New Integral Approach to the Specification of STPU-Solutions

Abstract

This paper is aimed at proposing some new formal system of a fuzzy logic – suitable for representation the “before” relation between temporal intervals. This system and an idea of the integral-based approach to the representation of the Allen’s relations between temporal intervals is later used for a specification of a class of solutions of the so-called Simple Temporal Problem under Uncertainty and it extends the classical considerations of R. Dechter and L. Khatib in this area.

1 Introduction

In [2] R. Dechter introduced the so-called Simple Temporal Problem as a restriction of the framework of Temporal Constraint Satisfaction Problems, tractable in polynomial time. In order to address the lack of expressiveness in standard STPs, Khatib in [10] proposed some extended version of STP – the so-called Simple Temporal Problem with Preferences (STPP). The lack of flexibility in execution of standard STPs was a motivation factor to introduce the so-called Simple Temporal Problem under Uncertainty (STPU) in [14]. In order to capture both the possible situations of acting with preferences and under uncertainty, the Simple Temporal Problem with Preferences under Uncertainty (STPPU) was described in [13]. Due to – [2] – The Simple Temporal Problems(STPs) is a kind of such a Constraints Satisfaction Problem, where a constraint between time-points $X_{i}$ and $X_{j}$ is represented in the constraint graph as an edge $X_{i}\rightarrow X_{j}$, labeled by a single interval $[a_{ij}, b_{ij}]$ that represents the constraint $a_{ij} \le X_{j} - X_{i} \le b_{ij}$. Solving an STP means finding an assignment of values to variables such that all temporal constraints are satisfied. Due to [14] – The Simple Temporal Problem under Uncertainty extends STP by distinguishing contingent events, whose occurrence is controlled by exogenous factors often referred to as “Nature”.

Independently of this research path, H-J. Ohlbach proposed in [11] a new integral-based approach to the fuzzy representation of the well-known Allen relations between temporal intervals¹– initially introduced by J. Allen in [1]. This paper analysis combines both research paths. In fact, we intend to propose a new-integral-based fuzzy logic system – capable of expressing the chosen relation “before” in terms of Ohlbach’s integrals – in this paper. The chosen “before” relation was chosen as some operationally “nice” and paradigmatic example among all Allen’s relations, which can be modeled in a similar way. This system is conceived as some extension of the Fuzzy Integral Logic of Pavelka-Hajek from [4] – developed in [8] and –for Allen’s relations in [9]. This manouvre is dictated by the second main paper purpose: to demonstrate how the integral-based approach to the modelling of Allen’s relations allows us to differentiate a potential class of STPU-solutions. Although the specification of a class of the STPU-solutions was made by means of some analytic tools, the introduced formalism supports this analysis, it constitutes its foundation and ensures – thanks the completeness theorem – a coherence between a description of STPU-problems in terms of the proposed formalism and the proposed semantics. An algebraic approach to some unique temporal problems such as scheduling with defects was proposed in [3].

1.1 Paper’s Motivation and Formulation of an Initial Problem

The main motivation factor of the current analysis is a lack of an approach to the STPU-solving – capable of elucidating of an “evolution” of solutions. In particular, there is no integral-based approach – in spite of integral-based representation of Allen relations between intervals. In addition, it seems that a theoretic, meta-logical establishing of the STPU² has not been discussed yet in a specialist literature. Some possibilities of modelling of preferences in fuzzy temporal contexts were, somehow, demonstrated by authors of this paper in [5–7], but without the explicit referring to STP and its extensions. From the more practical point of view this paper analysis are motivated by the following example of the STPU:

Example: Consider a satellite which performs a task to observe a volcano Etna in some time-interval [0; 80]. The cloudiness can take place in time interval j(x)= [20; 50], but it comes out gradually in this time-interval. When to begin the observation task (beginning from the initial time-point) in order to maximize a chance for finishing the satellite observation in a given time-interval [0; 80]?

We associate this main problem to the following (sub)problems supporting its solution in terms of the features of “before”-relation.

