- Artificial General Intelligence, Artificial Intelligence, Linguistics, Mathematics, Mathematical Logic, Logic And Foundations Of Mathematics, and 15 moreModal Logic, Category Theory, Computational Linguistics, Logic, Formal Systems, Philosophy of Logic, Philosophy of Mind, Philosophy of Science, Cognitive Science, Metaphilosophy, Metaphysics, Personal Identity, Philosophy of Artificial Intelligence, Consciousness, and Philosophy Of Mathematicsedit
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Research Interests: Computer Science, Artificial Intelligence, Human Computer Interaction, Architecture, Natural Language Processing, and 14 moreReinforcement Learning, Natural Language Generation, English language, Embodied Cognition, Software Architecture, Virtual Worlds, Artificial General Intelligence, Open Source, Natural language, Perception and Action, Software Framework, English Language, General Intelligence, and Virtual agent
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ABSTRACT The chiral bag model is a successful model for the static properties (masses and moments) of a nucleon. In its formulation a non-trivial vacuum of confined fermions plays an important role of maintaining integral baryon number... more
ABSTRACT The chiral bag model is a successful model for the static properties (masses and moments) of a nucleon. In its formulation a non-trivial vacuum of confined fermions plays an important role of maintaining integral baryon number and positivity of the Hamiltonian. The origin of non-zero expectation values for various fermion bilinears in this vacuum is understood as a Casimir effect and the values of these are computed. A discussion is made of the infinities arising in the calculation. An overview of the chiral bag model and some of its predictions are given.
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A novel approach to sentence generation – SegSim, Sentence Generation by Similarity Matching – is outlined, and is argued to possess a number of desirable properties making it plausible as a model of sentence generation in the human... more
A novel approach to sentence generation – SegSim, Sentence Generation by Similarity Matching – is outlined, and is argued to possess a number of desirable properties making it plausible as a model of sentence generation in the human brain, and useful as a guide for creating sentence generation components within artificial brains. The crux of the approach is to do
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A variety of speculations about the nature of quantum mechanics and wavefunction collapse. A number of “key principles” are set down; these must surely hold true. Holding onto these, a variety of mathematical effects are explored, to see... more
A variety of speculations about the nature of quantum mechanics and wavefunction collapse. A number of “key principles” are set down; these must surely hold true. Holding onto these, a variety of mathematical effects are explored, to see if or how they might be appropriate for describing QM, and its relationship to
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Research Interests: Mathematics and quotient
A new model of symbol grounding is presented, in which the structures of natural language, logical semantics, perception and action are represented categorically, and symbol grounding is modeled via the composition of morphisms between... more
A new model of symbol grounding is presented, in which the structures of natural language, logical semantics, perception and action are represented categorically, and symbol grounding is modeled via the composition of morphisms between the relevant categories. This model gives conceptual insight into the fundamentally systematic nature of symbol grounding, and also connects naturally to practical real-world AI systems in current research and commercial use. Specifically, it is argued that the structure of linguistic syntax can be modeled as a certain asymmetric monoidal category, as e.g. implicit in the link grammar formalism; the structure of spatiotemporal relationships and action plans can be modeled similarly using "image grammars" and "action grammars"; and common-sense logical semantic structure can be modeled using dependently-typed lambda calculus with uncertain truth values. Given these formalisms, the grounding of linguistic descriptions in spatiotempor...
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Given a continued fraction, we construct a certain function hat is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is... more
Given a continued fraction, we construct a certain function hat is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is 1/q2, and that, for higher orders, the gap appears to be perfectly ’randomly ’ distributed, in that it is Cauchy-dense on the unit square, and thus, this function h as a fractal measure of exactly 2. We find this result to be very intriguing, as we know f no other functions that have this property (There are many fractal urvesthat have this property, but not functions. That is, a space-filling curve can be used to enumerate R2by R but such space-filling curves have locality properties indu ced byR that the gap function appears not to have). When examining this function for small rationals, some very curious algebraic relationships appear to relate various rationals. This paper is part of a set of chapters that explore the relati onship between the real ...
