6 research outputs found
On Higher-Order Probabilistic Subrecursion
We study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like Godel's T with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of T essentially unaltered. The paper ends with some observations about the functional expressive power: expectedly, all the considered calculi capture the functions which T itself represents, at least when standard notions of observations are considered
On Higher-Order Probabilistic Subrecursion
We study the expressive power of subrecursive probabilistic higher-order
calculi. More specifically, we show that endowing a very expressive
deterministic calculus like G\"odel's with various forms of
probabilistic choice operators may result in calculi which are not equivalent
as for the class of distributions they give rise to, although they all
guarantee almost-sure termination. Along the way, we introduce a probabilistic
variation of the classic reducibility technique, and we prove that the simplest
form of probabilistic choice leaves the expressive power of
essentially unaltered. The paper ends with some observations about the
functional expressive power: expectedly, all the considered calculi capture the
functions which itself represents, at least when standard notions
of observations are considered
A Formalization of Polytime Functions
We present a deep embedding of Bellantoni and Cook's syntactic
characterization of polytime functions. We prove formally that it is correct
and complete with respect to the original characterization by Cobham that
required a bound to be proved manually. Compared to the paper proof by
Bellantoni and Cook, we have been careful in making our proof fully contructive
so that we obtain more precise bounding polynomials and more efficient
translations between the two characterizations. Another difference is that we
consider functions on bitstrings instead of functions on positive integers.
This latter change is motivated by the application of our formalization in the
context of formal security proofs in cryptography. Based on our core
formalization, we have started developing a library of polytime functions that
can be reused to build more complex ones.Comment: 13 page
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
Type Systems For Polynomial-time Computation
This thesis introduces and studies a typed lambda calculus with higher-order primitive recursion over inductive datatypes which has the property that all definable number-theoretic functions are polynomial time computable. This is achieved by imposing type-theoretic restrictions on the way results of recursive calls can be used.
The main technical result is the proof of the characteristic property of this system. It proceeds by exhibiting a category-theoretic model in which all morphisms are polynomial time computable by construction.
The second more subtle goal of the thesis is to illustrate the usefulness of this semantic technique as a means for guiding the development of syntactic systems, in particular typed lambda calculi, and to study their meta-theoretic properties.
Minor results are a type checking algorithm for the developed typed lambda calculus and the construction of combinatory algebras consisting of polynomial time algorithms in the style of the first Kleene algebra