37 research outputs found
Untangling Typechecking of Intersections and Unions
Intersection and union types denote conjunctions and disjunctions of
properties. Using bidirectional typechecking, intersection types are relatively
straightforward, but union types present challenges. For union types, we can
case-analyze a subterm of union type when it appears in evaluation position
(replacing the subterm with a variable, and checking that term twice under
appropriate assumptions). This technique preserves soundness in a call-by-value
semantics.
Sadly, there are so many choices of subterms that a direct implementation is
not practical. But carefully transforming programs into let-normal form
drastically reduces the number of choices. The key results are soundness and
completeness: a typing derivation (in the system with too many subterm choices)
exists for a program if and only if a derivation exists for the let-normalized
program.Comment: In Proceedings ITRS 2010, arXiv:1101.410
Search for Program Structure
The community of programming language research loves the Curry-Howard correspondence between proofs and programs. Cut-elimination as computation, theorems for free, \u27call/cc\u27 as excluded middle, dependently typed languages as proof assistants, etc.
Yet we have, for all these years, missed an obvious observation: "the structure of programs corresponds to the structure of proof search". For pure programs and intuitionistic logic, more is known about the latter than the former. We think we know what programs are, but logicians know better!
To motivate the study of proof search for program structure, we retrace recent research on applying focusing to study the canonical structure of simply-typed lambda-terms. We then motivate the open problem of extending canonical forms to support richer type systems, such as polymorphism, by discussing a few enticing applications of more canonical program representations
Refinement Types for Logical Frameworks and Their Interpretation as Proof Irrelevance
Refinement types sharpen systems of simple and dependent types by offering
expressive means to more precisely classify well-typed terms. We present a
system of refinement types for LF in the style of recent formulations where
only canonical forms are well-typed. Both the usual LF rules and the rules for
type refinements are bidirectional, leading to a straightforward proof of
decidability of typechecking even in the presence of intersection types.
Because we insist on canonical forms, structural rules for subtyping can now be
derived rather than being assumed as primitive. We illustrate the expressive
power of our system with examples and validate its design by demonstrating a
precise correspondence with traditional presentations of subtyping. Proof
irrelevance provides a mechanism for selectively hiding the identities of terms
in type theories. We show that LF refinement types can be interpreted as
predicates using proof irrelevance, establishing a uniform relationship between
two previously studied concepts in type theory. The interpretation and its
correctness proof are surprisingly complex, lending support to the claim that
refinement types are a fundamental construct rather than just a convenient
surface syntax for certain uses of proof irrelevance
A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions
The paper describes the refinement algorithm for the Calculus of
(Co)Inductive Constructions (CIC) implemented in the interactive theorem prover
Matita. The refinement algorithm is in charge of giving a meaning to the terms,
types and proof terms directly written by the user or generated by using
tactics, decision procedures or general automation. The terms are written in an
"external syntax" meant to be user friendly that allows omission of
information, untyped binders and a certain liberal use of user defined
sub-typing. The refiner modifies the terms to obtain related well typed terms
in the internal syntax understood by the kernel of the ITP. In particular, it
acts as a type inference algorithm when all the binders are untyped. The
proposed algorithm is bi-directional: given a term in external syntax and a
type expected for the term, it propagates as much typing information as
possible towards the leaves of the term. Traditional mono-directional
algorithms, instead, proceed in a bottom-up way by inferring the type of a
sub-term and comparing (unifying) it with the type expected by its context only
at the end. We propose some novel bi-directional rules for CIC that are
particularly effective. Among the benefits of bi-directionality we have better
error message reporting and better inference of dependent types. Moreover,
thanks to bi-directionality, the coercion system for sub-typing is more
effective and type inference generates simpler unification problems that are
more likely to be solved by the inherently incomplete higher order unification
algorithms implemented. Finally we introduce in the external syntax the notion
of vector of placeholders that enables to omit at once an arbitrary number of
arguments. Vectors of placeholders allow a trivial implementation of implicit
arguments and greatly simplify the implementation of primitive and simple
tactics