The new concept of semiknown values is used to represent uncertain knowledge. A semiknown value is one specific value of any domain
- which is not known precisely and unambiguously
    (in contrast to computer program values)   and
- which is not unknown
    (in contrast to unassigned variables).

FunLog++ is a programming language which includes denotation functions,
algebraic functions and evaluation functions for semiknown values.
Try to use semiknown values with the following examples in FunLog++.

Remark: Semiknown values could be implemented in almost any programming language. Especially object-oriented languages with polymorphic function identification are suitable (e.g. like JAVA). The reason to use FunLog++ was that this language allows to define prefix-, infix-, and postfix-operators. Thus algebraic operations could be extended to Semiknown Values comfortably.
 

Concept of semiknown values

concept - examples - basic functions - derived functions SE1 - SE2 - SE3 - SE4 - SE5 - SE6 - SE7 - SE8 - SE9 - SE10

Semiknown values of any domain are represented as
- possibilistic data   ( given as subsets )               or
- probabilistic data  ( given as stochastic variables )

As the concept of probability density function in stochastic variables is valid for discrete as well as for continuous variables we write p(x) for both kinds of data. The distribution function F_d(x₀) for discrete semiknown values is defined as:

On this Website there are examples of semiknown values which can be analyzed. In case of expressions with semiknown values the FunLog++ interpreter can be used to really calculate the expressions value.

Please try the following denotation functions with Semiknown Values:
- Interval Arithmetics
- Set Calculus
- Interval and Set Calculus
- Contextfree Grammers
- Probabilities
- Normal Distribution

Please try the algebraic functions with Semiknown Values:
- Bayes Theorem (qualitative symptoms)
- Bayes Theorem (quantitative symptoms)
- Bayes Theorem (intervals of symptoms)
- Statistical Testing (t-Test)

Basic denotation functions with FunLog++ Semiknown Values are the following denotation operators, which are described by the following specification:
  A -- B  ::=  ... A < X, X < B, X.
  < A     ::=  ... X < A, X.
  > A     ::=  ... A < X, X.
  A \/ B  ::=  ... ( X = A ; X = B ), X.

  A @ B   ::=  ... ( B = prob_density(X,A) ), X.

  moments(A,B,C,D)  ::=
               ( A = [Mean,Variance,Skewness,Kurtosis],
                 B = samplesize(X),
                 C = population(X),
                 D = interaction(X),
                 Z = prob_density(X,Y) ), X.

Basic algebraic functions are used to unify, order, summarize, or relate two FunLog++ Semiknown Values. Therefore we use the elementary operators =/2 (unification), </2 (ordring), -/2 (subtraction), and //2 (division) as well as the transcendental functions exp/1 and ln/1. These are given in a pseudo formal function definition:
  A = B   ::=  ... A is unifyable with B.
  A < B   ::=  ... A is less than B.
  A - B   ::=  ... A minus B.
  A / B   ::=  ... A divided by B.
  exp(A)  ::=  ... exponential of A.
  ln(A)   ::=  ... logarithm of A.

Basic evaluation functions are used to exploit Semiknown Values. The evaluation function is given in words as:
  prob_density(A,B) ::=
    ... calculates the probability density
    of any semiknown value A at position B.

Derived denotation functions are used to generate a FunLog++ Semiknown Value. For denotations we use the elementary denotation operators --/2, </1, >/1, \//2, @/2, and moments/4. Furthermore there are some derived denotation functions which are completely defined in terms of the elementary functions just to enhance programming with Semiknown Values. These are:
  <= A               ::=  <A \/ A.
  >= A               ::=  A \/ >A.
  A =-- B            ::=  A \/ A--B.
  A --= B            ::=  A--B \/ B.
  A =-= B            ::=  A \/ A--B \/ B.
  moments([M,V,S,K]) ::=  moments([M,V,S,K],_,_,_).
  moments([M,V,S])   ::=  moments([M,V,S,3]).
  moments([M,V])     ::=  moments([M,V,0,3]).
  moments            ::=  moments([0,1,0,3]).

  likelihood_ratio(A,B,C)  ::=
    prob_density(A,C) / prob_density(B,C).