Problem 1: Does the Allen relation “before” take a one or many values in the integral-based depiction? If many, show which values from [0,1]-interval can be taken by this relation in their integral-based depiction for linear functions.

Fig. 1.

STPU for observation task of the satellite

Problem 2: If the “before”-relation can be evaluated by values from [0,1], decide for which real parameters $C>0$ this relation takes values no smaller than 0,7?

2 Terminological Background

The proper analysis will be prefaced by introducing a terminological background regarding concepts of the fuzzy intervals, operations on them and the Ohlbach’s representation of Allen’s interval relations.

Definition 1 (Fuzzy Interval). Assume that $f:\mathcal {R}\mapsto [0.1]$ is a total integrable function (not necessary continuous). Than the fuzzy interval $i_{f}$ (corresponding to a function f) is defined as follows: $i_{f} =\{(x, y),\subseteq \mathcal {R}\times [0.1]\vert y\le f(x)\}$.

A fuzzy set (in a comparison with a crisp one) is illustrated on the picture (Fig. 2):

Fig. 2.

A crisp and a fuzzy interval

Operations of an intersection and a union of two fuzzy intervals are defined with a use of the appropriate t-norms. Classically: $(i\cap j)(x) =^{def} min\{i(x), j(x)\}$ and $(i\cup j)(x) =^{def} max\{i(x), j(x)\}$.

Some Basic Transformations on Fuzzy Sets. We can associate some additional transformation with fuzzy intervals – presented in details in [11, 12]. We restrict their list to the following, especially useful:

$$\begin{aligned}&identity(i) = ^{def}\quad i,\\&integrate^{+}(x) = ^{def} \displaystyle \int _{\lnot \infty }^{x}i(y)dy/\vert i\vert ,\\&integrate^{-}(x) = ^{def} \displaystyle \int _{x}^{+\infty }i(y)dy/\vert i\vert ,\\&cut_{x_{1}, x_{2}}(x) = 0,\text { if }x<x_{1}\text { or }x_{2}\le x; i(x) -\text { otherwise}. \end{aligned}$$

1. Before. In order to define this relation let us assume that some point-interval relation:’p before j’ is given and let us denote it by B(j). In order to extend B(j) to the interval-interval relation (for j and some interval i), we should average this point-interval before-relation over the interval i. Since fuzzy intervals form subsets of $R^{2}$, all these points satisfying this new relation before(i, j) are given by the appropriate integral, namely: $\displaystyle \int i(x)B(j) dx/\vert i\vert$. ($\vert i\vert$ normalizes this integral to be smaller than 1.)

Infinite Intervals: This general methods should be somehow modified w.r.t the situation when either i or j or both intervals are infinite. If i is $[a, \infty )$-type, than nothing can be after i, thus before(i, j) must yield 0. For a contrast, if j is $(-\infty , a]$-type, than nothing can be before j, what leads to the same value 0.

It remains the case, when i is $(-\infty , a]$-type, but j is finite or of $[a, \infty )$-type. In this case we should find some alternative, because $\displaystyle \int i(x)B(j) dx$ will be infinite. Therefore we take an intersection $i\cap _{min} j$ instead of the whole infinite i. Since j is not of a $(-\infty , a]$-type, the intersection $i\cap _{min} j$ must be finite and the before(i, j) is given by:

$$\begin{aligned} before(i, j) =^{def} \displaystyle \int (i(x)\cap _{min}B(j))dx/\vert i(x)\cap _{min} j(x)\vert . \end{aligned}$$

In results, for some point-interval relation B(j) the new interval-interval relation before(i, j) should be represented as below:

$$before(i, j)= {\left\{ \begin{array}{ll} 0 &{} if \, i=\emptyset \, or \, i = [a, \infty ) \, or \, j=\emptyset \\ 1 &{} if \, \, i=(\infty , a] \, and \, i\cap j= \emptyset \\ \displaystyle \int i(x)\cap _{min} B(j)/\vert i(x)\cap _{min} j(x)\vert &{} if \, i = (\infty , a]-type\\ \displaystyle \int i(x) B(j)/\vert i(x)\vert &{} otherwise\\ \end{array}\right. }$$

In order to solve this problem we will consider two fuzzy intervals i(x) and j(x). For simplicity (but without losing of generality) we can take into account a single Allen relation before(i, j)(x) between them localized w.r.t the y-axis as given on the picture (Fig. 3):

Fig. 3.

Fuzzy intervals i(x) and j(x)

3 Some Extension of the Fuzzy Integral Logic of Hajek for the Fuzzy Allen Relation “before”

3.1 Requirements of the Construction

We will extend the Fuzzy Integral Logic of Hajek from [4] in order to express the interval-interval relation “before”. In order to render it in a language of our system we need introduce a new relation symbol, say B(i, j) for atomic terms i, j (denoted by fuzzy intervals). In accordance with the Ohlbach’s definition of this relation, one also need introduce the following: a) a symbol, say B(i)(x) to represent the atomic interval-point relation before(i, x) (an interval i is’before’ a point x) and b) a constant for normalization factor N. The point-interval relation B(i)(x) etc. will be denoted by a symbol: $\hat{B}^{i}_{x}$. Because of the need of a clear distinction between the FLI -syntax and its semantics with Allen’s relations – the fuzzy intervals i(x), j(x) will be represented in the FLI -syntax by formulas $\phi ^{i}_{x}$ and $\phi ^{j}_{x}$ (resp.). In results, we will write: $\displaystyle \int \psi ^{i}_{t}\hat{B}^{j}_{t}dt$ instead of the Ohlbach’s formula: $\displaystyle \int i(x) B(j)(x) dx$ etc.

3.2 Syntax and Semantics

Language. For these purposes we introduce our FLI in an appropriate language L of Lukasiewicz Propositional Logic (LukPL) with the following connectives and constants: $\rightarrow , \lnot , \iff , \wedge$ (weak conjunction), $\otimes$ (strong conjunction), $\vee$ (weak disjunction), $\oplus$ (strong disjunction) and propositional constants 0 and 1. We extend by new constants: $\hat{r}_{1}, \hat{r}_{2}, \hat{r}_{3}\ldots$, representing in the language $\mathcal {L}(FLI)$ the rational numbers: $\hat{r}_{1}, \hat{r}_{2}, \ldots , s_{1}, s_{2}\ldots$ etc. We enrich this language by $\exists$– and $\forall$- quantifiers to the full language of Rational Pavelka Predicate Logic RPL$\forall$.

The alphabet of $\mathcal {L}$(FLI) consists of³:

• propositional variables: $\phi , \chi , \psi ,\ldots , a_{i}, b_{i},\ldots x, y, t\ldots$

• functional symbols: $\phi _{t}, \phi _{x-t},\chi _{t}, \chi _{x-t},\ldots$

• predicates (of point-interval relations): $\hat{B}^{i}_{t}$, $\hat{D}^{i}_{t}$, $\hat{M}^{i}_{t}$, $\hat{S}^{i}_{t}$, $\hat{F}^{i}_{t}\ldots$.

• rational constant names: $\widehat{r_{1}},\widehat{r_{2}},\ldots \widehat{0},\widehat{1}$, scalar constants: $\widehat{N}, \widehat{M}\ldots$ etc.

• quantifiers: $\forall , \exists \arrowvert \displaystyle \int ( )dx, \displaystyle \int \displaystyle \int ( )dx dy, \displaystyle \int _{0}^{\infty }( )dt, \displaystyle \int _{t_{0}}^{t_{1}}( )dt\ldots$

• operations: $\rightarrow , \lnot , \vee , \wedge ,\bullet , \oplus ,\otimes , =.$

