Possibilistic time

Possibility theory can be used to model and manage temporal knowledge that involves imprecisely known information. This approach uses points as the temporal primitive. Imprecisely known dates are modeled by a fuzzy set with a unimodal possibility distribution over the temporal axis. Fuzzy temporal intervals are derived from fuzzy dates that limit the time span during which an event occurs. Fuzzy durations are treated in a similar way. A typical possibilistic temporal interval representing a fuzzy duration may be approximated by a trapezoidal shape. A temporal interval , which is represented as {a'(0), a(1), b(1), b'(0)}, has a duration certainly between a and b and most plausibly between a' and b'. The approach originally proposed by Dubois and Prade allows for

• Representation of uncertain precedence relations (such as "much before" or "closely after") between events,
• Comparison, ranking and ordering of fuzzy dates and durations, and
• Propagation of qualitative fuzzy temporal constraints.

The approach proposed by Dubois and Prade establishes a very solid foundation for applying possibility theory for temporal reasoning by providing a possibilistic representation of temporal primitives.

Trapezoidal approximation of possibilistic distributions does not provide enough flexibility for modeling of mathematical functions. They are overly restrictive to the resulting function and are hardly useful for piece-wise approximation. For the purposes of modeling functions by approximation and to minimize the number of elementary intervals participating in the approximation, we propose to represent possibilistic distributions using alternative trapezoidal shapes. It is different from the trapezoids presented earlier - alternative trapezoids have their vertical edges parallel to each other, lower edge coincides with the horizontal axis, and the last edge (slope) is arbitrary. Such an alternative trapezoidal shape T is determined by four parameters: T = {a, b, h₁, h₂}.
Depending on the inclination of the slope (i.e. the sign of (h₁ - h₂)) we will distinguish between L-trapezoidal (h₁ < h₂) and R-trapezoidal (h₁ > h₂) shapes. Instances of these trapezoidal shapes include rectangular shape (h₁ = h₂), L-triangular (h₁ = 0) and R-triangular (h₂ = 0) shapes.

It is possible to approximate any function using only rectangular shapes, but a better approximation will require a very fine quantification and therefore will result in many elementary rectangles used to approximate the function. Approximations using several trapezoidal shapes are closer to the real functions and require less elementary shapes participating in such approximations.

An arbitrary function f(x) representing a possibilistic distribution may be modeled as a disjunction of k elementary distributions m_i(x).

In general, a possibilistic distribution can be represented as a sequence of n+1 adjacent points and their corresponding possibility measures. It takes the form of

Suppose that possibilistic distributions of the temporal distance from t₁ to t₂, and the temporal distance from t₂ and t₃ are respectively f(x) and g(y). Possibilistic distribution of temporal distance from t₁ to t₃, say F(z), can be computed from f(x) and g(y). Let z₁ be x₁ + y₁. A combination of possibilistic values f(x₁) and g(y₁) is min(f(x₁), g(y₁)) for z₁ = x₁ + y₁. Taking all the possible combinations for z₁ as a summation of x and y, we get F(z₁) as
max(min(f(x_k), g(y_k)| z₁ = x_k + y_k). This can be generalized as

Consider two functions, modeled by their respective possibilistic distributions

and

According to the results detailed in my dissertation, functions f(x) and g(y) can be combined together to form combination f(x)⊕g(y). Such a combination is a double union of all possible pairs of each elementary distribution from f(x) with each elementary distribution from g(y):