173 research outputs found
The recursion hierarchy for PCF is strict
We consider the sublanguages of Plotkin's PCF obtained by imposing some bound
k on the levels of types for which fixed point operators are admitted. We show
that these languages form a strict hierarchy, in the sense that a fixed point
operator for a type of level k can never be defined (up to observational
equivalence) using fixed point operators for lower types. This answers a
question posed by Berger. Our proof makes substantial use of the theory of
nested sequential procedures (also called PCF B\"ohm trees) as expounded in the
recent book of Longley and Normann
Extensional Collapse Situations I: non-termination and unrecoverable errors
We consider a simple model of higher order, functional computation over the
booleans. Then, we enrich the model in order to encompass non-termination and
unrecoverable errors, taken separately or jointly. We show that the models so
defined form a lattice when ordered by the extensional collapse situation
relation, introduced in order to compare models with respect to the amount of
"intensional information" that they provide on computation. The proofs are
carried out by exhibiting suitable applied {\lambda}-calculi, and by exploiting
the fundamental lemma of logical relations
Resource modalities in game semantics
The description of resources in game semantics has never achieved the
simplicity and precision of linear logic, because of a misleading conception:
the belief that linear logic is more primitive than game semantics. We advocate
instead the contrary: that game semantics is conceptually more primitive than
linear logic. Starting from this revised point of view, we design a categorical
model of resources in game semantics, and construct an arena game model where
the usual notion of bracketing is extended to multi- bracketing in order to
capture various resource policies: linear, affine and exponential
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
Bar recursion is not computable via iteration
We show that the bar recursion operators of Spector and Kohlenbach,
considered as third-order functionals acting on total arguments, are not
computable in Goedel's System T plus minimization, which we show to be
equivalent to a programming language with a higher-order iteration construct.
The main result is formulated so as to imply the non-definability of bar
recursion in T + min within a variety of partial and total models, for instance
the Kleene-Kreisel continuous functionals. The paper thus supplies proofs of
some results stated in the book by Longley and Normann.
The proof of the main theorem makes serious use of the theory of nested
sequential procedures (also known as PCF Boehm trees), and proceeds by showing
that bar recursion cannot be represented by any sequential procedure within
which the tree of nested function applications is well-founded.Comment: 43 pages, 5 figure
Effectivity and Density in Domains A Survey
AbstractThis article surveys the main results on effectivity and totality in domain theory and its applications. A more abstract and informative proof of Normann's generalized density theorem for total functionals of finite type over the reals is presented
On Berry's conjectures about the stable order in PCF
PCF is a sequential simply typed lambda calculus language. There is a unique
order-extensional fully abstract cpo model of PCF, built up from equivalence
classes of terms. In 1979, G\'erard Berry defined the stable order in this
model and proved that the extensional and the stable order together form a
bicpo. He made the following two conjectures: 1) "Extensional and stable order
form not only a bicpo, but a bidomain." We refute this conjecture by showing
that the stable order is not bounded complete, already for finitary PCF of
second-order types. 2) "The stable order of the model has the syntactic order
as its image: If a is less than b in the stable order of the model, for finite
a and b, then there are normal form terms A and B with the semantics a, resp.
b, such that A is less than B in the syntactic order." We give counter-examples
to this conjecture, again in finitary PCF of second-order types, and also
refute an improved conjecture: There seems to be no simple syntactic
characterization of the stable order. But we show that Berry's conjecture is
true for unary PCF. For the preliminaries, we explain the basic fully abstract
semantics of PCF in the general setting of (not-necessarily complete) partial
order models (f-models.) And we restrict the syntax to "game terms", with a
graphical representation.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main
changes for this version: 4.1: proof of game term theorem corrected, 7.: the
improved chain conjecture is made precise, more references adde
PCF extended with real numbers: a domain-theoretic approach to higher-order exact real number computation
We develop a theory of higher-order exact real number computation based on Scott domain theory. Our main object of investigation is a higher-order functional programming language, Real PCF, which is an extension of PCF with a data type for real numbers and constants for primitive real functions. Real PCF has both operational and denotational semantics, related by a computational adequacy property.
In the standard interpretation of Real PCF, types are interpreted as continuous Scott domains. We refer to the domains in the universe of discourse of Real PCF induced by the standard interpretation of types as the real numbers type hierarchy. Sequences are functions defined on natural numbers, and predicates are truth-valued functions. Thus, in the real numbers types hierarchy we have real numbers, functions between real numbers, predicates defined on real numbers, sequences of real numbers, sequences of sequences of real numbers, sequences of functions, functionals mapping sequences to numbers (such as limiting operators), functionals mapping functions to numbers (such as integration and supremum operators), functionals mapping predicates to truth-values (such as existential and universal quantification operators), and so on.
As it is well-known, the notion of computability on a domain depends on the choice of an effective presentation. We say that an effective presentation of the real numbers type hierarchy is sound if all Real PCF definable elements and functions are computable with respect to it. The idea is that Real PCF has an effective operational semantics, and therefore the definable elements and functions should be regarded as concretely computable. We then show that there is a unique sound effective presentation of the real numbers type hierarchy, up to equivalence with respect to the induced notion of computability. We can thus say that there is an absolute notion of computability for the real numbers type hierarchy.
All computable elements and all computable first-order functions in the real numbers type hierarchy are Real PCF definable. However, as it is the case for PCF, some higher-order computable functions, including an existential quantifier, fail to be definable. If a constant for the existential quantifier (or, equivalently, a computable supremum operator) is added, the computational adequacy property remains true, and Real PCF becomes a computationally complete programming language, in the sense that all computable functions of all orders become definable.
We introduce induction principles and recursion schemes for the real numbers domain, which are formally similar to the so-called Peano axioms for natural numbers. These principles and schemes abstractly characterize the real numbers domain up to isomorphism, in the same way as the so-called Peano axioms for natural numbers characterize the natural numbers. On the practical side, they allow us to derive recursive definitions of real functions, which immediately give rise to correct Real PCF programs (by an application of computational adequacy). Also, these principles form the core of the proof of absoluteness of the standard effective presentation of the real numbers type hierarchy, and of the proof of computational completeness of Real PCF.
Finally, results on integration in Real PCF consisting of joint work with Abbas Edalat are included
An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus
This paper brings together two lines of research: implicit characterization
of complexity classes by Linear Logic (LL) on the one hand, and computation
over an arbitrary ring in the Blum-Shub-Smale (BSS) model on the other. Given a
fixed ring structure K we define an extension of Terui's light affine
lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K. We
show that this calculus captures the polynomial time function class FP(K):
every typed term can be evaluated in polynomial time and conversely every
polynomial time BSS machine over K can be simulated in this calculus.Comment: 11 pages. A preliminary version appeared as Research Report IAC CNR
Roma, N.57 (11/2004), november 200
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