Set of Formulas FOR: The class of well-formed formulas FOR of $\mathcal {L}$(FLI) form propositional variables and rational constants as atomic formulas. The next - formulas obtained from given $\phi ,\chi \in FOR$ by operations $\lnot , \vee ,\wedge ,\rightarrow ~,\oplus ,\otimes ,\forall ,\exists ,\displaystyle \int ( )dx$ and the formulas obtained from $\phi _{i},\chi _{i}\in FOR$ by operations $\lnot ,\vee ,\wedge ,\rightarrow ,\oplus ,\otimes ,\bullet ,\forall ,\exists ,\displaystyle \int _{0}^{\infty }( )dt, \displaystyle \int _{0}^{t_{0}}( )dt,\displaystyle \int _{t_{0}}^{t_{1}}( )dt$. Finally, formulas obtained from $\phi _{i}\in$ FOR and rational numbers by operations $\lnot ,\vee ,\wedge ,\rightarrow ,\oplus ,\otimes ,\bullet$ belong to FOR as well. These classes of formulas exhaust the list of FOR of $\mathcal {L}$(FLI).

Example: $\displaystyle \int _{0}^{\infty }\phi _{t}\bullet \chi _{x-t} dt\rightarrow \widehat{r}\in FOR$, but $\displaystyle \int \phi dx\rightarrow \displaystyle \int _{0}^{\infty }\chi _{t} dt$ does not.

The mentioned system FLI arises in $\mathcal {L}$(FLI) by assuming the following

Axioms: – partially considered by Hajek in [4]:

$\displaystyle \int (\lnot \phi ) dx = \lnot \displaystyle \int \phi dx$, $\displaystyle \int (\phi \rightarrow \chi ) dx\rightarrow (\displaystyle \int \phi dx\rightarrow \displaystyle \int \chi dx)$

$\displaystyle \int (\phi \otimes \chi ) dx = ((\displaystyle \int \phi dx\rightarrow \displaystyle \int (\phi \wedge \chi ) dx)\rightarrow \displaystyle \int \chi dx))$

$\displaystyle \int \displaystyle \int \phi dx dy=\displaystyle \int \displaystyle \int \phi dy dx$ ⁴ (Fubini theorem):

and new axioms defining the algebraic properties of convolutions⁵:

$\displaystyle \int _{0}^{\infty }\phi _{t}\bullet \chi _{x-t} dt=\displaystyle \int _{0}^{\infty }\phi _{x-t}\bullet \chi _{t} dt$

$\widehat{r}\displaystyle \int _{0}^{\infty }\phi _t\bullet \chi _{x-t} dt = \displaystyle \int _{0}^{\infty }(\widehat{r}\phi _t)\bullet \chi _{x-t} dt, ($r$-constant)$ (associativity)⁶

$\displaystyle \int _{0}^{\infty }\phi _{t}\bullet (\chi _{x-t}\oplus \psi _{t}) dt = \displaystyle \int _{0}^{\infty }\phi _{t}\bullet \chi _{x-t} dt \oplus \displaystyle \int _{0}^{\infty }\phi _{t}\bullet \psi _{x-t} dt$ (distributivity)

As inference rules we assume Modus Ponens, generalization rule for $\displaystyle \int -symbol$ and two new specific rules: $\frac{\phi }{\displaystyle \int \phi dx}, \frac{\phi \rightarrow \chi }{\displaystyle \int \phi dx\rightarrow \displaystyle \int \chi dx}$

and the same rules for indexed formulas and convolution integrals.

Semantics. Our intention is to semantically represent $\displaystyle \int$-formulas of $\mathcal {L}$(FLI) by ‘semantic’ integrals. Because all of the considered point-interval relations D(p, j), M(p, j) etc. are functions for the fixed j, than such “semantic” integrals can be defined on a class of the appropriate functions.

More precisely, we will understand such integrals I as a mapping $I: f\in Alg\mapsto If(x)\in [0,1]$ (where Alg is an algebra of functions from $M\ne \emptyset$ to [0.1] containing each rational function $r\in [0.1]$ and closed on $\Rightarrow$ (see: [4], p. 240)) satisfying the conditions corresponding to the presented axioms of $\mathcal {L}$(FLI). For example, it holds:

$$\begin{aligned} I(1 - f) dx = 1 - I f dx, I(f\Rightarrow g) dx\le (If dx\Rightarrow Ig dx) \end{aligned}$$

(1)

$$\begin{aligned} I(If dx) dy = I(If dy) dx \end{aligned}$$

(2)

$$\begin{aligned} I(f(t)g(x - t)dt = Ig(t)f(x - t)dt \end{aligned}$$

(3)

$$\begin{aligned} rIf(t)g(x - t)dt = I(rf(t))g(x - t)dt \end{aligned}$$

(4)

We omit the whole presentation of these corresponding conditions. They can be found in [8] and partially in [4].

Interpretation. Let assume that $Int = (\bigtriangleup , \Vert \phi \Vert )$ with $\bigtriangleup \ne \emptyset$ and a (classical fuzzy) truth-value interpretation–function: $\Vert \Vert$ of formulas of $\mathcal {L}$(FLI). The propositions of Łukasiewicz logic are interpreted in the sense of $\Vert \Vert$ as follows: $\Vert \lnot (\psi )\Vert = 1 -x$, $\Vert \rightarrow (\phi , \psi )\Vert = min\{1, 1-x+y\}$, $\Vert \wedge (\phi , \psi )\Vert = min\{x, y\}$, $\Vert \wedge (\phi , \psi )\Vert = max\{x, y\}$, $\Vert \otimes (\phi , \psi )\Vert = max\{0, x+y-1\}$, and $\Vert \oplus (\phi , \psi )\Vert = min\{1, x+y\}$ for any $x, y\in$ MV-algebra A⁷.

We inductively expand now this interpretation for new elements of the grammar $\mathcal {L}$(FLI) as below.

syntax ($\phi \in \mathcal {L}(\textit{FLI})$)	fuzzy semantics $(\Vert \phi \Vert _{FLI})$
$a_{i}, b_{i}$	objects $A_{i}, B_{i}$ for $i\in \{1,\ldots , k\}$
$\phi _{i}$	functions $f(i) \mathrm {for} i\in \{x, t, x-t, t-x\}$
$\displaystyle \int \phi dx,(\displaystyle \int _{0}^{\infty }\phi dx$)	$If dx, (I_{0}^{\infty }f dx)$
$\phi _{i}\bullet \chi _{i}$	$\Vert \phi _{i}\Vert \star \Vert \chi _{i}\Vert$ (i like above)
$\phi \otimes \chi$	$min\{1, \Vert \phi \Vert + \Vert \chi \Vert \}$
$(\Vert \phi _{i}\otimes \chi _{i}\Vert )$	$min\{1, \Vert \phi _{i}\Vert +\Vert \chi _{i}\Vert \}$
$\displaystyle \int _{0}^{\infty }\phi _{t}\bullet \chi _{x-t} dt$	$I_{0}^{\infty }g(t)\star f(x- t)) dt$
$\widehat{r}\displaystyle \int _{0}^{\infty } dt$	$\Vert \widehat{r}\Vert \star \Vert I_{0}^{\infty }f\Vert dt = rI_{0}^{\infty }f dt$
$\frac{\displaystyle \int \phi _{t}^{i}\bullet \widehat{B}^{j}_{x-t} dt}{\widehat{N}}$	$I \frac{i(t)B(j)(x-t)\star f(x- t)) dt}{N}$

Definition of the Model: We define a model M as a n-tuple of the form: $M=\langle \vert M\vert , \{r_{0}, r_{1},..\}, f_{i}, If_{i} dx, I_{0}^{\infty }f_{i}g_{j} dt \rangle$ where $\vert M\vert$ is a countable (or finite) set $\{r_{0}, r_{1},\ldots \}, f_{i}$ are respectively: a set of rational numbers belonging to $\vert M \vert$, and atomic integrable functions. $If_{i}$ are integrals on the algebra Alg of subsets of $\vert M\vert$ and $I_{0}^{\infty }f_{i}g_{j} dt$ are convolutions of $f_{i}$ and $g_{j}$.

FLI turns out to be complete w.r.t such a model and undecidable. If a model M is given, we write $\Vert \phi \Vert _{M, v}$ for a denotation of the truth value under evaluation v for each formula of $\mathcal {L}(FLI)$ as a function: $\mathcal {L}(FLI)\rightarrow [0, 1]$. If M is a model and v is a valuation, than: $\Vert \widehat{r}\Vert _{M, v} = r, \Vert x\Vert _{M, v}= a\in [0,1]$ for a variable x and for a predicate $Pred(t_{1}, \ldots , t_{k})$ it holds: $\Vert Pred(t_{1},\ldots t_{k})\Vert _{M, v} = Rel(\Vert t_{1}\Vert _{M, v}, \ldots \Vert t_{k}\Vert _{M, v})$ for Rel interpreting Pred in a model M.

Example 1: Consider two intervals i(x) and j(x) such that $i, j\subseteq [a, b]$ and two actions A and B associated with i(x) and j(x) (resp.) Let denote this by $i(x)^{A}$ and $j(x)^{B}$ The fact that action A is parallel to B can be represented by an integral-based $during(i(x)^{A}, j(x)^{B})$-relation and interpreted by a model $\mathcal {M} = \langle [a, b], i^{A}, j^{B}, \displaystyle \int _{a}^{b} i^{A}D(i(x)^{A}, j(x)^{B})dx/\vert i^{A}\vert \rangle$.

Completeness Theorem for FLI: For each theory T over predicate $\mathcal {L}$ (FLI) and for each formula $\phi \in \mathcal {L}(FLI)$ it holds: $\vert \phi \vert _{FLI} = \Vert \phi \Vert _{FLI}.$

Proof is very similar to the completeness proof for the Hajek’s integral logic from [4], so it will be omitted here.

4 Solving of the Problem

In this section we intend to solve the main problem – defined and depicted on a picture in the introductory section with two problems associated to it. Anyhow, we preface this solution by considerations focused on analytic features of before(i, j)-relation in terms of integrals. In particular, we present a graph of the function representing this relation provided that the atomic point-interval relation is linear. We decide on this linearity assumption because of a simplicity of the further analysis.

4.1 A Formal Depiction of the Problem and Some Introductory Assumptions

We begin with the formal depiction of the presented problems. For that reason, let us note that the interval-interval definition of Allen’s relation before(i, j)(x) is of the type:

$$\begin{aligned} before(i, j)(x)= \frac{\displaystyle \int i(x)j(x)dx}{max_{a}\displaystyle \int i(x-a)j(x) dx} \end{aligned}$$

(5)

According to the above requirement let us consider their unique form for i(x) and j(x) given by linear functions, i.e. ${\left\{ \begin{array}{ll} i(x) = &{} Ax, A>0, B<0,\\ j(x) = &{} Bx, A<0, B>0\\ \end{array}\right. }$ (see: Fig. 1) Than:

$$\begin{aligned} (1) = \frac{AB\displaystyle \int x^{2} dx}{AB\displaystyle \int (x-a) x dx} = \frac{\displaystyle \int x^{2} dx}{\displaystyle \int (x^{2} - ax) dx} = \frac{x^{3}}{3[\frac{x^{3}}{3} - \frac{ax^{2}}{2}]}= \frac{x^{3}}{3[\frac{2x^{3} -3ax^{2}}{6}]} = \frac{2x^{3}}{2x^{3}-3ax^{2}} \end{aligned}$$

(6)

for some done $a\in R$.

It remains now to investigate the function $f(x) = \frac{2x^{3}}{2x^{3}-3ax^{2}}$ in order to find its values in the fuzzy interval [0,1].

4.2 Investigation of Properties of the Considered Allen’s Before-Relation in the Integrals-Based Representation

In this subsection we check the analytic properties of the function $\frac{2x^{3}}{2x^{3}-3ax^{2}}$ representing the considered Allen’s relation before(i, j)(x) for fuzzy intervals i(x) and j(x) given by linear functions.

(a) Domain of f(x).

$2x^{3} - 3ax^{2} \ne 0\iff x\ne 0$ or $x \ne 3/2a$, so $x\in R/\{0, 3/2a\}$.

(b) Limits:

$\lim _{x\rightarrow -\infty }\frac{3x^{3}}{2x^{3} -3ax^{2}} = [\frac{-\infty }{-\infty }] = 1$, $\lim _{x\rightarrow \infty }\frac{3x^{3}}{2x^{3} -3ax^{2}} = [\frac{\infty }{\infty }] = 1$,

$\lim _{x\rightarrow (\frac{3}{2}a)^{-}}\frac{2x^{3}}{2x^{3} -3ax^{2}} = [\frac{2(\frac{3a}{2})^{3}}{0^{-}}] = -\infty$, $\lim _{x\rightarrow (\frac{3}{2}a)^{+}}\frac{2x^{3}}{2x^{3} -3ax^{2}} = [\frac{2(\frac{3a}{2})^{3}}{0^{+}}] = \infty$.

$\lim _{x\rightarrow (\frac{0}{0})^{-}}\frac{2x^{3}}{2x^{3} -3ax^{2}} = [\frac{0}{0^{-}}] = 0$, $\lim _{x\rightarrow (\frac{0}{0})^{+}}\frac{2x^{3}}{2x^{3} -3ax^{2}} = [\frac{0}{0^{+}}] = 0$.

It allows us to visualize the graph of the function as follows: Therefore, $0 \le \frac{2x^{3}}{2x^{3} -3ax^{2}} \le 1$ for $x\in (-\infty , 0)$ (Fig. 4).

Fig. 4.

An outline of the function $\frac{2x^{3}}{2x^{3} -3ax^{2}}$

4.3 Some Modification of the Initial Assumptions

Let us note that the above solution holds by assumption that intervals i(x) and j(x) meets in a point $x = 0$ as on the picture. One needs therefore a function $F(x-B) = \frac{2(x-B)^{3}}{2(x-B)^{3} - 3a(x-B)^{2}}$. Its graph stems from the earlier graph of $\frac{2x^{3}}{2x^{3} -3ax^{2}}$ via translation by a vector (0, B). It looks like this: We are now interested in the part of this graph between $x_{0} = 0$ and $x_{1}$ = B. Immediately from the graph one can see that for $x = B$ the function $F(x -B)$ is not defined, but $\lim _{B\rightarrow 0} F(x-B) = 0$. On can easily compute that $F(0-B) = \frac{-2B^{3}}{-2B^{3}-3B^{2}a} = \frac{2B^{3}}{-B^{3} + 3B^{2}a} < 1$. It can be visualized as follows: Therefore, the investigated function takes the values from the interval $I = (0; \frac{2B^{3}}{2B^{3}+ 3B^{2}a})$ (Figs. 5 and 6).

Fig. 5.

A diagram of a function f(x) in the required vector translation.

Fig. 6.

The fragment of a graph of a function $F(x-B)$, which we are interested in – as a visual representation of our problem solution.

Example: For B = 1 we obtain an interval $I_{1} = (0, \frac{2}{2+3a})$ for done $a\in R$.

4.4 Further Properties of This Integral-Based Representation of Allen’s Before-Relation

At the end we intend to show that our function $\frac{2x^{3}}{2x^{3} -3ax^{2}}$ is uniformly continuous. It means that the change the fuzzy values (one for another) is “lazy” and non-radical in the whole interval (0, B) (for arbitrary pairs of $x_{1}$ and $x_{2}$ from this interval if only $\vert x_{1} -x_{2}\vert <\rho$ for some arbitrary $\rho >0$.)

For this purpose let us consider its module of continuity:

$$\begin{aligned} \vert \frac{2x_{1}^{3}}{2x_{1}^{3}-3ax_{1}^{2}} - \frac{2x_{2}^{3}}{2x_{2}^{3}-3ax_{2}^{2}}\vert = \vert \frac{2x_{1}^{3}(2x_{2}^{3} -3ax_{2}^{2})-2x_{2}^{3}(2x_{1}^{3}-3ax_{1}^{2})}{(2x_{1}^{3}-3ax_{1}^{2})(2x_{2}^{3}-3ax_{2}^{2})}\vert = \end{aligned}$$

(7)

$$\begin{aligned} \vert \frac{-6ax_{1}^{3}x_{2}^{2} + 6ax_{2}^{3}x_{1}^{3}}{(2x_{1}^{3}-3ax_{1}^{2})(2x_{2}^{3}-3ax_{2}^{2})}\vert = \vert \frac{6ax_{1}^{2}x_{2}^{2}(x_{2} - x_{1})}{(2x_{1}^{3}-3ax_{1}^{2})(2x_{2}^{3}-3ax_{2}^{2})}\vert \le \frac{6a\vert x_{1} -x_{2}\vert }{M} = \frac{6a\rho }{M}, \end{aligned}$$

(8)

where M is the appropriate lower bound of the last denominator. Hence $\forall \epsilon > 0: \vert \frac{2x_{1}^{3}}{2x_{1}^{3}-3ax_{1}^{2}} - \frac{2x_{2}^{3}}{2x_{2}^{3}-3ax_{2}^{2}}\vert < \epsilon$ if only assume $\rho \le \frac{M\epsilon }{6a}$, what justifies a desired uniform continuity of our function.

4.5 Solving of the Main Problem

The arrangements, presented above, allow us to solve the main problem with the observation task of a satellite and the problems associated to it in the introductory part. In order to make it let us recall that:

$$\begin{aligned} before(i, j) = \frac{\displaystyle \int i(x)Bef(j)(x) dx}{max_{a}\displaystyle \int i(x-a)Bef(j)(x)dx}, \end{aligned}$$

(9)

for some point-interval relation Bef(j)(x). Meanwhile, we have just shown that for linear functions defining the fuzzy intervals this general definition can be given by:

$$\begin{aligned} before(i, j) = \frac{2x^{3}}{2x^{3}-3ax^{2}} \end{aligned}$$

(10)

and $0\le before(i, j)(x) \le 1$ holds for $x\in (0; \frac{2C^{3}}{2C^{3}+3C^{2}a})$. Nevertheless, by our assumption $C= 20$ (min) we obtain that: $x\in (0; \frac{2\bullet 20^{3}}{2\bullet 20^{3}+3\bullet 20^{2}a})=(0; \frac{2\bullet 8000}{2\bullet 8000+1200a})$. Assuming for simplicity $a = 1$ we can get $x\in (0; \frac{16000}{17200})= (0; 0,9302)$.

Therefore, our function takes values from (0; 0, 9302).

Problem 2: For which parameters $C>0$ the before(i, j)(x) -relation takes values no smaller than 0,7?

Solution: $0,7\le \frac{2C^{3}}{2C^{3} + 3C^{2}}$. It is equivalent to $0\le \frac{2C^{3}}{2C^{3} + 3C^{2}} - \frac{0,7\bullet (2C^{3}+ 2C^{2})}{2C^{3} + 3C^{2}} = \frac{1,3(C^{3} -2,1C^{2})}{2C^{3}+3C^{2}}$. Let’s consider the equation $\frac{1,3(C^{3} -2,1C^{2})}{2C^{3}+3C^{2}} = 0$ (for $C\ne 0$). It leads to the equation $1,3(C^{3} -2,1C^{2}= 0\iff C^{2}(1,3C-2,1) = 0\iff 1,3C=2,1\iff C =\frac{2,1}{1.3}$. Therefore, our unequality holds for $C\in (-\infty ; 0)\bigcup (\frac{21}{13}; \infty )$. Because we are only interested in $C>0$, so the only solution is given by an interval $(\frac{21}{13}; \infty )$.

5 Concluding Remarks

It has emerged that the integral-based approach to the Allen temporal relations allows us to specify the class of STPU-solutions. It also appears that the intuitively graspable point-solutions are preserved as the appropriate ones – as a board case solution in considered situations. Finally, the construction of a fuzzy logic system and its completeness ensures that models of STPU (in terms of before-relation) really refer to the formal descriptions of STPU in the appropriate languages. It seems that the similar procedures can be repeated for other Allen relations in the integral-based Ohlbach’s depiction.

Anyhow, the considered STPU-problem belongs to the class of relatively elementary problems. It seems that many similar problems, with a higher complication degree – such as STPPU-problems – could be investigated in the similar way. In this perspective, the analysis of the current paper seems to be open.