5,858 research outputs found
Phase Transitions for Gödel Incompleteness
Gödel's first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers
have been looking for natural examples of such assertions and breakthroughs have been obtained in the seventies by Jeff Paris (in part jointly with Leo Harrington and Laurie Kirby) and Harvey Friedman who produced first mathematically interesting
independence results in Ramsey theory (Paris) and well-order and well-quasi-order theory (Friedman).
In this article we investigate Friedman style principles of combinatorial well-foundedness for the ordinals below epsilon_0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.
For these independence principles we classify (as a part of a general research program) their phase transitions, i.e. we classify exactly the bounding conditions which lead from
provability to unprovability in the induced combinatorial
well-foundedness principles.
As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel's coding of finite sequences from his classical 1931 paper on his incompleteness results.
This choice makes the investigation highly non trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions.
For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem by Parameswaran which apparently is far remote from Gödel's theorem
Some natural zero one laws for ordinals below Δ0
We are going to prove that every ordinal α with Δ_0â>âαââ„âÏ^Ï satisfies a natural zero one law in the following sense. For αâ<âΔ_0 let Nα be the number of occurences of Ï in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any α with Ï Ï ââ€âαâ<âΔ_0 and any sentence Ï in the language of linear orders the asymptotic density of Ï along α is an element of â{0,1}. We further show that for any such sentence Ï the asymptotic density along Δ_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below Ï^Ï
Unprovability results involving braids
We construct long sequences of braids that are descending with respect to the
standard order of braids (``Dehornoy order''), and we deduce that, contrary to
all usual algebraic properties of braids, certain simple combinatorial
statements involving the braid order are true, but not provable in the
subsystems ISigma1 or ISigma2 of the standard Peano system.Comment: 32 page
Connecting the provable with the unprovable: phase transitions for unprovability
Why are some theorems not provable in certain theories of mathematics? Why are most theorems from existing mathematics provable in very weak systems? Unprovability theory seeks answers for those questions. Logicians have obtained unprovable statements which resemble provable statements. These statements often contain some condition which seems to cause unprovability, as this condition can be modified, using a function parameter, in such a manner as to make the theorem provable. It turns out that in many cases there is a phase transition: By modifying the parameter slightly one changes the theorem from provable to unprovable.
We study these transitions with the goal of gaining more insights into unprovability
Integrating Prosodic and Lexical Cues for Automatic Topic Segmentation
We present a probabilistic model that uses both prosodic and lexical cues for
the automatic segmentation of speech into topically coherent units. We propose
two methods for combining lexical and prosodic information using hidden Markov
models and decision trees. Lexical information is obtained from a speech
recognizer, and prosodic features are extracted automatically from speech
waveforms. We evaluate our approach on the Broadcast News corpus, using the
DARPA-TDT evaluation metrics. Results show that the prosodic model alone is
competitive with word-based segmentation methods. Furthermore, we achieve a
significant reduction in error by combining the prosodic and word-based
knowledge sources.Comment: 27 pages, 8 figure
Prosody-Based Automatic Segmentation of Speech into Sentences and Topics
A crucial step in processing speech audio data for information extraction,
topic detection, or browsing/playback is to segment the input into sentence and
topic units. Speech segmentation is challenging, since the cues typically
present for segmenting text (headers, paragraphs, punctuation) are absent in
spoken language. We investigate the use of prosody (information gleaned from
the timing and melody of speech) for these tasks. Using decision tree and
hidden Markov modeling techniques, we combine prosodic cues with word-based
approaches, and evaluate performance on two speech corpora, Broadcast News and
Switchboard. Results show that the prosodic model alone performs on par with,
or better than, word-based statistical language models -- for both true and
automatically recognized words in news speech. The prosodic model achieves
comparable performance with significantly less training data, and requires no
hand-labeling of prosodic events. Across tasks and corpora, we obtain a
significant improvement over word-only models using a probabilistic combination
of prosodic and lexical information. Inspection reveals that the prosodic
models capture language-independent boundary indicators described in the
literature. Finally, cue usage is task and corpus dependent. For example, pause
and pitch features are highly informative for segmenting news speech, whereas
pause, duration and word-based cues dominate for natural conversation.Comment: 30 pages, 9 figures. To appear in Speech Communication 32(1-2),
Special Issue on Accessing Information in Spoken Audio, September 200
Listening in large rooms : a neurophysiological investigations of acoustical conditions that influence speech intelligibility
Thesis (M.S.)--Massachusetts Institute of Technology, Whitaker College of Health Sciences and Technology, 1997.Includes bibliographical references (p. 34-37).by Benjamin Michael Hammond.M.S
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