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Register a new userTypal Heterogeneous Equality Types
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Added by
Andrew Pitts
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Andrew M. Pitts
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The usual homogeneous form of equality type in Martin-Lof Type Theory contains identifications between elements of the same type. By contrast, the heterogeneous form of equality contains identifications between elements of possibly different types. This paper introduces a simple set of axioms for such types. The axioms are equivalent to the combination of systematic elimination rules for both forms of equality, albeit with typal (also known as propositional) computation properties, together with Streichers Axiom K, or equivalently, the principle of uniqueness of identity proofs.
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The $lambda$-calculus is a handy formalism to specify the evaluation of higher-order programs. It is not very handy, however, when one interprets the specification as an execution mechanism, because terms can grow exponentially with the number of $beta$-steps. This is why implementations of functional languages and proof assistants always rely on some form of sharing of subterms. These frameworks however do not only evaluate $lambda$-terms, they also have to compare them for equality. In presence of sharing, one is actually interested in the equality---or more precisely $alpha$-conversion---of the underlying unshared $lambda$-terms. The literature contains algorithms for such a sharing equality, that are polynomial in the sizes of the shared terms. This paper improves the bounds in the literature by presenting the first linear time algorithm. As others before us, we are inspired by Paterson and Wegmans algorithm for first-order unification, itself based on representing terms with sharing as DAGs, and sharing equality as bisimulation of DAGs.
This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.
In verified generic programming, one cannot exploit the structure of concrete data types but has to rely on well chosen sets of specifications or abstract data types (ADTs). Functors and monads are at the core of many applications of functional programming. This raises the question of what useful ADTs for verified functors and monads could look like. The functorial map of many important monads preserves extensional equality. For instance, if $f, g : A rightarrow B$ are extensionally equal, that is, $forall x in A, f x = g x$, then $map f : List A rightarrow List B$ and $map g$ are also extensionally equal. This suggests that preservation of extensional equality could be a useful principle in verified generic programming. We explore this possibility with a minimalist approach: we deal with (the lack of) extensional equality in Martin-Lofs intensional type theories without extending the theories or using full-fledged setoids. Perhaps surprisingly, this minimal approach turns out to be extremely useful. It allows one to derive simple generic proofs of monadic laws but also verified, generic results in dynamical systems and control theory. In turn, these results avoid tedious code duplication and ad-hoc proofs. Thus, our work is a contribution towards pragmatic, verified generic programming.
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Bogdan Aman (Romanian Academy
, Institute of Computer Science andn A.I.Cuza University
, Iasi
2019
This paper deals with the probabilistic behaviours of distributed systems described by a process calculus considering both probabilistic internal choices and nondeterministic external choices. For this calculus we define and study a typing system which extends the multiparty session types in order to deal also with probabilistic behaviours. The calculus and its typing system are motivated and illustrated by a running example.
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Lof type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.
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