4,888 research outputs found
Global semantic typing for inductive and coinductive computing
Inductive and coinductive types are commonly construed as ontological
(Church-style) types, denoting canonical data-sets such as natural numbers,
lists, and streams. For various purposes, notably the study of programs in the
context of global semantics, it is preferable to think of types as semantical
properties (Curry-style). Intrinsic theories were introduced in the late 1990s
to provide a purely logical framework for reasoning about programs and their
semantic types. We extend them here to data given by any combination of
inductive and coinductive definitions. This approach is of interest because it
fits tightly with syntactic, semantic, and proof theoretic fundamentals of
formal logic, with potential applications in implicit computational complexity
as well as extraction of programs from proofs. We prove a Canonicity Theorem,
showing that the global definition of program typing, via the usual (Tarskian)
semantics of first-order logic, agrees with their operational semantics in the
intended model. Finally, we show that every intrinsic theory is interpretable
in a conservative extension of first-order arithmetic. This means that
quantification over infinite data objects does not lead, on its own, to
proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories
are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were
used to characterize major computational complexity classes Their extensions
described here have similar potential which has already been applied
A uniform approach to fundamental sequences and hierarchies
In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one
On axiom schemes for T-provably Δ1 formulas
This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1 provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp implies BΣ1 to a purely recursion-theoretic question.Ministerio de Ciencia e Innovación MTM2008–0643
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Short Proofs for Slow Consistency
Let denote the finite
consistency statement "there are no proofs of contradiction in with
symbols". For a large class of natural theories , Pudl\'ak
has shown that the lengths of the shortest proofs of
in the theory
itself are bounded by a polynomial in . At the same time he conjectures that
does not have polynomial proofs of the finite consistency
statements . In contrast we show that Peano arithmetic
() has polynomial proofs of
,
where is the slow consistency statement for
Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also
obtain a new proof of the result that the usual consistency statement
is equivalent to iterations
of slow consistency. Our argument is proof-theoretic, while previous
investigations of slow consistency relied on non-standard models of arithmetic
A Categorical Construction of Bachmann-Howard Fixed Points
Peter Aczel has given a categorical construction for fixed points of normal
functors, i.e. dilators which preserve initial segments. For a general dilator
we cannot expect to obtain a well-founded fixed point, as the
order type of may always exceed the order type of . In the present
paper we show how to construct a Bachmann-Howard fixed point of , i.e. an
order with an "almost" order preserving collapse
. Building
on previous work, we show that -comprehension is equivalent to the
assertion that is well-founded for any dilator .Comment: This version has been accepted for publication in the Bulletin of the
London Mathematical Societ
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
- …