An algebra for qualitative modeling

Eric Fimbel

Technical report RT020205

Department of Electrical Engineering

École de Technologie Supérieure

February 2002

An algebra for qualitative modeling Eric Fimbel Department of Electrical Engineering École de Technologie Supérieure Abstract.- Qualitative simulations are useful for the study of natural systems, whenever the experimental evidence is insufficient for determining quantitative equations. They generate qualitative behaviors, which are sequences of coarse descriptions of the system called qualitative states. The transitions between consecutive states may correspond to progressive variations or to real discontinuities, which amplify the effects of small fluctuations, as it happens in biological control systems. This paper introduces a qualitative algebra that allows representing the detail of a transition by means of infinitesimal and/or uncertain qualitative values. In complement, we propose a qualitative transition equation, which produces all the possible transitions from a qualitative state. The algebra and the transition equation are general tools and may have applications in qualitative simulation, but also in image processing or for inference on uncertain or undetermined logical propositions.

2

I. INTRODUCTION I.1 Qualitative simulation A qualitative model is a coarse description of a system in terms of properties and relationships that are 1) generic and 2) easy to observe. For instance, the property for a variable to be positive and decreasing applies to a broad range of situations, and it can be observed without precise, quantitative measurement. Qualitative computer simulations produce descriptions of the behavior of a system, by means of a sequence of qualitative states. In a qualitative behavior, the timing and the values of the variables and derivatives are not quantified, or only partially quantified. This kind of approach is of special interest for the initial stages of the study of natural systems, when there is not enough experimental evidence to determine quantitative equations and laws. When the scientific study is « mature » (Smolen et al., 2000), the knowledge and the experimental data are more abundant and consistent, quantitative models can be elaborated to explain or complete the existing data. Qualitative computer simulation originates in theories such as qualitative physics (de Kleer, Brown, 1984) and qualitative process theory (Forbus, 1984). These works gave way to general tools such as QSIM (Kuipers, 1986) or QPE (Forbus, 1990), which have been used to develop experimental applications in several domains, including molecular biology (Ironi, Stefanelli, 1995, Heidke, Shutze-Kremer, 1998). Besides, qualitative computational techniques have been used in the mathematical biology community to model biological systems for years. Because of the switch-like properties of gene expression and enzymatic activity, continuous biological control systems can be modelled by means of logical descriptions (Glass, Kaufmann, 1973, Glass, 1975a, Thomas, d’Ari, 1990, Thomas, 1995). These logical descriptions allow to predict qualitatively the dynamical behavior of a system. Models based on piecewise linear differential equations combine the continuous and logical approaches and allow a more precise (but still qualitative) characterization of the possible dynamics (Glass, 1975a, Glass, Pasternack, 1978). In the qualitative simulations, the behavior of a system is described as a sequence of qualitative states, which are coarse descriptions of significant variables and their variations (e.g., x is positive, dx/dt is null ... ). Such a description corresponds to a qualitative domain in the phase space, i.e., the 2n-dimensional space where each point defines the position and the variation of the n significant variables. Within a qualitative domain, the real state of the system is unknown, i.e., it may be anything such that the qualitative description remains the same. A qualitative transition (or qualitative change) occurs any time that a variable or a derivative crosses some threshold. For instance, a qualitative transition for some variable x could correspond to the sequence of states x negative, x null, x positive (here, the threshold is zero). A qualitative behavior can be seen as a sequence of qualitative transitions from a qualitative domain to the next one.

3

Several concepts of non-linear dynamics such as steady states, orbits and limit cycles can be transposed to qualitative behaviors (Glass, Pasternack, 1978). It is worth noting that these qualitative concepts can be obtained directly from quantitative simulations. For instance, the existence and stability of the equilibria and limit cycles, the trajectory flows, the geometry of the basins of attractions, etc., can be extracted automatically (Zhao, 1994). Even higher-level features, such as the existence of multistability, bifurcations, etc., can be obtained automatically (Doedel, 1981). However, this requires the existence of a model where the functional relationships are well defined and constant, and only a small number of parameters of the model are unknown. In spite of producing highly significant results, qualitative simulations have several drawbacks. Whenever there is uncertainty on the exact timing of a behavior and/or the values of the significant variables, the usefulness of the results may be impaired. Besides, as the uncertainty impedes direct comparison with experimental data, spurious behaviors (i.e., behaviors that do not occur in the real world) are difficult to detect. Spurious behaviors are commonplace in simulation, either qualitative or quantitative, but in the latter case, they are easy to detect from experimental evidence. Spurious behaviors may be artifacts due to the simulation algorithms, but they may also have more fundamental causes, namely an inaccurate or uncertain model or theory. Qualitative models generally are accurate, in the sense that they do not contain precise but wrong information about the system. In fact, they trade accuracy for uncertainty, which can take two forms: parametric indeterminism (i.e., the exact values of the parameters such as the coefficients of the equations remain unknown) or functional indeterminism (i.e., the relationships between variables are unknown to the extend that they cannot be represented as a family of functions) (Kay, 1998). Because of the uncertainty of the model, many behaviors are predicted, most of which are spurious. As the model becomes more complex, the uncertainty generally increases, and this is a strong caveat to the scalability of qualitative simulation techniques (i.e. to their ability to handle large models). As spurious behaviors in qualitative simulation are unavoidable and, at the same time hard to detect a posteriori, their elimination a priori is an important issue. This can be done by means of restrictions on the possible transitions as in QSIM (Kuipers, 1986), and/or by a reduction of the uncertainty of the model: integration of qualitative and quantitative information (Berleant, 1989, Forbus, Falkenheimer, 1990, Berleant 1997), temporal constraints (Shults, Kuipers, 1997, Brajnik, Clancy, 1997). As an alternative, the behaviors may be classified by order of decreasing frequencies, assuming some distributions of probabilities on the parameters and the functional relationships (Brajnik, 1997). This statistical approach rests importance to the existence of spurious behaviors (which are assumed to have low frequencies). This article deals with two specific causes of spurious behaviors, namely the ambiguous nature of qualitative transitions, and the uncertainty on the next qualitative transition from a given state.

4

I.2 The ambiguous nature of qualitative transitions As seen before, a qualitative transition describes the crossing of some threshold for a variable or a derivative. However, the detail of what happens in the real system is unknown. The transition may correspond to a real discontinuity, or to a progressive change, which at some point crosses the threshold. When the system is close to a discontinuity, small fluctuations of the variables may have a strong effect on the behavior of the system, because they may cause a qualitative change. This phenomenon is especially important for biological control systems, because 1) many homeostasis correspond to points that are close to some threshold (the so-called Singular Stationary Points, see Snoussi, Thomas, 1993) 2) the concentrations of biochemical species present intrinsic fluctuations, which may be due to fluctuations of the rate of production and/or of molecular transport (Smolen et al., 2000). It has been suggested that the phenotypic variations which appear in isogenic populations may be due to the effect of such fluctuations (Mc Adam, Arkin, 1997). Quantitative stochastic simulations have been used to predict the effect of fluctuations in biological control systems (Smolen et al., 1999). In a more analytical approach, the switching zones between domains can be described by means of generic sigmoid functions with a variable steepness (Plahte et al., 1998). In a similar fashion, ramp functions can be used to parametrize the smoothness of the transitions and the width of the switching zones (de Jong et al., 2001). However, in absence of quantitative data, these approaches only allow general, qualitative predictions. Consequently, general qualitative techniques, which do not use a specific family of functions (step functions, sigmoids, etc.) remain a valuable alternative. In order to allow a more precise (although qualitative) representation of the transitions, we propose to represent infinitesimal values within a qualitative algebra. As in (Plahte et al., 1998), a qualitative transition is a zero-crossing for some variable or derivative, but the switching zone is now considered as infinitesimal and nothing is known about its quantitative features. The concept of infinitesimal corresponds to the concrete notion of undetectable phenomenon, i.e., something so small or transitory that it cannot be observed with the same techniques as the variables of the system. Infinitesimals are poorly represented in classical numerical analysis. For instance, the only infinitesimal real number (i.e., a number that is infinitely close from zero) is zero itself. This does not allow the representation of measurements that are several orders of magnitude smaller than others, like molecular fluctuations vs. cellular events. On the other hand, Non Standard Analysis provides a simple way of representing infinitesimals, under the form of hyperreal numbers (Robinson 1974). Around each real number, there are infinitely close hyper-reals, which can be observed through an "infinite microscope" (Keisler 1994).

5

We extended a three-valued qualitative algebra (see for instance de Kleer, Brown, 1984) in order to represent infinitesimals. The possible values of this new algebra are null, infinitesimal, finite, unbounded and infinite (positive or negative). For instance, a continuous transition on some variable x may correspond to: x finite negative, infinitesimal negative, null, infinitesimal positive and finite positive. The time interval between two states may be itself infinitesimal. It is worth noting that the result of the algebraic operations (e.g., sum, product) is sometimes uncertain. For instance infinitesimal negative + infinitesimal positive is: infinitesimal negative, null or infinitesimal positive. The null value, which corresponds to a single numerical value (namely zero), describes a state that is precisely known. The other elementary qualitative values such as infinitesimal positive or finite positive correspond to states that are only partially known. Complex qualitative values, which are unions of elementary qualitative values correspond to higher level of uncertainty. For instance, a variable x may be null or infinitesimal. The union of all the elementary qualitative values corresponds to the highest possible uncertainty, i.e., the absence of any information. The algebra proposed here is a general tool and could be used to represent qualitative information in different applications. In image processing, it could be used to represent undetermined pixels and elementary features. In neuromotricity, it could be used to represent physiological or biomechanical temporal data to detect motor events. In inference engines, it could be used to perform inference on uncertain and/or undetermined logical propositions. I.3 Uncertainty on the next qualitative transition Because a system may be anywhere within a qualitative domain, in the general case, it is impossible to know which qualitative transition will happen next, i.e., which of the variables and the derivative will reach a threshold first. Moreover, several variables and derivatives may cross a threshold at the same time. Even if this situation is highly improbable, it cannot be discarted a priori. The number of possible qualitative transitions increases exponentially with the number of variables of the model, which may cause combinatorial explosions during the simulations. On the contrary, the cost of quantitative simulations may be high, but it generally increases linearly with the number of variables, the size of the equations and the frequency of the simulation (i.e., inverse of the time step). In QPT (Forbus 1984), QSIM (Kuipers 1986) and their derived systems, the possible transitions are described on a case-by-case basis, and are implemented by a variety of specific transition rules. A simple method is suggested here to deal with the uncertainty on the qualitative transitions. We propose to use a unique and general mechanism in all the possible situations instead of a variety of specific cases. The mechanism produces a unique qualitative state that represents the union of the results of all the possible transitions. The next state may be chosen randomly within the result.

6

The general qualitative transition equation proposed here is used for each variable of a model. From the qualitative values of the variable, its derivative and the time increment, the equation produces the union of the next possible qualitative states of the variable. This equation is accurate in the sense that its result contains all the next possible states, but it may be uncertain, for instance, the variable X may be infinitesimal negative, null or positive. In this case, the next state may be chosen randomly. Therefore, simulations have to be repeated from the same initial conditions in order to produce a significant subset of all the possible behaviors. The repetition of simulations can be computationally cheaper than a complete exploratory algorithm, because each simulation can be done at a very low computational cost. In general terms, repetition of simulations is necessary because a single simulation can correspond to a spurious behaviors, hence, it cannot be considered as a significant prediction. The transition equation is a useful tool for qualitative simulation because it is easy to interpret and to implement. It has been used in a prototype of qualitative simulator (not presented here), and applied on simple biochemical networks and mechanical systems. The format of this paper is as follows:. A qualitative algebra is presented in section II. The qualitative transition equation is presented in section III. Possible applications and concluding remarks are presented in section IV. II. QUALITATIVE ALGEBRA The concepts and operators used to represent qualitative values are now defined. The content of this section is organized as follows: • The numerical values are defined as hyper-reals. Sets of numerical values are used to represent the possible values of variables and expressions. • Nine qualitative properties (or q-properties) are defined. They are used to describe the sets of numerical values (e.g., bounded and positive). • Qualitative values (or q-values) are defined as sets of q-properties. There are 29 q-values. There are 9 elementary q-values, which are the singletons corresponding to the q-properties. • The qualitative description (or q-description) of a set of numerical values is defined. It is a q-value composed of the properties verified by some subset. For instance, the q-description of ]0,1] is: infinitesimal positive or finite positive. • The four qualitative operations (+, *, -, /) are defined for the q-properties by means of tables, and they are extended to all the q-values by means of a distributivity rule. • Finally, the correspondence between the qualitative and the numerical operations is described.

7

II.1 Numerical values and sets of numerical values Numerical values. The set of numerical values is identified with the hyper-real numbers, *R, i.e., the real numbers, infinitesimals and infinites. NV = *R. Remark. Infinitesimal, real and infinite numbers are incommensurable, which means that they cannot be compared precisely, because their orders of magnitude are too distant. Complementary definitions. We call PNv the sets of numerical values PNv=P(Nv). These sets are important here, because they allow to represent the possible values of the variables and expressions of a model . Remark. The usual set operations (∪, ∩, ~, i.e., complement) are defined on PNv. Remark. The usual arithmetic operations ( +, - , *, / ) are defined on PNv. For instance, E1+E2 = {x+y | x, y ∈ E1 x E2}. II.2 Qualitative properties of the sets of numerical values Qualitative properties (q-properties). There are 9 qualitative properties, which are logical relations on PNv. The set of qualitative properties, Qp, is defined as follows: Qp = {q0, q1, q2, q3, q4, q5, q6, q7, q8 } Table 1. Qualitative properties. X stands for any set of PNv. qi(X) = true iff. the condition given in the column definition is satisfied. q-property symbol description definition infinite negative -∞ X = {-∞} q0 -unbounded negative ∀ a ∈ R ∀ x ∈X, x < a q1 finite negative ∃ a,b ∈ R2,∀ x∈X, a < x < b < 0 q2 0infinitesimal negative ∀ a ∈ R-,∀ x ∈X, a< x < 0 q3 0 null X = {0} q4 0+ infinitesimal positive ∀ a ∈ R+,∀ x ∈X,0 < x < a q5 + finite positive ∃ a,b ∈ R2,∀ x∈X, 0 < a < x < b q6 ++ unbounded positive ∀ a ∈ R ∀ x ∈X, x > a q7 infinite positive +∞ X = {+∞} q8

Remark. The distinction between unbounded and infinite does not exist in NonStandard Analysis. It has been introduced for technical reasons, because it allows to define the inverse of 0, 1/0 = ±∞. Remark. It can be seen from Table 1 that the q-properties are mutually exclusive. Besides, the q-properties are not empty, which means that any q-property is verified by at least a subset of Nv (note that there infinite, unbounded and infinitesimal have only hyper-reals examples. For instance, the only infinitesimal real number is zero):

∀ qi, qj ∈ Qp2 , ∀ xi ∈ PNv, qi(xi) ∧ ( j ≠ i ⊃ ¬qj(x) )

8

(1)

∀ qi ∈ Qp , ∃ xi ∈ PNv, qi(xi) II.3 Qualitative values Qualitative values (or q-values). The set of qualitative values, Qv, is defined as the subsets of Qp, Qv=P(Qp). Remark. As there are 9 q-properties, there are 29 = 512 q-values. Remark. The usual set operations (∪, ∩, ~) are defined on Qv. Complementary definitions. We call empty q-value the empty set ∅, elementary qvalues the singletons {qi}, complex (uncertain) q-values the q-values containing several elements and universal q-value Qp itself. The universal q-value will be noted: “?”. II.4 Qualitative descriptions The qualitative description (q-description) of a set of numerical values x is the set of the q-properties that some subset of x verifies. For instance, the q-description of X=[0,1] is {0,0+,+}, because X contains zero, some infinitesimal and some positive reals. More examples of q-descriptions are given in. Table 2. Formally: (2)

qd : PNv -> Qv qd(X) = {q ∈ Qp | ∃ Y ⊆ X, q(Y) }

Table 2. Examples of q-descriptions set X

∅ {-∞} [-∞, -a], a > 0 {-a} , a > 0 ]-a, 0[, a > 0 [0, a], a > 0 ]-a, b[ , a,b > 0 {1/n, n ∈ N+ } Z R R+

q-description(X)

∅ {-∞} {-∞, --, -} {-} {-, 0-} {0, 0+, + } {-,0-, 0, 0+, + } {0+, + } { --, -, 0, +, ++ } { -- , - , 0-, 0, 0+, + , ++ } { 0, 0+, +, ++ }

9

Remark. the q-description is distributive for the union:

∀ X1, X2 ∈ PNv2 , qd ( X1 ∪ X2 ) = qd ( X1) ∪ qd (X2)

(3)

Complementary definition. We define the q-description of a numerical value as the qdescription of the corresponding singleton:

∀ x ∈ Nv, qd ( x ) = qd ( { x } )

(4)

Remark. The q-description of a numerical value can only be an elementary q-value. This comes from the fact that the q-properties are mutually exclusive. II.5 Qualitative operations The algebraic operations, i.e., the sum (+), the product (*), the pseudo-opposite (-) and the pseudo-inverse (/) are first defined on the q-properties by means of tables (see Table 3Table 4). They are extended to the q-values by means of a distributivity rule. For instance, qv1+qv2 is defined as ∪ { (qi1+qi2) | qi1 ∈ qv1, qi2 ∈ qv2}. Table 3. Definition of sum and pseudo-opposite on the q-properties. qi...qj represents the set of the q-properties qk, k in [i,j]. ↓X Y→

-∞

--

-

0-

0

0+

+

++

+∞

X

-∞ -00 0+ + ++ +∞

-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞...+∞

-∞ --------...++ +∞

-∞ --...+ ++ +∞

-∞ -000-...0+ + ++ +∞

-∞ -00 0+ + ++ +∞

-∞ -0-...0+ 0+ 0+ + ++ +∞

-∞ --...+ + + + + ++ +∞

-∞ --...++ ++ ++ ++ ++ ++ ++ +∞

-∞...+∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞

-∞ -00 0+ + ++ +∞

- X +∞ ++ + 0+ 0 0--∞

Table 4. Definition of product and pseudo-inverse on the q-properties. "qi...qj" represents the set of the q-properties qk, k in [i,j]. "qi,qj" represents the set of q-properties "{qi,qj}". ↓X Y→

-∞

--

-

0-

0

0+

+

++

+∞

X

/ X

-∞ -00 0+ + ++ +∞

+∞ +∞ +∞ +∞ -∞..+∞ -∞ -∞ -∞ -∞

+∞ ++ ++ 0+...++ 0 --...0---∞

+∞ ++ + 0+ 0 0--∞

+∞ 0+...++ 0+ 0+ 0 00--...0-∞

-∞...+∞ 0 0 0 0 0 0 0 -∞...+∞

-∞ --...0000 0+ 0+ 0+...++ +∞

-∞ -00 0+ + ++ +∞

-∞ ----...00 0+...++ ++ ++ +∞

-∞ -∞ -∞ -∞ -∞...+∞ +∞ +∞ +∞ +∞

-∞ -00 0+ + ++ +∞

0 0--∞,+∞ ++ + 0+ 0

Extension of the operations to Qv . The arithmetic operations are extended to the qvalues by means of a distributivity rule:

10

∀ q1, q2 ∈ Qv2 q1 ω2 q2 = ∪ { ( qi1 ω2 qi2 ) | qi1 ∈ q1, qi2 ∈ q2} ω 1 ( q ) = ∪ { ω 1 ( qi ) | qi ∈ q } where ω2 is +, -, / or *, and ω1 is - or /

(5)

Remark. The distributivity rule grants that the arithmetic operations are distributive with the union:

∀ q1, q2, q3 ∈ Qv3 (q1 ∪ q2) ω2 q3 = (q1 ω2 q3) ∪ (q2 ω2 q3) ω1 (q1 ∪ q2) = ω1 (q1 ) ∪ ω1 (q2) where ω2 stands for +, -, / or *, and ω1 stands for - or /

(6)

Remark. From the previous definitions, it can be verified that the sum and the product are commutative, associative and distributive. Remark. It can be seen from the previous definitions that the neutral elements of the sum and the product are respectively qd(0) = {0} and qd(1) = {+}. Remark. As the sum and the product can introduce some uncertainty, it is false that q-q={0} and q/q={+}. The pseudo-opposite and the pseudo-inverse only verify the weak properties: {0} ∈ q - q {+} ∈ q / q

(7)

Complementary definitions. The difference (-) and the division (/) are defined from the previous operations: (8)

q 1 - q 2 = q 1 + ( - q2 ) q1 / q2 = q1 * (/ q2 ) II.6 Correspondence between the qualitative and numerical operations

Numerical values. As the q-descriptions of the numerical values are elementary qvalues, the correspondence between the qualitative and numerical operations is straightforward. The following properties can be verified directly from Table 3 andTable 4:

∀ x,y ∈ Nd2

(9)

q d ( x + y ) = qd ( x ) + q d ( y ) q d ( x - y ) = qd ( x ) - q d ( y ) q d ( x * y ) = qd ( x ) * q d ( y ) if y ≠ 0, qd ( x / y ) = qd ( x ) / qd ( y ) Sets of numerical values. As the q-descriptions of the sets of numerical values can be uncertain q-values, the correspondence between the qualitative and numerical operations is weaker. Nonetheless, it can be proven that:

11

∀ X,Y ∈ PNd2

(10)

qd (- X) = - qd(X) qd (X + Y) ⊆ qd(X) + qd(Y) qd (X - Y) ⊆ qd(X) - qd(Y) qd (X * Y) ⊆ qd(X) * qd(Y) qd (X / Y) ⊆ qd(X) / qd(Y) Scheme of Proof. These results are consequences of the properties (9). For instance, consider the case of the sum: let q be any element of qd(X+Y). By definition, for some x,y of X,Y, q= qd(x+y) = qd(x)+qd(y) (properties (9) hence q ∈ qd(X)+qd(Y) The proof of the other properties is similar, including the 2-steps proof of qd(-X)=qd(X). III. QUALITATIVE TRANSITION EQUATION This section addresses the specific issue of qualitative simulation, and is organized as follows: 1. The qualitative models considered here are presented. They use ordinary differential equations and constraints. 2. the concepts of qualitative simulation are defined more precisely: qualitative states, behaviors, transitions, and qualitative steady states. 3. The qualitative transition equation is presented. This equation gives the result of all the possible transitions from a given step. In case of non-determinism, the next state is chosen randomly within this result. 4. The qualitative transition equation is accurate, but for practical reasons, spurious behaviors cannot be avoided, otherwise, the simulation is too uncertain to be informative. 5. Qualitative constants are not numerical constants. The rationale below the concept is that if it cannot be quantified, it is impossible to know that it is constant.

III.1 Qualitative models: ordinary differential equations and constraints.

12

Qualitative model. The qualitative models considered here are composed of a set of ordinary differential equations (ODEs) and a set of constraints. An example of qualitative model can be found in Figure 1. The modelled system is a simple, reversible chemical reaction. System

Variables S: concentration of substrate P: concentration of product Vf: rate of direct reaction Vr: rate of reverse reaction

Equations and constraints d/dt S = 0 d/dt P = Vf - Vr Vf = Kf * S Vr = Kr * P Kf > 0 Kr > 0

Figure 1. A simple chemical reaction. The direct rate Vf is proportional to the concentration S of the substrate (which is constant) and the inverse rate Vr is proportional to the concentration P of the product.