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The RelEx semantic relation extractor (Ross et al., 2005) is an open-source dependency parser built on top of the CMU Link Grammar Parser (Sleator and Temperley., 1991; Sleator and Temperley, 1993). Starting with a parse generated by Link... more
The RelEx semantic relation extractor (Ross et al., 2005) is an open-source dependency parser built on top of the CMU Link Grammar Parser (Sleator and Temperley., 1991; Sleator and Temperley, 1993). Starting with a parse generated by Link Grammar, it applies a set of ...
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The concept of self-similarity is central to the notion of a fractal, but the actual symmetry group that generates that self-similarity is rarely named, and it seems fractals are almost never studied according to their symmetries. Yet, in... more
The concept of self-similarity is central to the notion of a fractal, but the actual symmetry group that generates that self-similarity is rarely named, and it seems fractals are almost never studied according to their symmetries. Yet, in other branches of mathematics and physics, it is well understood that symmetry provides a powerful mechanism for understanding systems. In this paper, we identify the symmetry group of period-doubling maps as being a monoid (semigroup) of the modular group PSL(2,Z). To anchor this assertion, we work out an explicit, exactly-solvable fractal curve, the Takagi or Blancmange Curve, as transforming under the three-dimensional representation of the (monoid of the) modular group. By replacing the triangular shape that generates the Blancmange curve with a polynomial, we find that the resulting curve transforms under the n + 2 dimensional representation of the monoid, where n is the degree of the polynomial. We also find that the (ill-defined) derivative ...
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The beta transformation is the iterated map $\beta x\mod1$. The special case of $\beta=2$ is known as the Bernoulli map, and is exactly solvable. The Bernoulli map provides a model for pure, unrestrained chaotic (ergodic) behavior: it is... more
The beta transformation is the iterated map $\beta x\mod1$. The special case of $\beta=2$ is known as the Bernoulli map, and is exactly solvable. The Bernoulli map provides a model for pure, unrestrained chaotic (ergodic) behavior: it is the full invariant shift on the Cantor space $\{0,1\}^{\omega}$. The beta transformation defines a subshift: iterated on the unit interval, it singles out a subspace of the Cantor space, in such a way that it is invariant under the action of the left-shift operator. That is, lopping off one bit at a time gives back the same subspace. The beta transform seems to capture something basic about the multiplication of two real numbers: $\beta$ and $x$. It offers a window into understanding the nature of multiplication. Iterating on multiplication, one would get $\beta^{n}x$ - that is, exponentiation; although the mod 1 of the beta transform contorts this in interesting ways. The work presented here is a research diary: a pastiche of observations and some ...
This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another definition, it is the transfer operator (the... more
This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another definition, it is the transfer operator (the Frobenius-Perron operator) of the Bernoulli map, also variously known as the doubling map or the sawtooth map. The map is interesting for multiple reasons. One is that the set of infinite binary strings is the Cantor set; this implies that the Bernoulli operator has a set of fractal eigenfunctions. These are given by the Takagi (or Blancmange) curve. The set of all infinite binary strings can also be understood as the infinite binary tree. This binary tree has a large number of self-similarities, given by the dyadic monoid. The dyadic monoid has an extension to the modular group PSL(2,Z), which plays an important role in analytic number theory; and so there are many connections between the Bernoulli map and various number-theoretic functions, includ...
This paper presents a review of the Gauss-Kuzmin-Wirsing (GKW) operator. The GKW operator is the transfer operator of the Gauss map, and thus has connections to the theory of continued fractions – specifically, it is the shift operator... more
This paper presents a review of the Gauss-Kuzmin-Wirsing (GKW) operator. The GKW operator is the transfer operator of the Gauss map, and thus has connections to the theory of continued fractions – specifically, it is the shift operator for continued fractions. The mathematical literature on this operator is fairly slim; even so, this text is not a complete review; it is partly an exposition, and partly a diary of research results. Connections to the Minkowski Question Mark Function are probed. In particular, the Question Mark is used to define a transfer operator which is conjugate to the GKW. This conjugate operator is solvable, and can be shown to have fractal eigenfunctions. However, the spectrum of this operator is not at all the same as that of the GKW. This is because the Jacobian of the transformation relating the two is given by (?′◦?−1)(x) , which is wellknown as the prototypical “multi-fractal measure”. Nonetheless, conjugacy allows the eigenfunctions of the one to be used...
This short note presents some empirical data on the distribution of word pairs obtained from English text. Particular attention is paid to the distribution of mutual information.
