60 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Towards Optimal Depth-Reductions for Algebraic Formulas
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973)
and Brent (JACM 1974) show that any algebraic formula of size s can be
converted to one of depth O(log s) with only a polynomial blow-up in size. In
this paper, we consider a fine-grained version of this result depending on the
degree of the polynomial computed by the algebraic formula. Given a homogeneous
algebraic formula of size s computing a polynomial P of degree d, we show that
P can also be computed by an (unbounded fan-in) algebraic formula of depth
O(log d) and size poly(s). Our proof shows that this result also holds in the
highly restricted setting of monotone, non-commutative algebraic formulas. This
improves on previous results in the regime when d is small (i.e., d<<s). In
particular, for the setting of d=O(log s), along with a result of Raz (STOC
2010, JACM 2013), our result implies the same depth reduction even for
inhomogeneous formulas. This is particularly interesting in light of recent
algebraic formula lower bounds, which work precisely in this ``low-degree" and
``low-depth" setting. We also show that these results cannot be improved in the
monotone setting, even for commutative formulas
Towards Optimal Depth-Reductions for Algebraic Formulas
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula.
Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas.
This improves on previous results in the regime when d is small (i.e., d = s^o(1)). In particular, for the setting of d = O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this "low-degree" and "low-depth" setting.
We also show that these results cannot be improved in the monotone setting, even for commutative formulas
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Integrative Levels of Knowing
Diese Dissertation beschĂ€ftigt sich mit einer systematischen Organisation der epistemologischen Dimension des menschlichen Wissens in Bezug auf Perspektiven und Methoden. Insbesondere wird untersucht inwieweit das bekannte Organisationsprinzip der integrativen Ebenen, das eine Hierarchie zunehmender KomplexitĂ€t und Integration beschreibt, geeignet ist fĂŒr eine grundlegende Klassifikation von Perspektiven bzw. epistemischen Bezugsrahmen. Die zentrale These dieser Dissertation geht davon aus, dass eine angemessene Analyse solcher epistemischen Kontexte in der Lage sein sollte, unterschiedliche oder gar konfligierende Bezugsrahmen anhand von kontextĂŒbergreifenden Standards und Kriterien vergleichen und bewerten zu können. Diese Aufgabe erfordert theoretische und methodologische Grundlagen, welche die BeschrĂ€nkungen eines radikalen Kontextualismus vermeiden, insbesondere die ihm innewohnende Gefahr einer Fragmentierung des Wissens aufgrund der angeblichen InkommensurabilitĂ€t epistemischer Kontexte. Basierend auf JĂŒrgen Habermasâ Theorie des kommunikativen Handelns und seiner Methodologie des hermeneutischen Rekonstruktionismus, wird argumentiert, dass epistemischer Pluralismus nicht zwangslĂ€ufig zu epistemischem Relativismus fĂŒhren muss und dass eine systematische Organisation der Perspektivenvielfalt von bereits existierenden Modellen zur kognitiven Entwicklung profitieren kann, wie sie etwa in der Psychologie oder den Sozial- und Kulturwissenschaften rekonstruiert werden. Der vorgestellte Ansatz versteht sich als ein Beitrag zur multi-perspektivischen Wissensorganisation, der sowohl neue analytische Werkzeuge fĂŒr kulturvergleichende Betrachtungen von Wissensorganisationssystemen bereitstellt als auch neue Organisationsprinzipien vorstellt fĂŒr eine KontexterschlieĂung, die dazu beitragen kann die AusdrucksstĂ€rke bereits vorhandener Dokumentationssprachen zu erhöhen. Zudem enthĂ€lt der Anhang eine umfangreiche Zusammenstellung von Modellen integrativer Wissensebenen.This dissertation is concerned with a systematic organization of the epistemological dimension of human knowledge in terms of viewpoints and methods. In particular, it will be explored to what extent the well-known organizing principle of integrative levels that presents a developmental hierarchy of complexity and integration can be applied for a basic classification of viewpoints or epistemic outlooks. The central thesis pursued in this investigation is that an adequate analysis of such epistemic contexts requires tools that allow to compare and evaluate divergent or even conflicting frames of reference according to context-transcending standards and criteria. This task demands a theoretical and methodological foundation that avoids the limitation of radical contextualism and its inherent threat of a fragmentation of knowledge due to the alleged incommensurability of the underlying frames of reference. Based on JĂŒrgen Habermasâs Theory of Communicative Action and his methodology of hermeneutic reconstructionism, it will be argued that epistemic pluralism does not necessarily imply epistemic relativism and that a systematic organization of the multiplicity of perspectives can benefit from already existing models of cognitive development as reconstructed in research fields like psychology, social sciences, and humanities. The proposed cognitive-developmental approach to knowledge organization aims to contribute to a multi-perspective knowledge organization by offering both analytical tools for cross-cultural comparisons of knowledge organization systems (e.g., Seven Epitomes and Dewey Decimal Classification) and organizing principles for context representation that help to improve the expressiveness of existing documentary languages (e.g., Integrative Levels Classification). Additionally, the appendix includes an extensive compilation of conceptions and models of Integrative Levels of Knowing from a broad multidisciplinary field
Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists
an explicit -variate and degree polynomial such
that if any depth four circuit of bounded formal degree which computes
a polynomial of bounded individual degree , that is functionally
equivalent to , then must have size .
The motivation for their work comes from Boolean Circuit Complexity. Based on
a characterization for circuits by Yao [FOCS, 1985] and Beigel and
Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that
functions in can also be computed by algebraic
circuits (i.e., circuits of the form -- sums
of powers of polynomials) of size. Thus they argued that a
"functional" lower bound for an explicit
polynomial against circuits would imply a
lower bound for the "corresponding Boolean function" of against non-uniform
. In their work, they ask if their lower bound be extended to
circuits.
In this paper, for large integers and such that , we show that any circuit of
bounded individual degree at most that
functionally computes Iterated Matrix Multiplication polynomial
() over must have size . Since Iterated
Matrix Multiplication over is functionally in
, improvement of the afore mentioned lower bound to hold for
quasipolynomially large values of individual degree would imply a fine-grained
separation of from
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