OK one more #mathematics & #formalLanguages question, not unrelated to the stupid equations below:

Surely (he said, in the form of a question) somebody has explored the "compressibility" of random arithmetic, trigonometric and other symbolic math expressions under standard simplification rules. Right?

For example, here's a random S-expression, simplified by sage to something shorter (and with fewer terms).

Have we talked about probability distributions of simplified tree sizes?

Theme One Program • Exposition 1.2
• https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

The Idea↑Form Flag

The graph-theoretic data structures used by the program are built up from a basic data structure called an “idea-form flag”. That structure is defined as a pair of Pascal data types by means of the following specifications.

Figure 1. Type Idea = ^Form
• https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-type-idea-5eform.png

Figure 2. Code Box
• type idea = ^form;
• form = record
• sign: char;
• as, up, on, by: idea;
• code: numb
• end;

An “idea” is a pointer to a “form”.
• A “form” is a record consisting of:
• A “sign” of type “char”;
• Four pointers, “as”, “up”, “on”, “by”, of type “idea”;
• A “code” of type “numb”, that is, an integer in [0, max integer].

Represented in terms of digraphs, or directed graphs, the combination of an idea pointer and a form record is most easily pictured as an arc, or directed edge, leading to a node labeled with the other data, in this case, a letter and a number.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Exposition 1.1
• https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

Theme One is a program for constructing and transforming a particular species of graph‑theoretic data structures, forms designed to support a variety of fundamental learning and reasoning tasks.

The program evolved over the course of an exploration into the integration of contrasting types of activities involved in learning and reasoning, especially the types of algorithms and data structures capable of supporting all sorts of inquiry processes, from everyday problem solving to scientific investigation. In its current state, Theme One integrates over a common data structure fundamental algorithms for one type of inductive learning and one type of deductive reasoning.

We begin by describing the class of graph-theoretic data structures used by the program, as determined by their local and global features. It will be the usual practice to shift around and view these graphs at many different levels of detail, from their abstract definition to their concrete implementation, and many points in between.

The main work of the Theme One program is achieved by building and transforming a single species of graph-theoretic data structures. In their abstract form these structures are closely related to the graphs called cacti and conifers in graph theory, so we’ll generally refer to them under those names.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 6
• https://inquiryintoinquiry.com/2022/08/20/theme-one-program-motivation-6-2/

Comments I made in reply to a correspondent’s questions about delimiters and tokenizing in the Learner module may be worth sharing here.

In one of the projects I submitted toward a Master’s in psychology I used the Theme One program to analyze samples of data from my advisor’s funded research study on family dynamics. In one phase of the study observers viewed video-taped sessions of family members (parent and child) interacting in various modes (“play” or “work”) and coded qualitative features of each moment’s activity over a period of time.

The following page describes the application in more detail and reflects on its implications for the conduct of scientific inquiry in general.

Exploratory Qualitative Analysis of Sequential Observation Data
• https://oeis.org/wiki/User:Jon_Awbrey/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

In this application a “phrase” or “string” is a fixed-length sequence of qualitative features and a “clause” or “strand” is a sequence of such phrases delimited by what the observer judges to be a significant pause in the action.

In the qualitative research phases of the study one is simply attempting to discern any significant or recurring patterns in the data one possibly can.

In this case the observers are tokenizing the observations according to a codebook that has passed enough intercoder reliability studies to afford them all a measure of confidence it captures meaningful aspects of whatever reality is passing before their eyes and ears.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 5
• https://inquiryintoinquiry.com/2022/08/20/theme-one-program-motivation-5-2/

Since I’m working from decades-old memories of first inklings I thought I might peruse the web for current information about Zipf’s Law. I see there is now something called the Zipf–Mandelbrot (and sometimes –Pareto) Law and that was interesting because my wife Susan Awbrey made use of Mandelbrot’s ideas about self-similarity in her dissertation and communicated with him about it. So there’s more to read up on.

Just off-hand, though, I think my Learner is dealing with a different problem. It has more to do with the savings in effort a learner gets by anticipating future experiences based on its record of past experiences than the savings it gets by minimizing bits of storage as far as mechanically possible. There is still a type of compression involved but it’s more like Korzybski’s “time-binding” than space-savings proper. Speaking of old memories …

The other difference I see is that Zipf’s Law applies to an established and preferably large corpus of linguistic material, while my Learner has to start from scratch, accumulating experience over time, making the best of whatever data it has at the outset and every moment thereafter.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 4
• https://inquiryintoinquiry.com/2022/08/19/theme-one-program-motivation-4-2/

From Zipf’s Law and the category of “things that vary inversely with frequency” I got my first brush with the idea that keeping track of usage frequencies is part and parcel of building efficient codes.

