Multiple Reductions Revisited

2019

I discuss Steinhart’s argument against Benacerraf’s famous multiple reductions argument to the effect that numbers cannot be sets. Steinhart offers a mathematical argument according to which there is only one series of sets to which the natural numbers can be reduced (namely, the finite von Neumann ordinals), and thus attacks Benacerraf’s assumption that there are multiple reductions of numbers to sets. I will argue that Steinhart’s argument is problematic and should not be accepted.

Information Processing Letters, 1991

We show that log-bounded rudimentary reductions de ned and studied by Jones in 1975 characterize Dlogtime-uniform AC 0 .

Computational Complexity, 2001

We build on the recent progress regarding isomorphisms of complete sets that was reported in . In that paper, it was shown that all sets that are complete under (non-uniform) AC 0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC 0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a "Gap Theorem", showing that all sets complete under AC 0 reductions are in fact already complete under NC 0 reductions. The following questions were left open in that paper:

2006

This is a first tentative examination of the possibility of reinstating reduction as a valid candidate for presenting relations between mental and physical properties. Classical Nagelian reduction is undoubtedly contaminated in many ways, but here I investigate the possibility of adapting to problems concerning mental properties an alternative definition for theory reduction in philosophy of science. The definition I offer is formulated with the aid of non-monotonic logic, which I suspect might be a very interesting realm for testing notions concerning localized mental-physical reduction. The reason for this is that non-monotonic reasoning by definition is about appeals made not only to explicit observations, but also to an implicit selection of back- ground knowledge containing heuristic information. The flexibility of this definition and the fact that it is not absolute, i.e. that the relation of reduction may be retracted or allowed to shift without fuss, add at least an interesting alternative factor to current materialist debates.

2016

The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement.

Journal de Théorie des Nombres de Bordeaux, 1991

L'accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ [Y] T. YOUSSEFI, Revêtements des courbes algébriques et réductions, Thèse, Université de Bordeaux 1, 1990.

What would be sufficient to show of some apparently higher-level property that it is 'nothing over and above' some complex configuration of more basic properties? This paper defends a new method for justifying reductions by demonstrating its comparative advantages over two methods recently defended in the literature. Unlike its rivals, what I'll call ''the semantic method'' makes a reduction's truth epistemically transparent without relying on conceptual analyses.

2007

In writing this thesis, I am greatly indebted to Robin F. Hendry, for many insightful conversations during the development of the ideas in this thesis and for helpful comments on the text. I am also indebted to my Japanese friend Daisuke Kaida, who I have learnt many things from. I would also like to thank commentators and participants in the First Lisbon Colloquium for the Philosophy of Science, The Unity of Science: Non-Traditional Approaches, the First Philosophy Graduate Conference at the Central European University in Budapest, and the Fifth European Congress for Analytic Philosophy in Lisbon for their helpful comments and discussions on some parts of this thesis. Also I would like to thank those members of the Durham Philosophy Department who, through their care and patience, have made the writing of this thesis possible.

Applicable Algebra in Engineering, Communication and Computing, 2023

In order to save the junior reader the trouble of referring to other works, I have put together here those elementary principles with regard to elimination, of which use has been made in the preceding pages. The device that follows, which, it may be hoped, finally eliminates from algebraic geometry the last traces of elimination-theory A. Weil Foundations of Algebraic Geometry A.M.S. 1962 pg.31 Eliminate, eliminate, eliminate Eliminate the eliminators of elimination theory. Abhyankar's 'eliminate' rhythmic refrain scans the climax periods in the evolution of the approach to solving by the Computational Algebraic Geometry communities, those around conferences as MEGA, ISSAC, CoCoA, JAA and which mainly publish in journals as Commutative Algebra, JSC, AAC and also AAECC: • In 1750 Cramer gave

Theory of Computing Systems, 2010

Reductions and completeness notions form the heart of computational complexity theory. Recently non-uniform reductions have been naturally introduced in a variety of settings concerning completeness notions for NP and other classes. We follow up on these results by strengthening some of them. In particular, we show that under certain well studied hypotheses: