This short note provides a numerical exploration of the entropy of the Gauss-Kuzmin distribution,... 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.
The quark vacuum plays an essential role in making the chiral bag model consistent and well-defin... more The quark vacuum plays an essential role in making the chiral bag model consistent and well-defined. We compute vacuum (one-loop) contributions to the baryon density as a regulated sum over quark eigenmodes. The sum is evaluated in straightforward numerical fashion and yields results for the moments of the baryon density which are in agreement with those of Jezabek and Heller.
Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at od... more Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry's constant given by Ramanujan: ζ (3) = 7π 3 180 − 2 ∞ ∑ n=1 1 n 3 (e 2πn − 1) Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe's identities as corollaries. The resemblance of the general relation to the structure of theta functions and modular forms is briefly sketched.
A software architecture is described which enables a virtual agent in an online virtual world to ... more A software architecture is described which enables a virtual agent in an online virtual world to carry out simple English language interactions grounded in its perceptions and actions. The use of perceptions to guide anaphor resolution is discussed, along with the use of natural language generation to answer simple questions about the observed world. This architecture has been implemented within the larger PetBrain system, which is built on the OpenCog open-source AI software framework and architected based on the OpenCogPrime design for integrative AGI, and has previously been used for nonlinguistic intelligent behaviors such as imitation and reinforcement learning.
Using cranking techniques a variety of baryon observables are calculated within the framework of ... more Using cranking techniques a variety of baryon observables are calculated within the framework of the chiral bag model. Effects from valence quarks, mesons and the quark Dirac sea are included. Results for the NA mass splitting and nucleon magnetic moments are found to be relatively insensitive to the bag radius. E NUCLEAR STRUCTURE in 1 ~. IH; calculated Chiral bag model.
ABSTRACT The chiral bag model is a successful model for the static properties (masses and moments... 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.
A novel approach to sentence generation – SegSim, Sentence Generation by Similarity Matching – is... 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
A variety of speculations about the nature of quantum mechanics and wavefunction collapse. A numb... 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
Bulletin of the Novosibirsk Computing Center. Series: Computing Science., 2017
This document develops general concepts useful for extracting knowledge embedded in large graphs ... more This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as causeeffect-type relations. Almost no underlying assumptions are made, other than that the data can be presented in terms of pair-wise relationships between objects/events. This assumption is used to mine for patterns in the dataset, defining a reduced graph or dataset that boils-down or concentrates information into a more compact form. The resulting extracted structure or set of patterns are manifestly symbolic in nature, as they capture and encode the graph structure of the dataset in terms of a (generative) grammar. This structure is identified as having the formal mathematical structure of a sheaf. In essence, this paper introduces the basic concepts of sheaf theory into the domain of graphical datasets. DRAFT: This is an unfinished draft; the last 1/4th of the document needs a complete make-over.
A new model of symbol grounding is presented, in which the structures of natural language, logica... 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...
Given a continued fraction, we construct a certain function hat is discontinuous at every rationa... 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 ...
The RelEx semantic relation extractor (Ross et al., 2005) is an open-source dependency parser bui... 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 ...
The concept of self-similarity is central to the notion of a fractal, but the actual symmetry gro... 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 ...
The beta transformation is the iterated map $\beta x\mod1$. The special case of $\beta=2$ is know... 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 o... 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 report presents measurements of the quality of various different dictionaries learned from t... more This report presents measurements of the quality of various different dictionaries learned from two different language-learning pipelines; one is the ULL/Kolonin variant, the other is the Linas variant. The ULL/Kolonin dictionaries suggest two major breakthroughs: they suggest how the learning pipeline should be tuned, and they indicate how the pipeline is relatively insensitive to early stages of processing. The Linas variant results indicate that the sheer quantity of training data has a strong impact on the quality of the learned grammars. This report briefly reviews the dictionaries, the algorithms used to obtain them, and measurement results. It is assumed that the reader has a general familiarity with the project. 1 Baseline Measurements This provides the introduction and the baseline to the algorithm, the measurements, and the results.
This paper presents a review of the Gauss-Kuzmin-Wirsing (GKW) operator. The GKW operator is the ... 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 provides a numerical exploration of the entropy of the Gauss-Kuzmin distribution,... 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.
The quark vacuum plays an essential role in making the chiral bag model consistent and well-defin... more The quark vacuum plays an essential role in making the chiral bag model consistent and well-defined. We compute vacuum (one-loop) contributions to the baryon density as a regulated sum over quark eigenmodes. The sum is evaluated in straightforward numerical fashion and yields results for the moments of the baryon density which are in agreement with those of Jezabek and Heller.
Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at od... more Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry's constant given by Ramanujan: ζ (3) = 7π 3 180 − 2 ∞ ∑ n=1 1 n 3 (e 2πn − 1) Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe's identities as corollaries. The resemblance of the general relation to the structure of theta functions and modular forms is briefly sketched.
A software architecture is described which enables a virtual agent in an online virtual world to ... more A software architecture is described which enables a virtual agent in an online virtual world to carry out simple English language interactions grounded in its perceptions and actions. The use of perceptions to guide anaphor resolution is discussed, along with the use of natural language generation to answer simple questions about the observed world. This architecture has been implemented within the larger PetBrain system, which is built on the OpenCog open-source AI software framework and architected based on the OpenCogPrime design for integrative AGI, and has previously been used for nonlinguistic intelligent behaviors such as imitation and reinforcement learning.
Using cranking techniques a variety of baryon observables are calculated within the framework of ... more Using cranking techniques a variety of baryon observables are calculated within the framework of the chiral bag model. Effects from valence quarks, mesons and the quark Dirac sea are included. Results for the NA mass splitting and nucleon magnetic moments are found to be relatively insensitive to the bag radius. E NUCLEAR STRUCTURE in 1 ~. IH; calculated Chiral bag model.
ABSTRACT The chiral bag model is a successful model for the static properties (masses and moments... 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.
A novel approach to sentence generation – SegSim, Sentence Generation by Similarity Matching – is... 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
A variety of speculations about the nature of quantum mechanics and wavefunction collapse. A numb... 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
Bulletin of the Novosibirsk Computing Center. Series: Computing Science., 2017
This document develops general concepts useful for extracting knowledge embedded in large graphs ... more This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as causeeffect-type relations. Almost no underlying assumptions are made, other than that the data can be presented in terms of pair-wise relationships between objects/events. This assumption is used to mine for patterns in the dataset, defining a reduced graph or dataset that boils-down or concentrates information into a more compact form. The resulting extracted structure or set of patterns are manifestly symbolic in nature, as they capture and encode the graph structure of the dataset in terms of a (generative) grammar. This structure is identified as having the formal mathematical structure of a sheaf. In essence, this paper introduces the basic concepts of sheaf theory into the domain of graphical datasets. DRAFT: This is an unfinished draft; the last 1/4th of the document needs a complete make-over.
A new model of symbol grounding is presented, in which the structures of natural language, logica... 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...
Given a continued fraction, we construct a certain function hat is discontinuous at every rationa... 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 ...
The RelEx semantic relation extractor (Ross et al., 2005) is an open-source dependency parser bui... 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 ...
The concept of self-similarity is central to the notion of a fractal, but the actual symmetry gro... 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 ...
The beta transformation is the iterated map $\beta x\mod1$. The special case of $\beta=2$ is know... 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 o... 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 report presents measurements of the quality of various different dictionaries learned from t... more This report presents measurements of the quality of various different dictionaries learned from two different language-learning pipelines; one is the ULL/Kolonin variant, the other is the Linas variant. The ULL/Kolonin dictionaries suggest two major breakthroughs: they suggest how the learning pipeline should be tuned, and they indicate how the pipeline is relatively insensitive to early stages of processing. The Linas variant results indicate that the sheer quantity of training data has a strong impact on the quality of the learned grammars. This report briefly reviews the dictionaries, the algorithms used to obtain them, and measurement results. It is assumed that the reader has a general familiarity with the project. 1 Baseline Measurements This provides the introduction and the baseline to the algorithm, the measurements, and the results.
This paper presents a review of the Gauss-Kuzmin-Wirsing (GKW) operator. The GKW operator is the ... 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...
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