formuloval

Formuloval is a theoretical construct used to describe how formulas are assigned values under a given interpretation and variable assignment. It can be viewed as a generalization of standard expression evaluation, incorporating choices of operator semantics, domain restrictions, and contextual truth values.

In a formuloval framework, formulas are built from a fixed grammar, a domain D, and an interpretation I that assigns meaning to symbols (for example, functions, relations, and constants) over D. An evaluation rule, sometimes called Eval, takes a formula φ, a variable assignment σ that maps variables to elements of D, and the interpretation I, producing a value in D or a Boolean truth value. The same framework can support multiple interpretations to model different arithmetic or logical semantics, such as integer arithmetic, modular arithmetic, or floating-point semantics.

Formuloval is compositional: the value of a complex formula is determined by the values of its subformulas according to the connectives and operators used. It also supports substitution, so replacing a variable with a term or constant yields the evaluation of the resulting formula under the same interpretation.

Example: consider the arithmetic formula f = x * 3 + y with domain Z and the standard interpretation of + and ×. For the variable assignment x = 4, y = 5, the formuloval value is 4*3 + 5 = 17.

Applications include symbolic computation, formal verification, and teaching the semantics of formulas under alternative mathematical models. See also valuation, symbolic computation, and formal semantics.