Results 1 
7 of
7
Logic and the Challenge of Computer Science
, 1988
"... Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objec ..."
Abstract

Cited by 153 (16 self)
 Add to MetaCart
Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objects with bounded resources. This chapter consists of two independent parts. The first part is devoted to finite model theory; it is mostly a survey of logics tailored for computational complexity. The second part is devoted to dynamic structures with bounded resources. In particular, we use dynamic structures with bounded resources to model Pascal.
What Is an Algorithm?
, 2000
"... Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (o ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The ChurchTuring Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Arithmetic complexity
 ACM Transactions on Computational Logic
, 2007
"... My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worstcase time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worstcase time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that recursive programs are not a new computational paradigm, since their manifestation as HerbrandGödelKleene systems was present at the very beginning of the modern theory of computability, in 1934. But they have been dissed as tools for complexity analysis, and part of my mission here is to rehabilitate them. I will draw my examples primarily from van den Dries ’ [1] and the joint work in [3, 2], incidentally providing some publicity for the fine results in those papers. Some of these results are stated in Section 3; before that, I will set the stage in Sections 1 and 2, and in the last Section 4 of this abstract I will outline very briefly some conclusions about recursion and complexity which I believe that they support. 1 Partial Algebras
Algorithms vs. Machines
"... Yiannis Moschovakis argues that some algorithms, and in particular the mergesort algorithm, cannot be adequately described in terms of machines acting on states. We show how to describe the mergesort algorithm, on its natural level of abstraction, in terms of distributed abstract state machines. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Yiannis Moschovakis argues that some algorithms, and in particular the mergesort algorithm, cannot be adequately described in terms of machines acting on states. We show how to describe the mergesort algorithm, on its natural level of abstraction, in terms of distributed abstract state machines.
1 What Is an Algorithm?
"... When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue ..."
Abstract
 Add to MetaCart
When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue