*4.2. Monotonicity*

We formulate the definition of monotonicity in terms of intervals and integrals. The property that a function is monotone over an interval can be expressed in DC4F by the requirement that for two neighboring subintervals of the same length, the integral over the left one has smaller or equal value to the integral over the right one. It can be expressed by formula *M* of the form ∫ *f* = *x* ∧ - = *y*⌢ ∫ *f* = *z* ∧ - = *w* ∧ *y* = *w* ⇒ *x* ⩽ *z*. This property must hold for all subintervals of this kind; thus, we have the formula: ◻ *M*. Figure 2 shows a monotonically increasing function *f* .

**Figure 2.** Monotone function *f* (*t*).

Alternatively, we can define the monotonicity property in the following way:

<sup>∀</sup>*<sup>x</sup>*,*<sup>y</sup>* ◻ {(- > 0 ∧ ∫ *f* - = *x*)⌢ (- > 0 ∧ ∫ *f* - = *y*) ⇒ *x* ⩽ *y*} (3)

This formula, though it may seem complicated, is rather simple. It states that for a given interval and for all its adjacent subintervals, the normalized value of the integral for the left subinterval is lower than or equal to the normalized value of the integral for the right subinterval. It is illustrated in Figure 2. Intervals [*b*, *c*] and [*c*, *d*] are adjacent. The values of the integrals for these intervals have to be normalized by their length, as their lengths are different. We will abbreviate this property of function *f* as *MIncreasing*( *f* ). Analogously, we can define the property of monotonic decrease, *MDecreasing*( *f* ). We use a capital letter to indicate that both functions have a functional argumen<sup>t</sup> instead of a time value.

Although length operator - occurs in the characteristics of the first and of the second interval, it returns independent values. Due to transitivity of smaller or equal relation ⩽, the monotonicity property holds for all subintervals, not only for the adjacent ones. Thus, ∫ *ba f* - *dt* ⩽ ∫ *cb f* - *dt* ⩽ ∫ *dc f* - *dt* (see Figure 2).

#### *4.3. Limits and Amplitude*

In this subsection, we show how to define lower limits of functions and their minimal amplitudes. The definitions are based on temporal modalities. Upper limits and maximal amplitudes can be defined analogously. We use those definitions below when dealing with volcanic Strombolian effects.

We start with the case when a function *g*(*t*) is an upper approximation of a function *f* (*t*) or, equivalently, function *f* (*t*) is a lower approximation of function *g*(*t*) (see Figure 3). This property can be expressed by demanding that for each subinterval of a given interval, the integral of *f* is smaller than or equal to the integral of *g*:

$$
\Box \quad \int f \prec \int \mathbf{g} \tag{4}
$$

The property that *g* increases *c* times faster than *c* can be expressed in DC4F in a simple way: ◻∫ *c f* ⩽ *g*. It should be noted that we cannot directly express it in DC.

The diagram presented in Figure 4 shows function *h*, which is above value *a*. The property that, for a given interval, values of function *h* are above a certain limit *a* all the time can be expressed as follows:

$$
\Box \quad \ell a \lessdot \int h
$$

The integral of *a* is equal to the value *a*. Consequently, we demand that it remains smaller than the integral of *h* for all subintervals of the underlying interval. The fact that function *h* exceeds threshold *y* for some time can be expressed as follows:

$$\text{true} \stackrel{\frown}{\frown} \{ 0 < \ell \land \bigcirc \ell \not\subseteq \ell \} \stackrel{\frown}{\frown} \text{true} \tag{6}$$

This formula states that for some subinterval of a nonzero length, the value of the integral is not lower than the value of a multiplied by the length of the subinterval. The function shown in Figure 4 extends beyond line *y* for a time period. Equivalently, the property can be expressed as follows:

$$
\diamondsuit \ (0 < \ell \land \Box \ell \, y \lessdot \int h) \tag{7}
$$

The fact that an amplitude of a function is at least *d* can be expressed as follows:

$$\exists x y \{ \Diamond \{ 0 < \ell \land \Box \int h \lhd \ell \, x \} \urcorner \land \Diamond \{ 0 < \ell \land \Box \, \ell \, y \leqslant \int h \, \bigrhcorner \} \urcorner \} \vdash d \, \Diamond \, y \, \bot \, \text{ } \tag{8}$$

Formula (8) says that there exist two subintervals of nonzero length. The integral of *h* for the first one is smaller than or equal to - *x*, as it was expressed by formula (7). Analogously, the integral of *h* for the second one is larger than or equal to - *y*. Formula (8) requires also that the difference *y* − *x* is equal to at least *d*.

