Type (i) instead everywhere. It seems Prop complicates everything a lot. dickinson motel dickinson txWitryna3 paź 2024 · Impredicative types are an advanced form of polymorphism, to be contrasted with rank-N types. Standard Haskell allows polymorphic types via the use of type variables, which are understood to be universally quantified: id :: a -> a means "for all types a, id can take an argument and return a result of that type". citrix file for windowsWitrynaof impredicative type systems, since it makes it possible to interpret universes, intersection types and subtyping in the same framework. But the most interesting feature of the model is that the inhabitants of impredicative types are not interpreted in a syntactical way, but by the points of an arbitrary coherence space, provided it is a … dickinson motorsportsWitryna1 sty 2001 · These type theories combine the impredicative type of propositions2, from the calculus of constructions, , with the inductive types and hierarchy of type universes of Martin-Löf’s constructive type theory, . Intuitively there is an easy way to determine an upper bound on the proof theoretic strength. citrix file association windows 10