Time. The time line is identified to the hyper-reals *R. This means that there are infinite and infinitesimal time intervals. Variables. The system is represented by a set of n variables V= {x1..xn}. At any time, the variables have a set of possible numerical values and a numerical value, which is precisely known only if the set of possible values contains only one element. Remark. The q-description of the variables is defined at any time, i.e., the qdescription of their sets of possible values. Derivatives. The derivatives of the variables are noted dxi/dt. At any time, the derivatives have a set of possible numerical values and a unique numerical value. Derivatives are defined at any time, but they may be unknown. This occurs whenever the function is not derivable in a classical approach. Complementary definition. dxj/dt is a free derivative if there exist no equation of the form xi = dxj / dt (see below). Equations. The dynamics of the system are characterized by a set EQ of ODEs, containing usual numerical expressions, undetermined parameters and functions. For instance, the following ODE: dxi/dt = - k xi + f (xi, xj). contains an undetermined parameter k and an undetermined function f. Constraints. The ODEs are completed by a set C of constraints. Any equation that is not an ODE is considered as a constraint. For instance: x3 = x1+x2 , d/dt xi >= 0 and d/dt xi < 1 are constraints. Parameters. Any coefficient appearing in an expression, e.g., k in the equation dx/dt = -k x, is considered as a parameter of the model. A parameter has either a numerical value or a q-description which is defined by means of constraints, e.g., (k < 0), k=1 . Qualitative invariants vs numerical invariants. The parameters that have a numerical value are invariant. The parameters that have a q-description are qualitative invariants, i.e., their q-description is the same at any time. However, the numerical value of the

13

coefficient can vary in time. Here, we consider that unless the exact value is known, it is impossible to assume that it is constant. III.2 Qualitative states, behaviors and transitions Qualitative state (q-state). The state of the system at a given time is defined by the q-description of the n variables and derivatives. It corresponds to a qualitative domain in the phase space, i.e., the 2n-dimensional space where each point corresponds to a numerical value of the n variables and derivatives. Remark. The number of q-states is finite: as there are only 29 possible q-values, the number of possible states is about 29n. Qualitative behavior (q-behavior). A qualitative behavior is a sequence of consecutive q-states and qualitative time increments, where each q-state is compatible with the equations and constraints of the model, and each transition is compatible with the qualitative transition equation (see below). Remark. A infinite q-behavior contains a limit cycle, because the number of q-states is finite. Remark. A finite q-behavior may terminate in three ways: 1) with a qualitative steady state(see below), 2) when the number of steps reaches some limit, or 3) when it is impossible to find a transition to some next step. This case corresponds to a spurious behavior. Qualitative time increment. The time increment between two consecutive states of a given behavior is not fixed. Its qualitative value can be infinitesimal positive, finite positive or unbounded positive. Qualitative transition (q-transition). A qualitative transition is composed of two consecutive q-states, and a qualitative time increment. Qualitative transitions must follow the qualitative transition equation presented in the next section. Qualitative steady state. It is a q-state such that all the possible transitions remain in the same q-state, whatever the time increment. A qualitative steady state is not always a steady state. For instance: x > 0 and dx/dt infinitesimal is a qualitative steady state, but not a steady state. Visualization of a qualitative behavior. For each variable, a behavior corresponds to a qualitative time plot, showing its q-value as a function of time. The important features such as extremas, periodical changes, etc., appear on the plot, but no quantitative information is available about amplitudes or time instants. Figure 2 shows an example of qualitative time plot.

14

Figure 2. Qualitative time plot of the position in a spring-mass system. Horizontal: time (finite time increments are 4 times larger than infinitesimal time increments). Vertical: qualitative value. Thick line: the variable is steady (derivative is null); grey band: the variable is not steady.

III.3 The qualitative transition equation Qualitative transition equation (or QTE). Given the q-value of a variable and its derivative, and the q-value of the time increment, this equation produces the next possible qualitative value of the variable.

qXn+1 ⊆ qXn + qX′n,n+1 * q∆tn

(11)

where qXn and qXn+1 are the qualitative value of variable X at steps n and n+1, q∆tn is the qualitative value of the time increment at step n, and qX'n,n+1 is the broadest Qvalue of the derivative X' = dX/dt during ∆tn: qX'n,n+1 =

qX'n∪q1∪q3 if qX'n contains q2, (finite negative) qX'n∪q5∪q7 if qX'n contains q6, (finite positive) qX'n otherwise.

(12)

Remark. X’n is not used directly in the equation, because a value that is finite at any time during an interval ∆tn can eventually become infinitesimal or unbounded. For instance, if the time interval is [0,1] (∆t = 1), X’(t) = 1/t is finite at any time, but it is unbounded when t is infinitely close to 0. However, the q-value of 1/t will be finite positive at any time. Application of the qualitative transition equation. From a given state, the qualitative transitions are calculated as follows: 1) a qualitative time increment is fixed; 2) the next state of each variable is calculated by means of the QTE; 3) the next state of the free derivatives is randomly chosen; 4) the constraints Q are evaluated. If there is some inconsistency, a new trial is done with another qualitative time step. Remark. The constraint evaluation can eventually reduce the qualitative values calculated by the QTE. An inconsistency occurs when some q-value is reduced to ∅. Example. Starting from x=0, dx/dt=m, m>0 leads to x infinitesimal positive, finite positive or unbounded positive. A constraint such as (x <= 10) restricts x to infinitesimal positive or finite positive; a constraint such as (x <= 0) leads to an inconsistency.

15

Random reduction. In order to avoid an increasing uncertainty on the state of the system, the q-values of the variables and derivatives are randomly reduced to one of the possible elementary q-values. For instance, if X is negative, null or positive, one of these three elementary q-values will be chosen randomly. The reduction is accepted only if it is consistent with the constraints of the model. III.4 Spurious behaviors, or accuracy vs. usefulness Random reduction and spurious behaviors. The qualitative transition equation gives an accurate result, in the sense that this result is a description of the real next state of the system. However, this description may be too imprecise to be useful. When simulations are performed without reduction, the state of the system becomes quickly too ambiguous to be useful. On the other hand, random reduction can give spurious behaviors. Statistical analysis of repeated simulations. As single simulations may correspond to spurious behaviors, they are by no means absolute predictions. Henceforth, repeated simulations have to be performed and their results must be interpreted globally, for instance by means of simple statistics such as the ranking of the behaviors by order of decreasing frequency. The relationship between the random reduction and the possible statistics goes beyond the scope of the present report. III.5 What are qualitative constants? Qualitative constants vs. numerical constants. A surprising characteristic of qualitative behaviors is that from the same qualitative state, the system may present different transitions. For instance, in the example of Figure 1, the system may present damped oscillations around its equilibrium point, and its behavior may change completely from one oscillation to the next one. This is partly because the reaction coefficients Kf and Kr are qualitative constants, not numerical constants. From one state to the next one, their values may fluctuate widely. Similarly, the functional dependencies represented by the qualitative functions may change from step to step, if their qualitative properties remain the same. If it cannot be quantified, it is impossible to know that it is constant. This simple point helps to understand the results of qualitative simulation. If a physical parameter can be measured, or evaluated from what is known about similar systems, it has to be represented by a numerical value. It is represented by a qualitative value only if very little is known about it, and in this case, it cannot be assumed that it is constant. The situation is similar for the functional dependencies: if they are represented as qualitative functions, their invariance is not granted.

16

IV. POSSIBLE APPLICATIONS AND CONCLUDING REMARKS The qualitative algebra presented here allows representing the states and the transitions of a system in a precise, although qualitative, manner. When qualitative states and transitions are described by means of infinitesimal and uncertain qualitative values and time intervals, it is possible to discriminate equilibriums from nearequilibriums, and to have a good insight on the nature of the qualitative transitions, e.g., discontinuities or progressive qualitative changes. The qualitative transition equation is a more specific tool, because it deals only with dynamical systems described by means of ordinary differential equations. However, ODEs are a very general framework, and they are used to model a a broad range of biological, physical and engineered systems. The qualitative algebra and the qualitative transition equation have been originally designed for qualitative simulation. In this context, they allow an accurate prediction of the effects of infinitesimal fluctuations when a system is close to a qualitative change. The qualitative transition equation allows predicting accurately all the possible transitions from a given state. Therefore, many spurious behaviors that correspond to inaccurate predictions are eliminated. However, this accuracy goes along with uncertainty, and many behaviors which as predicted may be unobservable or spurious. As an alternative to heuristics or specific rules, we propose a constrained random selection of the possible transitions as the simplest way of reducing this uncertainty. As the random selection is guided by constraints, the available knowledge on the modeled system can be used for the elimination of spurious transitions (and behaviors). Of course, much remains to do in order to reach the computational efficiency of existing heuristics. To that end, the qualitative algebra and the qualitative transition equation have been implemented in a prototype of hybrid qualitative-quantitative simulator, which has been tested on simple biochemical and mechanical systems. The tools proposed here rely on a solid conceptual framework, namely Non Standard Analysis. It is worth noting that infinitesimal qualitative values have more representative power than classical numerical values: infinitesimal qualitative values can represent what is happening beyond the numerical resolution of any quantitative model. This opens a broad field of applications for the qualitative algebra. Image processing. When feature detection is performed on a noisy image, many artifacts may appear. Qualitative values may be used to represent elementary features (as in contour detection) as an alternative to binary values. The representation of uncertain (ambiguous) elementary features may help to enhance the performances of noise elimination algorithms, based on local properties (RMF, local convolutions, etc.). Neuromotricity. Temporal physiological and biomechanical data, such as EMG recording, movement recording, etc. are generally noisy, and the detection of significant events, such as threshold crossing, inversion of derivative, etc. is generally done on a case by case basis. The conversion of the quantitative data (original measurement and successive derivatives) to qualitative value gives way to general event detection techniques, based on the recognition of temporal patterns of qualitative values. For instance, a derivative inversion may be described as dx/dt positive (..) dx/dt negative. Uncertain inference. As an alternative to boolean and fuzzy inference, qualitative values could be used to represent uncertain (or undermined) propositions. The inference

17

rules on boolean propositions can be easily represented in the qualitative algebra. For instance, the q-value of x ∧ y corresponds to the intersection of the q-values of x and of y. The foregoing remarks give tracks for further applications of the tools proposed here. However, much remains to do in order to explore the potential and the limits of these general qualitative tools.