In Figure 4, the intervals correspond to the parts of diagrams of *h* located below the *x* and above the *y* line, respectively, are indicated by bold lines. Below, we abbreviate formula (8) by *Amplitude*(*h*, *d*). Formula *PeakAbove*(*h*, *y*) is a shorthand of formula (6) and expresses the fact that function *h* is above threshold *y* for some time period.

**Figure 3.** Functional limit.

**Figure 4.** Constant limits and amplitude.

#### **5. Syntax and Semantics of DC4F**

In this section, we define the formal syntax of DC4F and describe its informal semantics. We list some basic properties of the proposed logic and show how to apply it in the multidimensional case, such as those involving two-dimensional images.

#### *5.1. Syntax of DC4F*

Every logic has its proper language, which has to be formally defined. In this subsection, we define the formal syntax of DC4F. This syntax is close to that of DC. The crucial difference consists in the fact that we allow all unary integrable functions, not only the Boolean-valued ones, and their integrals. Basically, there are constants, variables, and function symbols corresponding to the integrable functions, which can be added and multiplied by constants. There is also an integral operator. Formulas are either of the atomic form, when real numbers are compared, or are composed from other formulas.

To define the syntax formally, designated sets of variables, functions, and relation symbols are required. There are global variables, which remain unchanged. We assume that **VSym** is an unbound set of real-valued variables denoted by letters *x*, *y*, *z*, . . .We also assume that there is a set **CSym** = {*0*, *1*,..., *true*, *false*,... } of constant symbols. The set **FSym** contains unary functions symbols *f*, *g*, *h*,... corresponding to integrable functions. The set **FlSym** contains functional symbols *X*, *Y*, *Z*,... corresponding to functionals with arguments in the form of time intervals and values of type real R (cf. Appendix A).

The syntax of the logic comprises three categories of expressions: functions *Fn* containing integrable functions, terms *Real* containing expressions of type R, and formulas *Fo* containing logical formulas in the proper sense of the word. We assume in the definition below that *f* ∈ **FSym**,*C* ∈ **CSym**, *x* ∈ **VSym**, *X* ∈ **FlSym**.


The first category includes constants and functions of type real; the constants are treated here as functions with arity equal to 0. The functions can be multiplied by constants and added. There is also the maximum operator ∨; it denotes their maximum and is a reminiscent of DC (cf., e.g., [4]). In DC, there is an indefinable negation operator ¬ that applies to state functions (cf., e.g., [4]). It is characterized by the following property: *f* (*x*) = 1 if, and only if, ¬*f* (*x*) = 0. We can define the negation ¬ on integrable functions such that it coincides with negation on state functions: ¬*f* =*def* 1 − *f* (*x*) ∨ *f* (*x*) − 1. Consequently, if *f* (*x*) = 1, then ¬*f* (*x*) =*def* 1 − 1 ∨ 1 − 1 = 0. If *f* (*x*) = 0, then ¬*f* (*x*) = 1 − 0 ∨ 0 − 1 = 1, since ∨is the maximum.

The second category includes constants, variables, and applications of the integral operator. There are also auxiliary operators *X* on intervals with values of type real. Complex terms can be constructed from simpler ones using arithmetic operations.

The third category comprises formulas as such. The atomic formulas include constants *true* and *false*; the atomic formulas are also formed from relational symbol applied to terms, such as *u* < *v*. Complex formulas are composed from other formulas by the application of logical operators: logical negation, alternative, existential quantification, concatenation, the chop operator, and iteration. Thus, if *F* and *G* are formulas, then the alternative *F* ∨ *G*, the chop *F* ⌢ *G*, the iteration *F*<sup>∗</sup>, and the quantification ∃*xF* are formulas as well.

As an example of the above syntax, we can present formula ∫ *sin* = 2⌢ (∫ *sin* = <sup>−</sup><sup>2</sup>), provided that *sin* belongs to **FSym**. The formula is obtained by the application of chop to two atomic subformulas. The first subformula is obtained by the application of equation = to terms ∫ *sin* and 2; the first term is obtained from function *sin* by application of the integral operator. The other subformula is obtained in a similar way.

It should be noted that relations *u* ⩽ *v* and *u* > *v* can be defined in terms of < and = using logical operators ¬ and ∨, e.g., in the first case, we have *u* < *v* ∨ *u* = *v*. Similarly, the conjunction *F* ∧ *G* is defined as ¬(¬*F* ∨ <sup>¬</sup>*G*). The implication *F* ⇒ *G* is defined as ¬*F* ∨ *G*. The general quantifier ∀*xFo* is defined as ¬∃*x*<sup>¬</sup>*Fo*. As mentioned in Section 3.2, modalities ◻, ◊ are defined using chop and negation. ◻ *F* is defined as ¬(*true*⌢ (¬*Fo*⌢ *true*)) and ◊ *F* as *true*⌢ *F* ⌢ *true*. The length operator - is defined as ∫ 1 (see Section 3.1). We assume that ⌢ binds stronger than ∨ and ∧. ∨ and ∧ bind stronger than quantifiers ∃ and ∀. Quantifiers bind stronger than implication ⇒. Implication binds stronger than modalities ◊ and ◻.

#### *5.2. Informal Semantics of DC4F*

In this subsection, we present the intuitive semantics of the previously defined syntax. The presentation is informal, but it can be easily formalized, as shown in the Appendix A. We discussed it for the three syntactic categories defined above: *Fn*, *Real* and *Fo*.

We assume here that the time domain Time is the set of non-negative real numbers R<sup>+</sup>. In the case of discrete time modeled by natural numbers N, for every discrete function *f* , we define the corresponding step function *g* on R+ by extending the values of *f* to the corresponding unit interval: *g*(*t*) = *f* (⌊*t*⌋), where ⌊*t*⌋ is the largest integer *n* ∈ N such that *n* ⩽ *t*. Thus, we can treat discrete functions, or discrete time series, as if they were defined for continuous time.

Elements belonging to category *Fn* are interpreted as time series, and more precisely as Riemann integrable functions with the domain R<sup>+</sup>. The functions may be multiplied by constants, added, and subtracted; thus, the set corresponds to a linear space. The function *f* (*t*) ∨ *g*(*t*) returns the maximum of *f* (*t*) and *g*(*t*).

The category *Real* contains terms of type R. The constants, e.g., 0 or 1, are interpreted as the corresponding real numbers. Variable symbols are evaluated by functions mapping them into the set R. For *f* ∈ *Fn* and an interval [*a*, *b*], where *a*, *b* ∈ R+ and *a* ⩽ *b*, term of the form ∫ *f* is interpreted as the integral ∫ *ba f* (*t*)*dt*. This operator is linear in respect to functions: ∫ *c f* = *c* ∫ *f* and ∫ *f* + *g* = ∫ *f* + ∫ *g*.

A term *t* depends on variable *x* ∈ **FSym** if *t* contains it. In general, the value of terms containing variables depend on the values of its variables. Valuations are mappings of variables into R. We can, for example, define valuation *val*1 as mapping of the form *x* ↦ 1, *y* ↦ 2, *z* ↦ 5. Let term *p* be of the form *x* + *y* + *z* + 2. The value of *p* for *val*1 is 10; it does not depend on an underlying interval. The value of a term depends on an underling interval if contains operator ∫ . In the case of term *q* of the form (∫ 1) + *z*, the value depends on the underlying interval and variable *z*; for interval *I*1 = [0, 2] and *val*1, the value is 7.

The category of formulas *Fo* plays a crucial role, as formulas are the proper objects of logical reasoning. In fact, formulas can be understood as Boolean-valued terms. The satisfaction of formulas, i.e., the fact whether formulas are true or false, is defined in respect to an underlying interval and a valuation of the variables they include. The values of these terms may be compared using <, =, given the underlying interval and the valuation. The result of this comparison is either true or false, depending on the value of the corresponding terms. For example, for interval *I*1 and valuation *val*1, formula *q* < *p* is true but formula *q* = *p* is false.

Let *I* = [*a*, *b*] be an interval and *val* be a valuation. Formula ¬*F* is satisfied in interval *I* for valuation *val*, if formula *F* is false in interval *I* for *val*. Similarly, the alternative *F* ∨ *G* is satisfied if *F* or *G* is satisfied in *I* for *val*.

In case of chop, *F* ⌢ *G* is satisfied in interval *I* for valuation *val* if there exists *c* ∈ [*a*, *b*] such that *F* is satisfied in [*a*, *c*] for *val* and *G* is satisfied in [*c*, *b*] for *val*. Iteration *F*∗ of formula *F* is satisfied in *I* for *val* if there is an *n* such that we can split the interval into subintervals [*a*, *<sup>a</sup>*1], [*<sup>a</sup>*1, *<sup>a</sup>*2],..., [*an*−1, *b*] such that *F* is satisfied in each subinterval for *val* (cf. [4] Section 4). An example of the iteration formula is presented below and its formal semantics are presented in the Appendix A.

Quantified formula ∃*xF* is satisfied in interval *I* for valuation *val* if there exists a valuation *val*′ which differs from *val* only for variable *x* such that *F* is satisfied in *I* for *val*. The fact that *val* and *val*′ differ only at *x* means that for every variable *y* different from variable *x*, it holds that *val*(*y*) = *val*′(*y*). For example, ∃*x x* + *y* > 25 is satisfied for valuation *val*1 defined above; in fact, we can define *val*′ as *val*1 but *val*′(*x*) = 30.
