statistical properties of random qualitative simulations with infinitesimal and uncertain values

Eric Fimbel

Technical report RT020409

Department of Electrical Engineering

École de Technologie Supérieure

April 2002

statistical properties of random qualitative simulations with infinitesimal and uncertain values Eric Fimbel Department of Electrical Engineering École de Technologie Supérieure Abstract.- Qualitative simulations are useful whenever the experimental evidence about a system is insufficient for determining quantitative equations. Random qualitative simulations (Fimbel, 2002) produce qualitative behaviors, i.e. sequences of states defined by means of qualitative values (q-values), e.g., null, infinitesimal, finite. At each step of a simulation, the next state is randomly chosen. Simulations are repeated in order to determine statistical properties of the variables and/or the behaviors. This paper focuses on the statistical properties of free simulations, i.e. simulations of unconstrained (free) variables. It is shown that the q-values of free variables follow bipolar distributions, where null and infinitesimal q-values have low probabilities. Bipolar distributions are found 1) in idealized simulations, seen as homogeneous Markovian processes, and 2) in real simulations, performed with an experimental software. This results provides a simple criterion for detecting significant statistical properties, namely that their probability of occurence is higher than in free simulations.

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I. INTRODUCTION I.1 Qualitative computer models 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 to be positive and decreasing applies to a broad range of physical variables and situations and can be observed without precise, quantitative measurement. 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 a scientific domain is « mature » (Smolen, 2000), accurate quantitative models can be elaborated because both the knowledge and the experimental data are abundant and consistent. These models should replace their qualitative counterparts, whose predictions are less precise. Qualitative computer models come from the artificial intelligence community, and originate in frameworks for common-sense (naive) reasoning, such as qualitative physics (de Kleer, 1984) and qualitative process theory (Forbus, 1984). Qualitative computer models have also been identified to personal cognitive models (Chandra, 1995), which could explain why they are more explicative than numerical models. However, whatever their fundamental nature is, qualitative computer models are useful scientific tools. For instance, such models have been developed and applied by the mathematical biology community for years. Because of the switch-like properties of gene expression and enzymatic activity, biological control systems can be modeled by means of equivalent logical networks (Glass, Kaufmann, 1973, Glass, 1975, Thomas, d’Ari, 1990, Thomas, Thieffry 1995) and/or piecewise linear differential equations (Glass, Pasternack, 1978) in order to predict qualitative properties of their dynamics. I.2 Qualitative simulation Qualitative simulation is a sub-field of qualitative computer modeling, aiming to produce the possible behaviors of a system. In this approach, general properties such as the existence and stability of equilibriums and limit cycles, the trajectory flows, the geometry of the basins of attraction, etc., are determined from the qualitative behaviors produced by the simulation process. On the other hand, such properties can be obtained directly from the analysis of the equations of the model and/or numerical simulations (Doedel, 1981, Zhao, 1994), but this requires quantitative information about the real system. When such information is not available, the model must be written in a simplified manner to allow non-trivial analysis, for instance using threshold functions and assuming asynchronous transitions (Glass, Kauffman, 1973), piecewise linear approximations (Sacks, 1990), or generic sigmoid functions (Plahte, 1998). Simulations can be done by means of general tools such as Envision (de Kleer, 1984), QSIM (Kuipers, 1986), QPE (Forbus, 1990) and SIMGEN (Forbus, Falkenheimer, 1990), more or less adapted for specific applications (Ironi, 1995, Heidke, 1998). Besides, simulations can be done by means of domain-specific tools (de Jong, 2001).

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The result of a simulation is a qualitative behavior i.e. a sequence of qualitative states (q-states), which are coarse descriptions of significant variables and their variations, typically under the form of qualitative values (q-values), e.g., positive, null. Each q-state corresponds to a qualitative domain in the phase space, i.e., the 2ndimensional 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 (q-transition) occurs any time that a variable or a derivative crosses some threshold. For instance, a q-transition for some variable x could correspond to the sequence: negative, null, positive (here, the threshold is zero). As a result, a new q-state is reached, and the simulation goes one step ahead. I.3 Qualitative transitions Because the system may be anywhere within a qualitative domain, in the general case, it is impossible to know which transition will happen next, i.e., which of the variables and their derivatives will cross 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 discarded a priori. The number of theoretically possible transitions increases exponentially with the number of variables of the model. In Envision (de Kleer, Brown, 1984), the set of possible transitions is defined by means of rules, assuming implicitely that the variables vary continuously, and that positive and negative feedbacks loops give well defined patterns of behaviors. However, a causal analysis must be performed for detecting such loops from the model. In QSIM and its derived systems (Kuipers, 1986), the transitions are also defined by means of rules, but these rules depend upon the nature of the current qualitative state. Complementary heuristics filter the possible transitions. For instance, identical states (the so-called Corresponding Values) lead to identical transitions. In QPE and SIMGEN (Forbus, Falkenheimer, 1990), the problem of defining the qualitative transitions is left to the modeler, which must write unambiguous state-transition procedures. All these approaches partially control the combinatorial explosions, but 1) they may also discard some unlikely, but still possible transitions, and 2) they may allow unrealistic transitions that cannot be detected formally. For instance, some transitions may violate general mathematical or physical rules; in this case, they can be filtered by means of additional heuristic controls. For instance, some transitions allowed by QSIM can be discarded because they violate the l’Hopital’s rule for continuous variables (Cem Say, 1998). Unrealistic transitions can also be detected by means of domain dependant knowledge. For instance, hybrid qualitative-quantitative simulators (Berleant, 1989, 1997, Forbus, Falkenheimer, 1990, Kay, 1998) discard many unrealistic transitions, because of the strong constraints introduced by quantitative information. To a lesser extend, the addition of temporal constraints to the model have the same effect (Shults, 1997, Brajnik, Clancy, 1997). However, all these approaches may require more information than what is generally available in the early stages of the study of a system.

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I.4 Random qualitative simulations As an alternative to the foregoing approaches, a random qualitative simulation algorithm has been proposed (Fimbel, 2002). The variables and their derivatives are represented within a qualitative algebra handling infinitesimal and uncertain q-values, e.g., null or infinitesimal, negative or positive, etc. At each step of a simulation, a unique qualitative transition equation (QTE) produces all the next possible q-states, and one of them is randomly chosen. In practical terms, the QTE is used to calculate the q-value of each variable and derivative after some time interval δt. whenever this q-value is uncertain, one of the possibilities is chosen randomly. For instance, if X is finite positive and dX/dt is finite negative, the QTE produces the uncertain q-value finite negative or ... or finite positive for X, and one of these elementary q-values is chosen, e.g. at the next step X will be finite negative. This non-deterministic algorithm generates a behavior at a low computational cost. However, individual behaviors are mere possibilities rather than predictions, because they may be spurious, which means that they are unrealistic, but they cannot be formally discarded from the information contained in the model. Notice that spurious behaviors can occur even in the absence of spurious transitions. For instance, for a damped spring mass system, a behavior containing the wrong number of oscillations is spurious, in spite of the fact that none of its individual transitions can be discarded without a quantitative, precise information. As individual behaviors are not significant, the simulation process is repeated from the same initial conditions in order to generate a set of possible behaviors. This set is considered accurate provided that it contains the real behaviors of the system. As the algorithm does not discard any possible behavior, its p-correctness (i.e. the property to be accurate with probability higher than p) is granted for any level p, provided that the number of repetitions is sufficient (Brassard, 1996). This rests importance to the problem of spuriousness, inasmuch as spurious behaviors are sufficiently infrequent. I.5 Statistical properties of qualitative behaviors From a qualitative point of view, the dynamics of a system can be characterized by means of significant patterns of behavior, such as uniform convergence, damped oscillations, undamped oscillations, convergence or divergence from an equilibrium, etc. Whenever the initial conditions may change, or the system is open or undeterministic, the probability of occurrence of different patterns of behavior give a useful characterization of the system. In (Brajnik, 1997), these probabilities are estimated from repeated numerical simulations. The patterns of behaviors are directly extracted from the simulations and their frequencies are estimated when the parameters of the model vary randomly following some given probability distributions. This approach assumes that the frequencies of the patterns of behaviors in the simulations correspond their probability of occurrence in the real system.

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In the case of random qualitative simulations, each pattern of behavior corresponds some property of the variables and their derivatives. For instance, a pattern of uniform convergence for X can be characterized by means of the following condition: dX/dt has the same sign for an arbitrary number of steps, then dX/dt remains null. It will be admitted without further justification that any pattern of behavior can be defined from the q-values of some variables and derivatives by means of a logical formula. The correspondence between the simulated and the real probability distributions does not hold for random qualitative simulations, because the qualitative model and the simulation algorithm contain no statistical or quantitative information about the real system. It cannot even be assumed that ranking the properties by order of decreasing frequency in the simulations produces a qualitative probability, in the sense that it is compatible (even approximately) with their real probabilities (Vogel, 1995). In a similar fashion, it cannot be assumed that ranking the quantifiable properties (such as degrees) by order of decreasing average value in the simulations produces a qualitative expectation (Clark, 2000). However, it is possible to detect the significant properties by means of their frequency in repeated simulations, under two conditions: 1) constrained simulations, performed with the desired model M, and 2) free simulations,performed without any constraints. As any difference between these two conditions is due to the model M, the significant properties are those whose frequency in constrained simulations is above their frequency in free simulations. This criterion allows discarding all the non-significant properties before any further analysis. The rest of this paper deals with the most elementary property, namely the property for a variable to take a given q-value. The extension of the results to more complex properties will be presented in the discussion. I.6 Distributions of the qualitative values of free variables Because a free variable is unconstrained, it should have no preferred values during repeated simulations. Hence, with an infinite number of simulations, its numerical value should follow a uniform probability distribution. For the same reason, its q-value is expected to follow a bipolar probability distribution, i.e. null and infinitesimal q-values should have low probabilities. It is worth noting that the correspondence between the distributions of qualitative and numerical values of a variable is intuitive rather than formal. The probability distributions of numerical values are defined on real numbers, which are null of finite but never infinitesimal. On the other hand, infinitesimal qualitative values do not correspond to real numbers: they correspond to hyper-real infinitesimal numbers, in the sense of Non Standard Analysis (Robinson, 1964). Informally speaking, infinitesimal numbers cannot be represented on the same line than finite real numbers, because they are “too small to be seen”.

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As a first step, the foregoing hypothesis will be checked in the case of idealized simulations. Such simulations can be considered as homogeneous Markovian processes, where the state of the system is the q-value of a free variable X. A transition probability matrix can be easily determined, provided that the q-value of the derivative dX/dt follows a fixed probability distribution. This kind of process has a unique stationary distribution, which corresponds to the asymptotic distribution of the q-values of X. Several probability distributions of dX/dt are examined, and all of them lead to a bipolar probability distribution of the q-values of X. The second step of this work is the verification of the hypothesis in practical conditions, because the effects of the simulation algorithm itself on the distributions of qvalues must be assessed. To that end, the relative frequencies of the q-values during repeated simulations of a free variable X have been measured directly, with an experimental simulation software. This empirical validation is conclusive: the frequencies of the q-values follow bipolar distributions. This paper is organized as follows. The qualitative algebra is presented in section II. The random qualitative simulation algorithm is presented in section III. The experimental simulation software is presented in section IV. Qualitative probability distributions are presented in section V. Asymptotic qualitative probability distributions of variables during ideal simulations are presented in section VI. The main result, namely the bipolarity of the asymptotic qualitative distributions of the free variables during ideal simulations, is presented in section VII. The empirical verification of the result is presented in section VIII. The application of this result for the characterization of significant statistical properties of repeated simulations is discussed in section IX. II. QUALITATIVE ALGEBRA The qualitative algebra introduced in (Fimbel, 2002) is founded on Non Standard Analysis (NSA). NSA provides a simple way of representing infinitesimals under the form of hyper-real numbers (Robinson 1974). Around each real number, there are infinitely close hyper-real numbers that can be observed through an "infinite microscope" (Keisler, 1994). At the other extremity, there exist infinite values. The qualitative value of the variables can be: null, infinitesimal, finite, unbounded or infinite (positive or negative). Moreover, the qualitative value can be a combination of these cases, e.g. null, infinitesimal positive, or finite positive. The qualitative algebra is now briefly described.