REFERENCES Berleant, D. (1989) "A unification of numerical and qualitative model simulation". Proceedings of the workshop on model-based reasoning, IJCAI-89. Berleant, D., Kuipers., B. (1997) "Qualitative and quantitative simulation: bridging the gap". Artificial Intelligence Journal, 95(2) pp. 215-255. Brajnik, G. (1997) "Statistical properties of qualitative behaviors". 11th International Workshop on Qualitative reasoning, Cortona, Siena, Italy. pp.233-240. Brajnik, G., Clancy, D.J. (1997) "Focusing qualitative simulation using temporal logic: theoretical foundations". Annals of Mathematics and Artificial Intelligence 22 pp.5986. de Jong, H., Page, M., Hernandez, C., Geiselmann, J. (2001) "Qualitative Simulation of Genetic Regulatory Networks: Method and Application". IJCAI 2001 pp.67-73. de Kleer, J., Brown, S. (1984) "A Qualitative Physics Based on Confluences". Artificial Intelligence 24 pp.7-83. Doedel, E. (1981). AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr. Num. 30 pp. 265-284. Forbus, K.D. (1984) "Qualitative Process Theory". Artificial Intelligence 24 pp.85-168. Forbus, K.D. (1990) "The Qualitative Process Engine". In Weld, D.S., de Kleer, J. Readings in Qualitative Reasoning about Physical Systems, Morgan Kaufman, San Mateo, CA, pp.220-235. Forbus, K.D., Falkenheimer, B. (1990) "Self-explanatory simulations: an integration of qualitative and quantitative knowledge". AAAI-90. Glass, L., Kauffman, S. (1973) "The logical analysis of continuous non-linear biochemical control networks". J. Theo. Biol. 39 pp. 103-129. Glass, L. (1975a) "Combinatorial and Topological Methods in Nonlinear Chemical Kinetics". J.Chem. Phys. 63 pp.1325-1335. Glass, L. (1975b) "Classification of Biological Networks by their Qualitative Dynamics". J. Theoretical Biol. 54 pp.85-107. Glass, L., Pasternack, J.S. (1978) "Prediction of limit cycles in mathematical models of biological oscillations". Bulletin of mathematical biology 40 pp.27-44. Heidke, K.R., Shutze-Kremer, S. (1998) "Design and implementation of a qualitative simulation model of λ-phage infection". Bioinformatics. 14(1) pp.81-91. Ironi, L., Stefanelli, M. (1995) “Generating explanations of pathophysiological system behaviors from qualitative simulation of compartmental models” in Barahona, Stefanelli, Wyatt (eds.), Lectures Notes in Artificial Intelligence, 934, Springer, pp. 115--126. Kay, H. (1998) "SQSIM: a simulator for imprecise ODE models". Computers and Chemical Engineering 23(1) pp.27-46.

18

Keisler, H.J. (1994) "The hyppereal line". In Real Numbers, Generalizations of the Reals, and Theories of Continua, ed. by P. Erlich, Kluwer Academic Publishers, pp. 207-237. Kuipers, B.J. (1986) "Qualitative Simulation". Artificial Intelligence 29 pp.289-338 Mc Adams, H., Arkin, A. (1997) "Stochastic mechanisms in gene expression". Proc. Natl. Acad. Sci. USA 94 pp. 814-819. Plahte, E., Mestl, T., Omholt, S. (1998) "A methodological basis for description and analysis of systems with complex switch-like interactions". J. math. Biol. 36 pp. 321348. Robinson, A. (1974) "Non Standard Analysis". Princeton Univ. Press (Reissued paperback 1996, ISBN 0-691-04490-2). Shults, B., Kuipers, B.J. (1997) "Proving properties of continuous systems: qualitative simulation and temporal logic". Artificial Intelligence 92 pp.91-129. Shults, B., Kuipers, B.J. (1997) "Proving properties of continuous systems: qualitative simulation and temporal logic". Artificial Intelligence 92 pp.91-129. Smolen, P.D., Baxter, A., Byrne, J.H. (1999) "Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems". Am. J. of Physiol. 277, pp. C777-C790. Smolen, P., Baxter, D., Byrne, J.H. (2000) "Modeling Transcriptional Control in Gene Networks - Methods, Recent Results and Future Directions". Bull. of Mathematical Biology 62 pp.247-292. Snoussi, E.H., Thomas, R, (1993) "Logical identification of all steady states. The concept of feedback loop characteristic state", Bull. Math. Biol. 55 pp. 973-991. Thomas, R., d’Ari, R. (1990) "Biological Feedback". CRC Press. Thomas, R., Thieffry, D., Kauffman, M. (1995) "Dynamic behaviour of biological regulatory networks-I Biological role of feedback loops and practical use of the concept of the loop-characteristic state". Bull. Math. Biol. 57 pp.247-276. Zhao, F. (1994) "Extracting and Representing Behaviors of Complex Systems in Phase Space". Artificial Intelligence 69(1,2) pp.51-92.

19