A report on the results of applying the principal component analysis (PCA) algo to the word-disjunct pairs, and some spot checks of the cosine similarity between words. Results: the cosine similarity looks excellent. Naive PCA and sparse... more
A report on the results of applying the principal component analysis (PCA) algo to the word-disjunct pairs, and some spot checks of the cosine similarity between words. Results: the cosine similarity looks excellent. Naive PCA and sparse PCA does not work. Also a very short note that points out that Link Grammar disjuncts are the same thing as the sheaves of sheaf theory. This has the side-effect of explaining why LG seems to be described by modal logic: apparently, the relation betwen sheaves and modal logic was noticed in 1965, and goes under the name of ’Kripke-Moyal semantics’. Now I know the precise mechanism of why the grammar of natural language also appears to be modal. Interesting.
XXX OBSOLETE XXX HALF-BAKED XXX The ideas described herein are confusingly presented, and are half-baked, incomplete, incoherent. Essentially all the ideas herein are presented more sensibly, more coherently in other papers on this... more
XXX OBSOLETE XXX HALF-BAKED XXX The ideas described herein are confusingly presented, and are half-baked, incomplete, incoherent. Essentially all the ideas herein are presented more sensibly, more coherently in other papers on this website. Much of what was originally in this paper has been edited out, and moved to another paper, which presents the ideas in a much clearer fashion: see [9]. XXX OBSOLETE XXX HALF-BAKED XXX This paper reviews connections between the Bernoulli map, the baker’s map and the Ising model. The Bernoulli map is possibly the simplest exactly solvable model of deterministic chaos. Its Frobenius-Perron operator or transfer operator has a set of well-known polynomial eigenvectors, given by the Bernoulli polynomials. Its also has a set of smooth but non-integrable eigenvectors given by the Hurwitz zeta function. Alternately, it has a set of fractal eigenvectors, of which the Blancmange or Takagi curve is one. An even larger set of functions that are eigenvectors o...
A short report on word-pair datasets for Chinese and English. Although these are strictly not comparable, because the pairs were formed between hanzi, and not words, for Chinese, the pair statistics are still .. interesting.
Extract from the language-learning diary, reporting on an initial dataset containing connector sets. This is a revised (6 August 2017) version of the original 11 May 2017 report. It re-analyzes and expands the original analysis on a... more
Extract from the language-learning diary, reporting on an initial dataset containing connector sets. This is a revised (6 August 2017) version of the original 11 May 2017 report. It re-analyzes and expands the original analysis on a newer, larger dataset. This was motivated in part due to several errors found and fixed in the processing pipeline in late June/early July. In retrospect, it appears these errors mostly did not affect the earlier analysis, as the most significant error was introduced after the inital analysis was made. None-the-less, it seemed prudent to redo the report. Sadly, several months were lost in the confusion, requiring large datasets to be discarded.
The distribution of rationals on the unit interval is filled w ith surprises. As a child, one is told that the rationals are distributed “unifo rmly” on the unit interval. If one considers the entire set Q, then yes, in a certain narrow... more
The distribution of rationals on the unit interval is filled w ith surprises. As a child, one is told that the rationals are distributed “unifo rmly” on the unit interval. If one considers the entire set Q, then yes, in a certain narrow sense, this is true. But if one considers just subsets, such as the subset of ratio nals with “small” denominators, then the distribution is far from uniform and fu ll of counter-intuitive surprises, some of which we explore below. This implies that using “intuition” to understand the rationals and, more generally, the real nu mbers is a dangerous process. Once again, we see the footprints of the set-theore tic representation of the modular groupSL(2,Z) at work. This paper is part of a set of chapters that explore the relati onship between the real numbers, the modular group, and fractals. 1 Distributions Of Rationals on the Unit Interval The entire field of classical calculus and analysis is based o n the notion that the real numbers are smoothly an...
The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by... more
The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by classical techniques that its derivative must vanish on all rationals. Since the Question Mark itself is continuous, one concludes that the derivative must be non-zero on the irrationals, and is thus a discontinuous-everywhere function. This derivative is the subject of this essay. Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then sho...