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Numerical values. The numerical values are hyper-real numbers, *R, i.e., the real numbers, augmented with infinitesimals and infinite numbers. The usual arithmetic operations (i.e., +, -, *, /) are defined on the sets of values. For instance, E1+E2 = {x+y | x, y ∈ E1xE2 }. Qualitative properties (q-properties). Any set of hyper-real numbers can be described by means of 9 predefined qualitative properties. The set of q-properties, Qp, is given in Table 1. Table 1. Qualitative properties. X stands for any set of hyper-real numbers. 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 = ±∞. On the other hand, the inverse of infinitesimal is unbounded. Qualitative values (q-values). The qualitative values are defined as set of qproperties, Qv= P(Qp). As there are 9 q-properties, there are 29 = 512 q-values. The elementary q-values correspond to the singletons, the uncertain q-values (also called complex q-values) contain several elements, and the universal q-value "?" is the union of all the properties of Qp. Qualitative descriptions (q-descriptions). The q-description of a set X is a q-value defined as the set of the q-properties that some subset of X verifies. For instance, the qdescription of X=[0,1], is {0, 0+, +} because X contains zero, infinitesimal and finite positive numbers. It is worth noting that if X is the set of possible values of some variable v, the more complex the q-description of X, the higher the uncertainty on v. When the qdescription is the universal q-value, nothing is known about v. Operations on q-values. 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 Tables 2, 3). Then, 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}.

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Table 2. 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 3. 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

III. THE RANDOM QUALITATIVE SIMULATION ALGORITHM The simulation process starts from a q-state corresponding to the initial conditions of the system. It stops whenever the first of the following conditions occurs: 1) a qualitative steady state is reached, which means that the qualitative description of the system cannot change any more; 2) a fixed number of steps is reached; or 3) an inconsistency occurs. In this case, the resulting behavior is discarded. At each step of the simulation, the transition is chosen as follows. 1) A qualitative time interval is fixed. 2) The next q-value of each variable is calculated by means of a general qualitative transition equation (QTE). 3) The next state of the free derivatives is randomly chosen. 4) The constraints of the model are evaluated. In case of inconsistency, a new trial is done with another qualitative time interval. 5) If the new state contains some uncertain q-values, one of them is randomly chosen and reduced, i.e. it is replaced by one of its elementary q-values. 6) The constraints are applied again, because the reduction can introduce inconsistencies. If this is the case, the uncertain value is restored. Steps 5 and 6 are repeated until no further reduction is possible. In the rest of this section, the main elements of the simulation algorithm are briefly presented. Qualitative model. The model used by the simulation is composed of a set of variables X1..Xn, a set of differential equations dXi/dt = fi( ... ), and a set of constraints upon the variables and the derivatives, e.g. (X1+X2>=1), (X>=0). Complementary definition. The derivatives dXi/dt that are not defined by means of some equation dXi/dt = ... are said to be free derivatives.

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Qualitative state A qualitative state of the system is defined by means of the q-values of the variables and their derivatives. The q-values must be such that the constraints are verified. For instance, if the model contains the constraint (X > 1), the q-value of X cannot be infinitesimal. Qualitative time step. It is the time interval during which the system remains in a given q-state. Its numerical value is undetermined, and it is only defined by means of its q-value: infinitesimal positive, finite positive or unbounded positive. Qualitative transition equation (QTE). This equation defines the next possible q-value of a variable:

qXn+1 ⊆ qXn + qX′n,n+1 * q∆tn

(1)

where qXn and qXn+1 are the q-values of X at steps n and n+1, q∆tn is the q-value of the time interval at step n, and qX'n,n+1 is the union of the possible q-values of the derivative X' = dX/dt during the time interval ∆tn. Notice that even if X' remains finite at any time during ∆tn, it can eventually become infinitesimal or unbounded. For instance, on the time interval [0,1] (∆t = 1), X’(t) = 1/(1-t) is finite positive at any time, but it is unbounded when t is infinitely close to 1. Formally: 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.

(2)

The qualitative transition equation as a binary operator. For each of the possible time intervals (infinitesimal, finite, unbounded), the QTE is equivalent to a binary operator QTEδt(X, dX/dt). These operators are presented in the following tables. Table 4. The qualitative transition equation for an infinitesimal time increment QTE0+(qi,qj).. qi...qj represents the set of the q-properties qk, k in [i,j]. dx/dt→ x↓ -∞ -00 0+ + ++ +∞

-∞

--

-

0-

0

0+

+

++

+∞

-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞...+∞

-∞ --------...++ +∞

-∞ --...+ ++ +∞

-∞ -000-...0+ + ++ +∞

-∞ -00 0+ + ++ +∞

-∞ -0-...0+ 0+ 0+ + ++ +∞

-∞ --...+ + + + + ++ +∞

-∞ --...++ ++ ++ ++ ++ ++ ++ +∞

-∞...+∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞

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Table 5. The qualitative transition equation for a finite time increment QTE+(qi,qj). "qi...qj" represents the set of the q-properties qk, k in [i,j]. dx/dt→ x↓ -∞ -00 0+ + ++ +∞

-∞

--

-

0-

0

0+

+

++

+∞

-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞...+∞

-∞ --------...++ +∞

-∞ --...+ ++ +∞

-∞ -000-...0+ + ++ +∞

-∞ -00 0+ + ++ +∞

-∞ -0-...0+ 0+ 0+ + ++ +∞

-∞ --...+ + + + + ++ +∞

-∞ --...++ ++ ++ ++ ++ ++ ++ +∞

-∞...+∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞

Table 6. The qualitative transition equation for an unbounded time increment QTE++(qi,qj).. "qi...qj" represents the set of the q-properties qk, k in [i,j]. dx/dt→ x↓ -∞ -00 0+ + ++ +∞

-∞

--

-

0-

0

0+

+

++

+∞

-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞...+∞

-∞ --------...++ +∞

-∞ --...+ ++ +∞

-∞ -000-...0+ + ++ +∞

-∞ -00 0+ + ++ +∞

-∞ -0-...0+ 0+ 0+ + ++ +∞

-∞ --...+ + + + + ++ +∞

-∞ --...++ ++ ++ ++ ++ ++ ++ +∞

-∞...+∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞ +∞

Constraint verification and inconsistencies. The constraints of the model are verified when the q-values of the variables have been produced by the QTE. The q-values are reduces until they are compatible with the constraints. For instance if the q-value of X is null, infinitesimal or finite positive, then the application of the constraint: (X>1) reduces the q-value to finite positive. Whenever a q-value becomes empty, an inconsistency occurs. For instance, X negative and (X>1) produce an inconsistency, Qualitative steady state. It is a q-state such that no transition leads to another qstate. A qualitative steady state is not always a steady state. For instance: x finite positive and dx/dt infinitesimal is a qualitative steady state, but not a steady state (a real steady state occurs only when all the derivatives are null).

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IV. EXPERIMENTAL SIMULATION SOFTWARE The prototype of simulation software has been implemented on a personal computer. In addition to the foregoing algorithm, this software uses several probabilistic heuristics in order to produce more significant results. These heuristics tend to produce preferentially some kind of behaviors, but as they are only probabilistic, any kind of behavior can occur. They are the following. Progressive variation of free derivatives. This heuristic tends to produce progressive variations of the q-values of the free derivatives. The choice of the q-value of a free derivative is a 2-steps process. 1) A q-value close from or identical to the previous one is randomly chosen, and 2) if step 1 fails, then any q-value is randomly chosen. Restriction a posteriori of the free derivatives. When the q-value of X at step n+1 qXn+1 is chosen, the q-value of its derivative at step n qX'n is restricted to its useful part, i.e. all its elementary q-values that cannot give qXn+1 from qXn are removed. For instance, if qXn is positive, qX'n = negative, null or positive, and qXn+1 is negative, qX'n is reduced to negative. Close state heuristic. This heuristic tries to produce q-states that change progressively. The reduction of the q-values of the variables qXn+1 and derivatives qX'n+1 is a 2-steps process. 1) One of the elementary q-values of qXn+1 (or qX'n+1), close from qXn (or qX'n), is randomly chosen, and 2) if step 1 fails, then any elementary q-value of qXn+1 (or qX'n+1) is randomly chosen. Progression heuristic. This heuristic tends to avoid trivial stationary behaviors. Whenever the result of the transition is identical to the current state, a new transition is done. This process stops whenever the first of the following conditions occurs: 1) a different state is produced; 2) the number of trials exceeds a given bound. In this case, the result is a qualitative steady state and the simulation stops. Time step heuristic. This heuristic tends to alternate finite and infinitesimal time steps. The q-values of the time step are tried in the following order: 1) if the previous time step was infinitesimal, try finite, infinitesimal, unbounded, 2) if the previous time step was finite, try infinitesimal, finite, unbounded. Unbounded time steps are tried when anything else failed, and they stop the simulation. Asymptotic qualitative steady state. After an unbounded time step, this heuristic tries to produce a possible asymptotic qualitative steady state. The new q-state is reduced to the largest qualitative steady state that it contains. For instance, if qX'n is finite positive, and Xn is not negative, after an unbounded time step, qXn will be reduced to unbounded positive. Visualization of the results. At the end of the simulation, the variables can be visualized as qualitative time plots, showing their q-value as a function of time. In order to represent all the q-values on the same axis, they correspond to exponentially increasing dots, from null to infinite: {0, ±4, ±16, ±64, ±256}. The same representation is used for the time intervals (i.e. finite intervals are 4 times larger than infinitesimals). These diagrams are only informal representations; hence they contain no quantitative information about amplitude or time (see Figure 1).

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Figure 1. 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.

V. QUALITATIVE PROBABILITY DISTRIBUTIONS Because there exist only nine elementary q-values, a probability function can be easily defined on the space of the q-values Qv. However, there exist no straightforward relationship between the probability distributions on the q-values and their numerical counterpart, because 1) a numerical probability distribution is classically defined on a subset of the real line, and 2) the qualitative values infinitesimal, unbounded and infinite cannot be represented in the real line. However there exist some informal correspondences between families of qualitative and numerical probability distributions. They are briefly presented in the rest of this section. Qualitative probability distributions. They are defined by means of any function pq on the q-properties Qp (i.e. the elementary q-values) such that: ∑qi∈Qp pq(qi) = 1

(3)

The probability of a q-value qv is defined as: pq(qv) = ∑qi∈qv pq(qi)

(4)

Uniform qualitative probability distributions. They are a family of qualitative probability distributions where a contiguous subset of q-properties have a constant probabilities. The distribution pqb is uniform if: ∃ k,l ∈ [0,8] |

j ∈ [k,l] ⊃ pqb(j) = 1/(l-k+1) j ∉ [k,l] ⊃ pqb(j) = 0

(5)

Complementary definition. Symmetrical uniform qualitative probability distributions are centered on the null q-value. There are 4 non-trivial symmetrical uniform qualitative probability distributions, defined as follows:

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Table 7. Symmetrical uniform qualitative probability distributions. pi -∞ -00 0+ + ++ +∞

Psu∞ 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

Psu∞

Psu++ 0 1/7 1/7 1/7 1/7 1/7 1/7 1/7 0

Psu+ 0 0 1/5 1/5 1/5 1/5 1/5 0 0

Psu0+ 0 0 0 1/3 1/3 1/3 0 0 0

Psu++

Psu+

Psu0+

Figure 2. Symmetrical uniform qualitative probability distributions.

Remark. Uniform qualitative probabilities do not correspond to uniform numerical probability densities. Rather, they favor "small" numerical values. The following figure illustrates this correspondence. The horizontal axis is a schematic representation of *R, where the q-values correspond to intervals of exponentially increasing size, from null to infinite: {[0,0], ±]0,4], ±]4,16], ±]16,64], ±]64,256]}.

Figure 3. Informal representation of a numerical probability density function (left) corresponding to the uniform qualitative probability distribution Psu∞ (right). Horizontal: regions corresponding to the elementary q-values.

