Formulloi

Formulloi is a formalism for representing mathematical formulas as tree-structured expressions. It is defined as a language-agnostic schema intended for symbolic computation, formula serialization, and formal verification. Expressions in formulloi are built from four categories of nodes: operator nodes, function nodes, variable nodes, and constant nodes. The formalism emphasizes explicit structure and a minimal, extensible set of constructors.

Syntax and data model: A formulloi expression is a rooted, finite tree. Each operator node specifies its arity and the operator symbol (for example, add, mul, pow). Function nodes carry a function symbol with a fixed arity. Leaves are either variables or constants. Canonical forms are obtained by normalization rules that flatten associative chains, sort commutative operands, and fold numeric constants when possible.

Semantics: Evaluation maps variables to values in a chosen domain (such as real numbers). The evaluation function is defined recursively; it may be total for closed expressions or partial when operands are undefined or domain errors occur. Formulloi supports equivalence by structural normalization, so two semantically equal formulas can be transformed into a common form under the normalization rules.

Examples: The expression x^2 + 3x + 2 can be represented in formulloi as (add (pow x 2) (mul 3 x) (const 2)). A historical note: in practice, formulloi expressions are serialized as compact trees and can be converted to more verbose textual or graphical forms for debugging and verification.

Origin and status: Formulloi is a synthetic concept used in theoretical discussions of formula representation. It has no single standard specification; several research projects explore prototype libraries, interoperability methods, and equivalence checking within the formulloi paradigm.