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This text provides a broad sketch of how deep-learning/neural-net approaches are quite similar to symbolic approaches to machine learning, knowledge representation and AI. Taken from the right viewpoint, they can be seen to be two... more
This text provides a broad sketch of how deep-learning/neural-net approaches are quite similar to symbolic approaches to machine learning, knowledge representation and AI. Taken from the right viewpoint, they can be seen to be two variants of the same structure. To keep the development focused and concrete, the presentation is limited models of natural language, and thus compares Word2Vec, SkipGram or AdaGram-style vector-space approaches to traditional symbolic linguistics approaches. To maintain concreteness, Link Grammar is used as a stand-in for a prototypical dependency grammar. Super cially, these systems appear to have nothing in common. On closer examination, it becomes evident that both employ a vector representation of words-incontext. The context is an N-gram, skip-gram or adagram, in the neuralnet case, and a dependency linkage disjunct in the symbolic case. The similarity becomes most apparent when the word+context is viewed as a bipartite graph, with words on the left ...
A novel approach to the fully automated, unsupervised extraction of dependency grammars and associated syntax-to-semantic-relationship mappings from large text corpora is described. The suggested approach builds on the authors' prior... more
A novel approach to the fully automated, unsupervised extraction of dependency grammars and associated syntax-to-semantic-relationship mappings from large text corpora is described. The suggested approach builds on the authors' prior work with the Link Grammar, RelEx and OpenCog systems, as well as on a number of prior papers and approaches from the statistical language learning literature. If successful, this approach would enable the mining of all the information needed to power a natural language comprehension and generation system, directly from a large, unannotated corpus.
A notable example of a discontinuous-everywhere function that is not traditionally integrable, yet, when properly defined, can be integrated, is derivative of the Minkowski Question Mark function. Some subject matter overlaps that of [6].... more
A notable example of a discontinuous-everywhere function that is not traditionally integrable, yet, when properly defined, can be integrated, is derivative of the Minkowski Question Mark function. Some subject matter overlaps that of [6]. This paper includes numerical results for the Fourier transform of the measure, its Mellin transform, and Poisson kernel. This document is a research diary noting various results, and is haphazardly structured. This note is a part of a set of papers that explore the relationship between the real numbers, the Cantor set, the dyadic monoid (a sub-monoid of the modular group SL(2,Z)), and fractals. 1. INTRO THIS IS A CLIP-BOOK or DIARY of PARTIALLY-EXPLORED RESULTS. The intro hasn’t been written yet, but if it was, it would work like this: 2. DISTRIBUTION OF THE RATIONAL NUMBERS IN THE FAREY TREE One way to enumerate all of the rationals is by placing them on the Farey tree (or the Stern-Brocot tree). It can be shown that the Farey tree enumerates all...
This short note provides a numerical exploration of the entropy of the Gauss-Kuzmin distribution, confirming that it seems to have a value of 3.432527514776... bits. Some information-theoretic questions regarding the distribution of... more
This short note provides a numerical exploration of the entropy of the Gauss-Kuzmin distribution, confirming that it seems to have a value of 3.432527514776... bits. Some information-theoretic questions regarding the distribution of rationals are explored. In particular, one may define a “de facto” entropy for fractions with a small denominator; it is not clear that this de-facto entropy approaches the above in the limit of large denominators.
This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another definition, it is the transfer operator (the... more
This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another definition, it is the transfer operator (the Frobenius-Perron operator) of the Bernoulli map, also variously known as the doubling map or the sawtooth map. The map is interesting for multiple reasons. One is that the set of infinite binary strings is the Cantor set; this implies that the Bernoulli operator has a set of fractal eigenfunctions. These are given by the Takagi (or Blancmange) curve. The set of all infinite binary strings can also be understood as the infinite binary tree. This binary tree has a large number of self-similarities, given by the dyadic monoid. The dyadic monoid has an extension to the modular group PSL(2,Z), which plays an important role in analytic number theory; and so there are many connections between the Bernoulli map and various number-theoretic functions, includ...
A series representation for the Riemann zeta function in terms of the falling Pochhammer symbol is derived from the polynomial representation of the GaussKuzmin-Wirsing (GKW) operator.
Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in... more
Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in turn describes the structure of the Cantor set. The infinite binary tree can be viewed as a certain subset of the modular group PSL(2;Z). The subset is essentially the dyadic groupoid or dyadic monoid. It provides the natural setting for the symmetry and self-similarity of many frac- tals, including those associated with period-doubling maps, with phase-locking maps, and with various dynamical systems in general. The aim of this text is to provide a simple exposition of the symmetry and its articulation. In the process, this paper attempts to clarify the relationships between a cluster of inter- related ideas from number theory: those surrounding the modular group, elliptic curves and the Cantor Set. It has long been widely known that the modular g...
Some random ruminations on the nature of algebra and symbolic reasoning.
This paper consists of the extended working notes and observations made during the development of a joint paper[?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to... more
This paper consists of the extended working notes and observations made during the development of a joint paper[?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably more content, although is looser and sloppier, and is an exploration of tangents, dead-ends, and ideas shooting off in uncertain directions. The finite differences or Newton series of certain expressions involving the Riemann zeta function are explored. These series may be given an asymptotic expansion by converting them to Norlund-Rice integrals and applying saddle-point integration techniques. Numerical evaluation is used to confirm the appropriateness of the asymptotic expansion. The results extend on previous results for such series, and a general form for Dirichlet L-functions is given. C...
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Extract from the language-learning diary, reporting on the first small dataset containing connector sets. This is the 11 June 2017 update of the original 11 May 2017 report. It includes more data and figures, and updates the figures for... more
Extract from the language-learning diary, reporting on the first small dataset containing connector sets. This is the 11 June 2017 update of the original 11 May 2017 report. It includes more data and figures, and updates the figures for readability (legibility). It also revises notation slightly.
This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and... more
This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and thus this note focuses on providing a numerical survey. These results indicate a broad class of previously unexamined functions may obey the Riemann hypothesis in general, and even share the non-trivial zeros in particular.
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Research Interests: Number Theory, Ergodic Theory, Dynamical Systems, Diophantine approximation, Graduate Student, and 12 moreContinued Fractions, Symbolic Dynamics, General linear group, Phase Locking, Fields with Finite Characteristics, Binary Tree, Modular group, Dynamic System, Metric Number Theory, Hausdorff Dimension, Transcendental Theory, and P adic Numbers
ABSTRACT The chiral bag model is a successful model for the static properties (masses and moments) of a nucleon. In its formulation a non-trivial vacuum of confined fermions plays an important role of maintaining integral baryon number... more
ABSTRACT The chiral bag model is a successful model for the static properties (masses and moments) of a nucleon. In its formulation a non-trivial vacuum of confined fermions plays an important role of maintaining integral baryon number and positivity of the Hamiltonian. The origin of non-zero expectation values for various fermion bilinears in this vacuum is understood as a Casimir effect and the values of these are computed. A discussion is made of the infinities arising in the calculation. An overview of the chiral bag model and some of its predictions are given.
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This short note provides a numerical exploration of the entropy of the Gauss-Kuzmin distribution, confirming that it seems to have a value of 3.432527514776... bits. Some information-theoretic questions regarding the distribution of... more
This short note provides a numerical exploration of the entropy of the Gauss-Kuzmin distribution, confirming that it seems to have a value of 3.432527514776... bits. Some information-theoretic questions regarding the distribution of rationals are explored. In particular, one may define a "de facto" entropy for fractions with a small denominator; it is not clear that this de-facto entropy approaches the above in the limit of large denominators.
This short note presents some empirical data on the distribution of word pairs obtained from English text. Particular attention is paid to the distribution of mutual infor-mation.
The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by... more
The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by classical techniques that its derivative must vanish on all rationals. Since the Question Mark itself is continuous, one concludes that the derivative must be non-zero on the irrationals, and is thus a discontinuous-everywhere function. This derivative is the subject of this essay. Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then sho...
Research Interests:
A novel approach to the fully automated, unsupervised extraction of dependency grammars and associated syntax-to-semantic-relationship mappings from large text corpora is described. The suggested approach builds on the authors' prior... more
A novel approach to the fully automated, unsupervised extraction of dependency grammars and associated syntax-to-semantic-relationship mappings from large text corpora is described. The suggested approach builds on the authors' prior work with the Link Grammar, RelEx and OpenCog systems, as well as on a number of prior papers and approaches from the statistical language learning literature. If successful, this approach would enable the mining of all the information needed to power a natural language comprehension and generation system, directly from a large, unannotated corpus.