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Bipolar qualitative probability distributions. In these distributions, the extreme qproperties have a higher probability than the others. A bipolar qualitative probability distribution pqb verifies the following condition: ∃ k,l ∈ [0,8] | j ∈ ]k,l[ ⊃ pqb(j) < max(pqb(k), pqb(l)) j ∉ [k,l] ⊃ pqb(j) = 0

(6)

Complementary definition. Strictly bipolar qualitative probability distributions are those that are 1) symmetrical, and 2) only their extreme q-properties have non-null probabilities. There are only 4 strictly bipolar qualitative probability distributions, defined in the following table. Table 8. Strictly bipolar qualitative probability distributions. pi -∞ -00 0+ + ++ +∞

Psb∞ 1/2 0 0 0 0 0 0 0 1/2

Psb∞

Psb++ 0 1/2 0 0 0 0 0 1/2 0

Psb+ 0 0 1/2 0 0 0 1/2 0 0

Psb0+ 0 0 0 1/2 0 1/2 0 0 0 Psb+

Psb++

Figure 4. Strictly bipolar qualitative probability distributions.

15

Psb0+

Remark. A subset of the bipolar symmetrical qualitative probability distributions correspond to the uniform probability distributions on the numerical values. The following figure illustrates this correspondence. The representation is the same as in Figure 1.

Figure 5. Left: informal representation of a uniform numerical probability density function. Horizontal: regions corresponding to the elementary q-values. Right: corresponding qualitative probability distribution.

16

VI. ASYMPTOTIC QUALITATIVE PROBABILITY DISTRIBUTIONS IN IDEAL SIMULATIONS This section focuses on the asymptotic probability distributions of the q-values of a variable X, i.e. the distributions followed by X during infinitely large simulations. These asymptotic distributions are studied in the case of ideal simulations, and only for free variables. Such a simulation can be considered a homogeneous Markovian process, where the state of the system is the q-value of the free variable X. The transition probabilities for X are constant whenever dX/dt follows a constant qualitative distribution, and the time step is fixed. A homogeneous process leads to a unique probability distribution for the q-values of X, called the asymptotic stationary distribution. Ideal simulation. For such a simulation, we assume the following. 1) The time step δt has the same q-value during all the simulation. 2) There exist some constant qualitative probability distribution pq. 3) The q-values of the free variables are chosen following the probability distribution pq. 4) The uncertain q-values are randomly reduced, following the probability distribution pq. . Asymptotic qualitative probability distribution. The asymptotic qualitative probability distribution of a variable X of the model during a simulation, p∞x, is defined as a probability distribution such that: p∞x (qi) = lim s->∞ ( ∑t=1..s (qi ∈ qXt ) )/ ( ∑t=1..s | qXt | )

(7)

where qi is the i-th elementary q-value, (qi ∈ qXt ) is 1 if the condition is true, 0 otherwise, and |qXt| is the number of elementary q-values contained in qXt. Free simulation distribution (FSD) of a variable X. It is the asymptotic qualitative probability distribution of X when the model is empty. Complementary definition. The free simulation probability (FSP) of the q-value q for a variable X is the probability of the event (q ∈ qX) in the free simulation distribution of X. Transition probability matrix. With the foregoing simplifications, the probability for X to reach any q-value qj at the next step, given its current q-value qi, is constant, and it can be calculated from the qualitative transition equation as follows: pδt (qXn+1=qj|qXn=qi)=∑k=0..8pq(qk).(qj∈QTEδt(qi, qk)).pq(qj)/pq(QTEδt(qi, qk)) .

(8)

where the term pq(qk) stands for the constant probability that the derivative is qk (i.e. qX'n=qk ), the expression (qj∈QTEδt(qi, qk)) is 1 if the result of the transition contains qj and 0 otherwise, and the term pq(qj)/pq(QTEδt(qi, qk)) stands for the probability that the random reduction of the q-value produced by the QTE gives qj. As an example, the transition probability matrix is given for a finite time step, and the symmetrical uniform qualitative probability distribution Psu∞. This matrix is calculated from Table 4.

17

Table 9. Matrix M0+su∞ of transition probabilities for a free variable, with an infinitesimal time step and a uniform qualitative probability distribution Psu∞. