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ternary.md

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Design of Ternary

This document just presents the design part of the MCP. There is also a rationale.

The Ternary type

The new build-in type Ternary is similar to Boolean:

type Ternary // Note: Defined with Modelica syntax although predefined
  TernaryType value; // Accessed without dot-notation
  parameter StringType quantity = "";
  parameter BooleanType start = unknown;
  parameter BooleanType fixed = true;
end Ternary;

New keyword unknown

The keyword unknown is introduced to refer to the unknown truth value.

Conversion from Boolean

Explicit conversion

The two known ternary truth values can be constructed by explicit conversion from the Boolean value x using the syntax Ternary(x). This is the primitive way of constructing the other two ternary values besides unknown.

Implicit conversion

A Boolean expression can be used wherever a Ternary is expected, via implicit conversion. For example true and unknown is a valid expression of type Ternary, since the operator and requires both operands to be Ternary as soons as one of them is.

Total order of values

There is a total order of the possible Ternary values given by Ternary(false) < unknown < Ternary(true). This allows Ternary to be used as array dimension, and gives meaning to the relational operators.

Expressions with Ternary

Logical operations on Ternary are defined according to Kleene, where unknown can be thought of as representinc uncertainty about being true or false. With this interpretation the Kleene logic tables then follow from the Boolean logic.

For example, to evaluate false or unknown, one has to consider both possible outcomes for the uncertain operand, meaning that the result might be ither false or false or false or true. Since the different outcomes don't agree, the result of the operation is uncertain, represented by unknown.

Unary operators

Unary operators on Ternary, where x is an expression of type Ternary:

Expression Result type
not x Ternary

Truth table for not x:

x not x
Ternary(false) Ternary(true)
unknown unknown
Ternary(true) Ternary(false)

Binary operators

Binary operators on Ternary, where x and y are an expressions of type Ternary:

Expression Result type
x and y Ternary
x or y Ternary

Truth table for x and y:

x and y y = Ternary(false) y = unknown y = Ternary(true)
x = Ternary(false) Ternary(false) Ternary(false) Ternary(false)
x = unknown Ternary(false) unknown unknown
x = Ternary(true) Ternary(false) unknown Ternary(true)

Truth table for x or y:

x or y y = Ternary(false) y = unknown y = Ternary(true)
x = Ternary(false) Ternary(false) unknown Ternary(true)
x = unknown unknown unknown Ternary(true)
x = Ternary(true) Ternary(true) Ternary(true) Ternary(true)

Boolean-valued relations

Relational operators for Ternary are defined according to the total order. Here, x and y are an expressions of type Ternary:

Expression Result type Equivalent to
x < y Boolean
x <= y Boolean not (y > x)
x > y Boolean y < x
x >= y Boolean not (x < y)
x != y Boolean x < y or y < x
x == y Boolean not (x != y)

The truth tables follow directly from the total order, so there is no point in listing them all. Here is an example which is just another way of saying Ternary(false) < unknown < Ternary(true):

x < y y = Ternary(false) y = unknown y = Ternary(true)
x = Ternary(false) false true true
x = unknown false false true
x = Ternary(true) false false false

Ternary equations

As a discrete-valued type, Ternary is treated analogous to Boolean in equations. In equations, there is no special meaning of unknown.

External language interface

External type corresponding to Ternary:

External language Input type Output type
"C" int int *
"FORTRAN 77" INTEGER INTEGER

Mapping of values:

Ternary External language
Ternary(false) -1
unknown 0
Ternary(true) 1

Unlike Boolean for which there is a natural interpretation of any non-zero value from C as meaning true, there is a unique way to represent each of the three different values of Ternary. This is analogous to the external language mapping for enumerations and helps detecting errors in the external language code.