Computability and Complexity Theory

Homer, Steven; Selman, Alan L.

doi:10.1007/978-1-4757-3544-4

Cited by 61 publications

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“…By complexity I mean that musical signals (like linguistic signals) are more complex than the various innate vocalization available in our species (groans, sobs, laughter and shouts). Although complexity can be measured and quantified in various ways (see, e.g., Homer & Selman, 2001;Shmulevich & Povel, 2000;Simon, 1972;Weng, Bhalla, & Iyengar, 1999), there is no single widely used metric applicable to all musics (Pressing, 1998), so it would be premature to specify any absolute threshold for ''complexity'' at present. Oddly, Hockett did not include complexity on his list of design features, although its necessity for language is implicit in his discussion.…”

Section: Linguistic Comparisons: Design Features Of Human Musicmentioning

confidence: 99%

The biology and evolution of music: A comparative perspective

2006

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Studies of the biology of music (as of language) are highly interdisciplinary and demand the integration of diverse strands of evidence. In this paper, I present a comparative perspective on the biology and evolution of music, stressing the value of comparisons both with human language, and with those animal communication systems traditionally termed ''song''. A comparison of the ''design features'' of music with those of language reveals substantial overlap, along with some important differences. Most of these differences appear to stem from semantic, rather than structural, factors, suggesting a shared formal core of music and language. I next review various animal communication systems that appear related to human music, either by analogy (bird and whale ''song'') or potential homology (great ape bimanual drumming). A crucial comparative distinction is between learned, complex signals (like language, music and birdsong) and unlearned signals (like laughter, ape calls, or bird calls). While human vocalizations clearly build upon an acoustic and emotional foundation shared with other primates and mammals, vocal learning has evolved independently in our species since our divergence with chimpanzees. The convergent evolution of vocal learning in other species offers a powerful window into psychological and neural constraints influencing the evolution of complex signaling systems (including both song and speech), while ape drumming presents a fascinating potential homology with human instrumental music. I next discuss the archeological data relevant to music evolution, concluding on the basis of prehistoric bone flutes that instrumental music is at least 40,000 years old, and perhaps much older. I end with a brief review of adaptive functions proposed for music, concluding that no one selective force (e.g., sexual selection) is adequate to explaining all aspects of human music. I suggest that questions about ARTICLE IN PRESSthe past function of music are unlikely to be answered definitively and are thus a poor choice as a research focus for biomusicology. In contrast, a comparative approach to music promises rich dividends for our future understanding of the biology and evolution of music.

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Section: Linguistic Comparisons: Design Features Of Human Musicmentioning

confidence: 99%

The biology and evolution of music: A comparative perspective

2006

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“…We adopt the standard definitions and notations for well-known complexity classes, and other notions of computational complexity, as can be found, e.g., in [6] and [24]. We use the standard paring function denoted a, b for binary strings a, b, where | a, b | = 2|a| + |b| + 2.…”

Section: Notationmentioning

confidence: 99%

Non-Uniform Reductions

et al. 2009

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We study properties of non-uniform reductions and related completeness notions. We strengthen several results of Hitchcock and Pavan (ICALP (1), Lecture Notes in Computer Science, vol. 4051, pp. 465-476, Springer, 2006) and give a tradeoff between the amount of advice needed for a reduction and its honesty on NEXP. We construct an oracle relative to which this trade-off is optimal. We show, in a more systematic study of non-uniform reductions, among other things that non-uniformity can be removed at the cost of more queries. In line with Post's program for complexity theory (Buhrman and Torenvliet in Bulletin of the EATCS 85, pp. 41-51, 2005) we connect such 'uniformization' properties to the separation of complexity classes.

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“…Since max(g 1 , g 2 , g 3 ) + 1 is a total computable function, there exists a fast function 1 g such that for all n, g(n) > max(g 1 (n), g 2 (n), g 3 (n)) (We refer to Proposition 7.3 of the text by Homer and Selman [HS01].) Also, G[g] ∈ P (Lemma 7.1, [HS01]).…”

Section: Claim 44 For Allmentioning

confidence: 99%

Canonical Disjoint NP-Pairs of Propositional Proof Systems

Glaßer

Selman

Zhang

2005

Mathematical Foundations of Computer Science 2005

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We prove that every disjoint NP-pair is polynomial-time, many-one equivalent to the canonical disjoint NP-pair of some propositional proof system. Therefore, the degree structure of the class of disjoint NP-pairs and of all canonical pairs is identical. Secondly, we show that this degree structure is not superficial: Assuming there exist P-inseparable disjoint pairs, there exist intermediate disjoint NP-pairs. That is, if (A, B) is a P-separable disjoint NP-pair and (C, D) is a P-inseparable disjoint NP-pair, then there exist P-inseparable, incomparable NP-pairs (E, F ) and (G, H) whose degrees lie strictly between (A, B) and (C, D). Furthermore, between any two disjoint NP-pairs that are comparable and inequivalent, such a diamond exists.

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Computability and Complexity Theory

Cited by 61 publications

References 0 publications

The biology and evolution of music: A comparative perspective

The biology and evolution of music: A comparative perspective

Non-Uniform Reductions

Canonical Disjoint NP-Pairs of Propositional Proof Systems

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