281 research outputs found
Retraction map categories and their applications to the construction of lambda calculus models
This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of D∞, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category C equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of C is R=(R, ⩽, i, j), where ⩽ is a partial order relation on the set |C| of all the objects of C, R is the category of the poset (|C|, ⩽), and i and j are functors from R to C and from Rop to C that satisfy the conditions: (1) j a, b ∘ i a, b ⩾ ida and (2) i a, b ∘ j a, b ⩽ idb for every arrow a, b: a → b in R (i.e., a⩽b). The ɛ-category E=E(C, R) of C w.r.t. R is the category whose objects are ideals of (|C|, ⩽) and whose arrows are ideals of (C, ⊑), where ⩽ is the partial order relation in R and ⊑ is the partial order relation defined by f ⊑ g iff dom(f)⩽dom(g), cod(f)⩽cod(g) in R and f⩽j a, b ∘ g ∘ i(a, b in C. We show that every ɛ-category E=E(C, R) is also an order-enriched ccc. Moreover when E and R satisfy a particular condition, E(C, R) has a reflexive object. For example, if there is an ideal U of (|C|, ⩽) satisfying the following conditions, then U is isomorphic to UU in E and a λ-algebra is constructed from E and U: (1) for every pair of a, b ∈ U, U contains ba, and (2) for every c ∈ U, there are a, b ∈ U such that c ∈ ba. We reconstruct Pω and D∞ using ɛ-categories
Categorical Realizability for Non-symmetric Closed Structures
In categorical realizability, it is common to construct categories of
assemblies and categories of modest sets from applicative structures. These
categories have structures corresponding to the structures of applicative
structures. In the literature, classes of applicative structures inducing
categorical structures such as Cartesian closed categories and symmetric
monoidal closed categories have been widely studied. In this paper, we expand
these correspondences between categories with structure and applicative
structures by identifying the classes of applicative structures giving rise to
closed multicategories, closed categories, monoidal bi-closed categories as
well as (non-symmetric) monoidal closed categories. These applicative
structures are planar in that they correspond to appropriate planar lambda
calculi by combinatory completeness. These new correspondences are tight: we
show that, when a category of assemblies has one of the structures listed
above, the based applicative structure is in the corresponding class. In
addition, we introduce planar linear combinatory algebras by adopting linear
combinatory algebras of Abramsky, Hagjverdi and Scott to our planar setting,
that give rise to categorical models of the linear exponential modality and the
exchange modality on the non-symmetric multiplicative intuitionistic linear
logic
Metatheorems about convertibility in typed lambda calculi : applications to CPS transform and "free theorems"
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.Includes bibliographical references (p. 98-96).by Jakov Kuc̆an.Ph.D
High-level signatures and initial semantics
We present a device for specifying and reasoning about syntax for datatypes,
programming languages, and logic calculi. More precisely, we study a notion of
signature for specifying syntactic constructions.
In the spirit of Initial Semantics, we define the syntax generated by a
signature to be the initial object---if it exists---in a suitable category of
models. In our framework, the existence of an associated syntax to a signature
is not automatically guaranteed. We identify, via the notion of presentation of
a signature, a large class of signatures that do generate a syntax.
Our (presentable) signatures subsume classical algebraic signatures (i.e.,
signatures for languages with variable binding, such as the pure lambda
calculus) and extend them to include several other significant examples of
syntactic constructions.
One key feature of our notions of signature, syntax, and presentation is that
they are highly compositional, in the sense that complex examples can be
obtained by assembling simpler ones. Moreover, through the Initial Semantics
approach, our framework provides, beyond the desired algebra of terms, a
well-behaved substitution and the induction and recursion principles associated
to the syntax.
This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi,
which, in turn, was directly inspired by some earlier work of
Ghani-Uustalu-Hamana and Matthes-Uustalu.
The main results presented in the paper are computer-checked within the
UniMath system.Comment: v2: extended version of the article as published in CSL 2018
(http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in
Section 1.5 of the paper; v3: small corrections throughout the paper, no
major change
Monoidal computer III: A coalgebraic view of computability and complexity
Monoidal computer is a categorical model of intensional computation, where
many different programs correspond to the same input-output behavior. The
upshot of yet another model of computation is that a categorical formalism
should provide a much needed high level language for theory of computation,
flexible enough to allow abstracting away the low level implementation details
when they are irrelevant, or taking them into account when they are genuinely
needed. A salient feature of the approach through monoidal categories is the
formal graphical language of string diagrams, which supports visual reasoning
about programs and computations.
In the present paper, we provide a coalgebraic characterization of monoidal
computer. It turns out that the availability of interpreters and specializers,
that make a monoidal category into a monoidal computer, is equivalent with the
existence of a *universal state space*, that carries a weakly final state
machine for any pair of input and output types. Being able to program state
machines in monoidal computers allows us to represent Turing machines, to
capture their execution, count their steps, as well as, e.g., the memory cells
that they use. The coalgebraic view of monoidal computer thus provides a
convenient diagrammatic language for studying computability and complexity.Comment: 34 pages, 24 figures; in this version: added the Appendi
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
A computable expression of closure to efficient causation
International audienceIn this paper, we propose a mathematical expression of closure to efficient causation in terms of lambda-calculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in lambda-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability
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