Appendix 615

system is different from 10, it may be given explicitly (for example, 10 ¯ 2 = 10₂ = 2). Arithmetic expressions are formed from various arithmetic operations with numbers as operands. Operations are classified by their syntactic form: unary operations ( -(3) or + (7) ); binary operations (7 - 6, 3/8 or 3 ­ 2 = 3²); and n-ary operations (3 + 8 + 6 or 5 x 6 x 2 x 3). Functions are defined as taking a list of numbers as operands (abs(3) or max(5, 7, - 12)). There is a counting function that takes any set or list of entities as inputs and produces their number (if X : = (Ma, Mb, Mc) then number(X) = 3). Abbreviations are introduced for many of the operations and functions.

11.2 number-set-name : = (digit ÷ F )-string

A special subset of (alternative) numbers may be defined by substituting a F for a digit. The F stands for any digit (of the base of the number).

EXAMPLE 01F = 010÷ 011 01F binary

A quantity is a number of units of a given dimension. A unit is defined by the dimension and the conversion between the given unit and other units of the same dimension. Conversions can be expressed either as the amount of the other unit for each of the given units (e.g., 1 minute is 60 seconds) or as the amount of the given unit per each of the other units (e.g., 1 minute is 1/60 per second = .0167 / second). When conversions are not linear, it is necessary to use functions of the other unit. Thus, for bits the conversion to states is log₂(states) (e.g., 128 states is equivalent to log₂(128) = 7 bits).

Each dimension has a base unit (e.g., seconds for the dimension of time). A dimension may also be given as a product of two other dimensions (e.g., [energy] is [force x distance]) or the ration of two other dimensions (e.g., [velocity] is [length / time]). We use the standard bracket notation to indicate dimension, (e.g., [l/t] for the dimension of velocity).

There are two primary boolean values, true and false. Boolean-variables, boolean-expressions, and relational-expressions are expressions that evaluate (potentially) to true or false. Boolean expressions are made up from the standard operations on truth values: negation (Ø ), implication (É ), equivalence (º ), conjunction (L ), disjunction (V), and exclusive-or (Å ). Relations are defined on numbers.

COMMENT More general definitions for entities (for = and ¹ ) and for ordered sets (for <, >, £ , and ³ ) are not needed.

Making use of the prior general conventions, PMS is developed systematically through the definitions of the various components: P. M, S, etc. Much of the development repeats common abbreviations and conventions, simply to provide a self-contained notational system.