The LFL Engine is designed to allow a computer to emulate the decision making process of humans where there is not a direct mapping between the input and output based on membership to a few logical groups as implemented in a cloud based server to allow a cloud based artificial intelligence system or cloud based soft AI for data analysis.

Where R is the set of Rule interval maximum/minimum pairs, I is the distance of the intervals, b is the percentage of belonging to a given interval rule, S is the set of inputs with x as any given input and y is the output for any given x input.

To understand how this system works;

**The syntax for representation of possibly dependent looping structures and associated arithmetic and logic in the aforementioned system,**

Given;

Using a combination of symbolism from Arithmetic, Logic, Set Notation and Sequences, it is possible to represent, in a compact notational form, a system of dependent (or a series of independent) looping structures which have a set (or sets) of inputs and the associated arithmetic and logic contained within those structures.

To represent the looping portion dealing with the input, the symbols ∀and ∈ from set notation can be used to indicate that the operations to follow should be performed for each of the inputs. In words, the first part of the example algorithm generically reads ‘For each input which is an element of the set of inputs’. This represents the looping portion of the structure by indicating that the operation(s) which follow should be performed for each input given. A comma is used to separate this portion from the operations which are to be performed on the input set. In words, this comma should be read as ‘do this’.

Another symbol which can represent a looping portion of the algorithm is the {} symbol with sub and super scripts followed by the word ‘with’ to represent the finite sequence or set with lower and upper bounds which is generated by the indicated operation(s). This can be used in combination with the above set notation symbols as seen in the given example. A standard subscripted variable is used to represent each individual element of the sequence or ‘set’ so generated as indicated by the

in the first part of the example algorithm. Using a multiple subscripted variable, it is possible to show the dependent nature of the looping structures as shown by the

in the second part of the example algorithm.

Using arithmetic notations the calculation operations to be performed can be shown and, if needed, this can be combined with logical operators to represent any logic test which may need to be represented. In the first part of the example algorithm, there is a simple subtraction calculation used to generate each element of the output set. In the second part, there is an ‘if and only if’ test (indicated by the ⇔ symbol) performed and only if this test is successful will the calculation indicated be done; otherwise, that element of the set will be set to zero as indicated be the ‘else’ portion of the statement. Any possible logical operators could be used and the logical operators could also be used without any arithmetic calculation indicated depending on the exact nature of the algorithm in question.

A comma is used to separate multiple operations to be performed on the same set of input(s) or any intermediate output(s); as indicated in the second part of the example algorithm, where a comma separates the generation of the

set and its use in generating the final output of the

set. In this context, the comma should be read as ‘and then do this’. As shown in the example algorithm, a semicolon is used to separate the different portions (or structures) of the algorithm as a whole.

Within this syntax, it is permissible to use any of the mathematical operators needed to represent the calculations or operations being performed. In the final portion of the example algorithm, the sigma notation is used in the calculation of the average of the

set to generate each element of the

set.

Using this syntax it is possible to show, in a relatively compact form, even more complex structures than the given example and most any calculation or logical operation. The given example should not be construed as showing all possibilities as there could be even more dependent or independent structures and even more inputs, calculations, operators or variables depending on the nature of any specific algorithm in question.

Given this we can go back and decompose that core LFL Engine algorithm into the language of your choice.