Tag Archives: Formal Languages

Cactus Language • Discussion 1

Posted on May 7, 2025 by Jon Awbrey

Re: Cactus Language • Preliminaries 9 Re: Cybernetics • Joe Bury JB: What does subcatenation and surcatenation mean?  Their definitions are not found in a dictionary.  I get the formulas you wrote but I don’t understand the meaning. Thanks for … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Survey of Theme One Program • 7

Posted on May 6, 2025 by Jon Awbrey

This is a Survey of resources relating to the Theme One Program I worked on all through the 1980s.  The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning.  I had earlier developed … Continue reading →

Posted in Algorithms, Animata, Artificial Intelligence, Automated Research Tools, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 80 Comments

Survey of Relation Theory • 9

Posted on May 5, 2025 by Jon Awbrey

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many … Continue reading →

Posted in Algebra, Algebra of Logic, C.S. Peirce, Category Theory, Combinatorics, Discrete Mathematics, Duality, Dyadic Relations, Formal Languages, Foundations of Mathematics, Graph Theory, Group Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Model Theory, Relation Theory, Semiotics, Set Theory, Sign Relational Manifolds, Sign Relations, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , | 10 Comments

Cactus Language • Preliminaries 9

Posted on April 25, 2025 by Jon Awbrey

We now have the materials in place to formulate a definition of our subject. The painted cactus language with paints in the set is the formal language defined as follows. In the idiom of formal language theory, a string is … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 9 Comments

Cactus Language • Preliminaries 8

Posted on April 22, 2025 by Jon Awbrey

Defining the basic operations of concatenation and surcatenation on arbitrary strings gives them operational meaning for the all‑inclusive language   With that in hand it is time to adjoin the notion of a more discriminating grammaticality, in other words, a … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Preliminaries 7

Posted on April 19, 2025 by Jon Awbrey

The array of syntactic operators may be put in more organized form by making a few additional conventions and auxiliary definitions. Concatenation The conception of concatenation permits extension to its natural prequel, the corresponding operator on zero operands. From that … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Preliminaries 6

Posted on April 16, 2025 by Jon Awbrey

The definitions of the syntactic connectives can be made a little more succinct by defining the following pair of generic operators on strings. Concatenation The concatenation of the sequence of strings is defined recursively as follows. Surcatenation The surcatenation of … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 4 Comments

Cactus Language • Preliminaries 5

Posted on April 13, 2025 by Jon Awbrey

The easiest way to define the language is to indicate the general run of operations required to construct the greater share of its sentences from the designated few which require a special election. To do that we introduce a family … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Preliminaries 4

Posted on April 11, 2025 by Jon Awbrey

The informal mechanisms illustrated in the preceding discussion equip us with a description of cactus language adequate to providing conceptual and computational representations for the minimal formal logical system variously known as propositional logic or sentential calculus. The painted cactus … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 5 Comments

Cactus Language • Preliminaries 3

Posted on April 9, 2025 by Jon Awbrey

A few definitions from formal language theory are required at this point. An alphabet is a finite set of signs, typically, A string over an alphabet is a finite sequence of signs from The length of a string is just … Continue reading →

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments