Grammars of Formal Uncertainty

https://arxiv.org/abs/2505.20047

#HackerNews #Grammars #of #Formal #Uncertainty #arxiv #FormalLanguages #UncertaintyTheory #HackerNews

TIL:

• There is a psychology.fandom.com wiki. someone made a wiki about psychology on fandom.com
• It has an article about the Chomsky hierarchy (which is usually a programming/CS topic)
• It has 34 thousand pages and is updated semi-regularily

Though it does seem quite a lot is taken from wikipedia.

#psychology #programming #formallanguages #wikipedia

Cactus Language • Overview 3.2
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/

Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again. Moreover, we need to implement both ways of proceeding in computational form. Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects. Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets. Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.

Resources —

Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory

Cactus Language • Overview 3.1
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/

In the development of Cactus Language to date the following two species of graphs have been instrumental.

• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).

It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage. Partly because the two species are so closely related and partly for the sake of brevity, we'll always use the genus name “PARC” to denote the corresponding cacti.

To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the “fibers” or “inverse images” of the propositions.

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory

Cactus Language • Overview 2
• https://inquiryintoinquiry.com/2025/03/06/cactus-language-overview-2/

In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between the two domains. If none of the formalisms readily available or in common use meet all the design requirements coming to mind then it is necessary to contemplate the design of a new language especially tailored to the purpose.

In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe which are indicated by them.

For computational purposes it is convenient to have a middle ground or an intermediate language for negotiating between the “koine” of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if that makes it necessary to introduce an artificial medium of exchange between the two domains.

If the necessary computations are to be carried out in an organized fashion, and ultimately or partially by familiar classes of machines, then the strings expressing logical propositions are likely to find themselves parsed into tree‑like data structures at some stage of the game. As far as their abstract structures as graphs are concerned, there are several species of graph‑theoretic data structures fitting the task in a reasonably effective and efficient way.

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#FormalLanguages

Cactus Language • Overview 1.2
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/

Resource —

For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to my Inquiry blog.

Part 1
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1

Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax

Part 2
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2

Generalities About Formal Grammars
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities

Part 3
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics

Cactus Language • Mechanics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics

Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics

Stretching Exercises
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises

References
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References

Document History
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory

Cactus Language • Overview 1.1
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/

❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.❞

— Herbert J. Bernstein • “Idols of Modern Science”

The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions. In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called “painted cacti”.

Painted cacti are a specialization of what graph‑theorists refer to as “cacti”, which are in turn a generalization of what they call “trees”. The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint. The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory

Theme One Program • Motivation 1
• https://inquiryintoinquiry.com/2024/06/03/theme-one-program-motivation-1-b/

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.

Resources —

Theme One Program • Overview
• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition
• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide
• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

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

Today (3PM GMT, May 15), I will be hosting this week's Formal Language and Neural Networks (FLaNN - https://flann.super.site/) seminar.

The speaker is Frank Drewes from Umeå University, who will be talking about Graph Extension Grammars.

This is a fantastic weekly seminar focusing on the interpretability and computational power of neural language models, especially as related to formal languages.

Check out the site for past and future talks!

#ML #Interpretability #FormalLanguages

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/2024/06/09/theme-one-program-exposition-1-b/

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
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism

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