1,752 research outputs found
Head reduction and normalization in a call-by-value lambda-calculus
Recently, a standardization theorem has been proven for a variant of Plotkin\u27s call-by-value lambda-calculus extended by means of two commutation rules (sigma-reductions): this result was based on a partitioning between head and internal reductions. We study the head normalization for this call-by-value calculus with sigma-reductions and we relate it to the weak evaluation of original Plotkin\u27s call-by-value lambda-calculus. We give also a (non-deterministic) normalization strategy for the call-by-value lambda-calculus with sigma-reductions
First Class Call Stacks: Exploring Head Reduction
Weak-head normalization is inconsistent with functional extensionality in the
call-by-name -calculus. We explore this problem from a new angle via
the conflict between extensionality and effects. Leveraging ideas from work on
the -calculus with control, we derive and justify alternative
operational semantics and a sequence of abstract machines for performing head
reduction. Head reduction avoids the problems with weak-head reduction and
extensionality, while our operational semantics and associated abstract
machines show us how to retain weak-head reduction's ease of implementation.Comment: In Proceedings WoC 2015, arXiv:1606.0583
A Type System For Call-By-Name Exceptions
We present an extension of System F with call-by-name exceptions. The type
system is enriched with two syntactic constructs: a union type for programs
whose execution may raise an exception at top level, and a corruption type for
programs that may raise an exception in any evaluation context (not necessarily
at top level). We present the syntax and reduction rules of the system, as well
as its typing and subtyping rules. We then study its properties, such as
confluence. Finally, we construct a realizability model using orthogonality
techniques, from which we deduce that well-typed programs are weakly
normalizing and that the ones who have the type of natural numbers really
compute a natural number, without raising exceptions.Comment: 25 page
Discriminating Lambda-Terms Using Clocked Boehm Trees
As observed by Intrigila, there are hardly techniques available in the
lambda-calculus to prove that two lambda-terms are not beta-convertible.
Techniques employing the usual Boehm Trees are inadequate when we deal with
terms having the same Boehm Tree (BT). This is the case in particular for fixed
point combinators, as they all have the same BT. Another interesting equation,
whose consideration was suggested by Scott, is BY = BYS, an equation valid in
the classical model P-omega of lambda-calculus, and hence valid with respect to
BT-equality but nevertheless the terms are beta-inconvertible. To prove such
beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey
information of the tempo in which the data in the BT are produced. Boehm Trees
are thus enriched with an intrinsic clock behaviour, leading to a refined
discrimination method for lambda-terms. The corresponding equality is strictly
intermediate between beta-convertibility and Boehm Tree equality, the equality
in the model P-omega. An analogous approach pertains to Levy-Longo and
Berarducci Trees. Our refined Boehm Trees find in particular an application in
beta-discriminating fixed point combinators (fpc's). It turns out that Scott's
equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a
variety of production schemes of which the simplest was found by Boehm, stating
that new fpc's are obtained by postfixing the term SI, also known as Smullyan's
Owl. We prove that all these newly generated fpc's are indeed new, by
considering their clocked BT's. Even so, not all pairs of new fpc's can be
discriminated this way. For that purpose we increase the discrimination power
by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257
New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized
The definitional equality of an intensional type theory is its test of type
compatibility. Today's systems rely on ordinary evaluation semantics to compare
expressions in types, frustrating users with type errors arising when
evaluation fails to identify two `obviously' equal terms. If only the machine
could decide a richer theory! We propose a way to decide theories which
supplement evaluation with `-rules', rearranging the neutral parts of
normal forms, and report a successful initial experiment.
We study a simple -calculus with primitive fold, map and append operations on
lists and develop in Agda a sound and complete decision procedure for an
equational theory enriched with monoid, functor and fusion laws
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