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 \mathfrak{P} = \{ p_j : j \in J \} is the formal language \mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^* defined as follows.

\begin{array}{ll} \text{PC 1.} & \text{The blank symbol}~ m_1 ~\text{is a sentence.} \\ \text{PC 2.} & \text{The paint}~ p_j ~\text{is a sentence for each}~ j ~\text{in}~ J. \\ \text{PC 3.} & \mathrm{Conc}^0 ~\text{and}~ \mathrm{Surc}^0 ~\text{are sentences.} \\ \text{PC 4.} & \text{For each positive integer}~ n, \\ & \text{if}~ s_1, \ldots, s_n ~\text{are sentences} \\ & \text{then}~ \mathrm{Conc}_{k=1}^n s_k ~\text{is a sentence} \\ & \text{and}~ \mathrm{Surc}_{k=1}^n s_k ~\text{is a sentence.} \end{array}

In the idiom of formal language theory, a string s is called a sentence of \mathfrak{L} if and only if it belongs to \mathfrak{L}, or simply a sentence if the language \mathfrak{L} is understood.  A sentence of \mathfrak{C} (\mathfrak{P}) is referred to as a painted and rooted cactus expression on the palette \mathfrak{P}, or a cactus expression for short.

Resources

cc: Academia.edu • BlueSky • Laws of Form • Mathstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

This entry was 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 and tagged , , , , , , , , , , , , , , , , . Bookmark the permalink.

9 Responses to Cactus Language • Preliminaries 9

  1. Pingback: Survey of Animated Logical Graphs • 7 | Inquiry Into Inquiry

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  4. ashkotin's avatar ashkotin says:
    May 12, 2025 at 4:32 am

    Interesting language. It’s unusual to treat ” ” as a sentence. Usually it is just a separator for other lexemes.

    Is these correct:

    One blank in brackets i.e. “( )” is a sentence.

    Two blanks in brackets i.e. “(  )” is not a sentence.

    Are the three strings below sentences?

    ( , , ) 

    (,,)

    ((),(),())

    And at last. You use your own notation to define formal language. Is it correct that this language is context-free?

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  5. ashkotin's avatar ashkotin says:
    May 13, 2025 at 5:05 am

    One more thing.

    String “()” can be derived using Conc0 and PC 4:

    Surc0 ≝ “(“·Conc0·”)”

    i.e. Surc0 can be defined on language, not for language definition. It seems we need not Surc0 in language definition.

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  6. Pingback: Cactus Language • Discussion 2 | Inquiry Into Inquiry

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