In its first application the environment the Learner has to learn is the usage behavior of its user, as given by finite sequences of characters from a finite alphabet, which sequences of characters might as well be called “words”, together with finite sequences of those words which might as well be called “phrases” or “sentences”. In other words, Job One for the Learner is the job of constructing a “user model”.

In that frame of mind we are not seeking anything so grand as a Universal Induction Algorithm but simply looking for any approach to give us a leg up, complexity wise, in Interactive Real Time.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 3
• https://inquiryintoinquiry.com/2022/08/18/theme-one-program-motivation-3-2/

Sometime around 1970 John B. Eulenberg came from Stanford to direct Michigan State’s Artificial Language Lab, where I would come to spend many interesting hours hanging out all through the 70s and 80s. Along with its research program the lab did a lot of work on augmentative communication technology for limited mobility users and the observations I made there prompted the first inklings of my Learner program.

Early in that period I visited John’s course in mathematical linguistics, which featured Laws of Form among its readings, along with the more standard fare of Wall, Chomsky, Jackendoff, and the Unified Science volume by Charles Morris which credited Peirce with pioneering the pragmatic theory of signs. I learned about Zipf’s Law relating the lengths of codes to their usage frequencies and I named the earliest avatar of my Learner program XyPh, partly after Zipf and playing on the xylem and phloem of its tree data structures.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 2.2
• https://inquiryintoinquiry.com/2022/08/17/theme-one-program-motivation-2-2/

As I mentioned, work on those two projects proceeded in a parallel series of fits and starts through interwoven summers for a number of years, until one day it dawned on me how the Learner, one of whose aliases was Index, could be put to work helping with sundry substitution tasks the Modeler needed to carry out.

So I began integrating the functions of the Learner and the Modeler, at first still working on the two component modules in an alternating manner, but devoting a portion of effort to amalgamating their principal data structures, bringing them into convergence with each other, and unifying them over a common basis.

Another round of seasons and many changes of mind and programming style, I arrived at a unified graph-theoretic data structure, strung like a wire through the far‑flung pearls of my programmed wit. But the pearls I polished in alternate years maintained their shine along axes of polarization whose grains remained skew in regard to each other. To put it more plainly, the strategies I imagined were the smartest tricks to pull from the standpoint of optimizing the program’s performance on the Learning task I found the next year were the dumbest moves to pull from the standpoint of its performance on the Reasoning task. I gradually came to appreciate that trade-off as a discovery.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 2.1
• https://inquiryintoinquiry.com/2022/08/17/theme-one-program-motivation-2-2/

A side-effect of working on the Theme One program over the course of a decade was the measure of insight it gave me into the reasons why empiricists and rationalists have so much trouble understanding each other, even when those two styles of thinking inhabit the very same soul.

The way it came about was this. The code from which the program is currently assembled initially came from two distinct programs, ones I developed in alternate years, at first only during the summers.

In the Learner program I sought to implement a Humean empiricist style of learning algorithm for the adaptive uptake of coded sequences of occurrences in the environment, say, as codified in a formal language. I knew all the theorems from formal language theory telling how limited any such strategy must ultimately be in terms of its generative capacity, but I wanted to explore the boundaries of that capacity in concrete computational terms.

In the Modeler program I aimed to implement a variant of Peirce’s graphical syntax for propositional logic, making use of graph-theoretic extensions I had developed over the previous decade.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Theme One Program • Motivation 1
• https://inquiryintoinquiry.com/2022/08/16/theme-one-program-motivation-1-2/

The main idea behind the Theme One program is the efficient use of graph-theoretic data structures for the tasks of “learning” and “reasoning”.

I am thinking of learning in the sense of learning about an environment, in essence, gaining information about the nature of an environment and being able to apply the information acquired to a specific purpose.

Under the heading of reasoning I am simply lumping together all the ordinary sorts of practical activities which would probably occur to most people under that name.

There is a natural relation between the tasks. Learning the character of an environment leads to the recognition of laws which govern the environment and making full use of that recognition requires the ability to reason logically about those laws in abstract terms.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

Anyone into #parsing #formallanguages willing to look at a solution for an exercise from the purple #dragonbook for me?

The exercise in question asks to write a recursive descent parser for what seems to me like an ambiguous left recursive grammar. I solved the ambiguity by completely ignoring the left recursion. Is this reasonable? Would you have taken a different approach? Short python solution & grammar at https://gitlab.com/-/snippets/2510459

This book is the most accessible introductory text I've ever read on #FormalLanguages theory.

I don't know Prof. Emre Sermutlu, the author. But he said in the preface that he is a long-time friend of Prof. Ali Selcuk, who is my long-time friend. Ali and I attended #ComputerScience grad school together in the early 1990s. We read Hopcroft and Ullman, then.

I wish I had this book those many years ago, when I thought Pumping Lemma meant something wholly unholy.

https://www.amazon.com/dp/B08K41YGHS?psc=1&ref=ppx_yo2ov_dt_b_product_details

This book is the most accessible introductory text I've ever read on #FormalLanguages theory.

I don't know Prof. Emre Sermutlu, the author. But he said in the preface that he is a long-time friend of Prof. Ali Selcuk, who is my long-time friend. Ali and I attended #ComputerScience grad school together in the early 1990s. There, I used Hopcroft and Ullman. I wish I had this book those many years ago, when I still believed Pumping Lemma meant something wholly unholy.

https://www.amazon.com/dp/B08K41YGHS?psc=1&ref=ppx_yo2ov_dt_b_product_details

There are a four invited speakers, but I am only personally familiar with three (Cyril Allauzen @ Google, Will Merrill @ NYU/Google, and Dana Angluin @ Yale).

These three talks should be fantastic, especially if you are interested in #automata, #FormalLanguages, and #Interpretability in neural language models!

(Plugging https://flann.super.site/ if those sound cool to you)

three days for nailing the correct order of parser combinators for a large language 🫠

I think I'll need to add Bison/Yacc to my toolbelt

#proglang #plt #formallanguages #compilers

So, I think I need some help from #mathstodon and the wider community with understanding of #computability and #recursiveenumeration.

If you go to

http://jdh.hamkins.org/alan-turing-on-computable-numbers/

there is a wonderful article by Dr. Joel David Hamkins, a mathematician whose work I deeply admire.

However, if you scroll down to the comments, you will notice a comment from Nathan Harvey (that’s me!) contesting some of the claims of the article. In particular, Dr. Hamkins makes the claim that with Turing’s original definition of computable numbers and functions, addition is not a computable function. He appears to view computable functions as consumers of the output of the programs that represent the reals, not as consumers of the programs themselves, and I give an example where the analysis changes and make reference to Turing’s definition.

But! I can be wrong here. Dr. Hamkins is the real stuff. I just keep coming back, after spending time to consider his points and trying to reframe them to ensure I understand, thinking that my point wasn’t refuted or even really addressed. And as I read the responses, I fail to see any comments about my example or my point about the difference between consuming outputs of the computation versus the actual program, … and I keep thinking the point is getting missed. But that’s dangerous territory that can lead one to crankdom and obstinate ignorance. I don’t want to do that to myself.

So if there are any mathematicians who enjoy the area of computable functions and want to give it a quick read, I would appreciate any comments on my point. Even if it’s just a comment “No, Nate, you’re wrong and deeply misguided” with no further explanation. After one or two of those from other mathematicians, I’ll take the L and shrink off to read more books on the topic.

And if you don’t know the answer but have followers who work in that or related areas of math, a boost would be appreciated. This is an area of math that is deeply interesting to me and I thought I understood it well, but self-taught people are known to go off the rails.

Some hashtags to meet the right eyes:
#math #mathematics #metamathematics #constructivism #Turing #formallanguages

Some attags, not to get their direct response (unless interested themselves in doing so), but if they find the discussion respectful and the topic interesting, a boost might benefit the discussion:
@ProfKinyon @MartinEscardo @BartoszMilewski

#DifferentialPropositionalCalculus • 2.4
• https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

#CactusCalculus Conventions —

The briefest expression for #LogicalTruth is the #EmptyWord, denoted \(\boldsymbol\varepsilon\) or \(\boldsymbol\lambda\) in #FormalLanguages, where it forms the #IdentityElement for #Concatenation. It may be given visible expression in this context by means of the logically equivalent form \(\texttt{((} ~ \texttt{))},\) or, especially if operating in an algebraic context, by a simple \(1.\)

#Survey of #AnimatedLogicalGraphs

• https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

This is a Survey of blog and wiki posts on #LogicalGraphs, encompassing several families of graph-theoretic structures originally developed by C.S. #Peirce as graphical #FormalLanguages or visual styles of #Syntax amenable to #Interpretation for #Logic.

In his papers on #QualitativeLogic, #EntitativeGraphs, and #ExistentialGraphs, Peirce developed several versions of a #GraphicalFormalism designed to be interpreted for logic.

The best thing about classes on #FormalLanguages and #TheoraticalCS is all the opportunities to annoy people with random facts about Noam #Chomsky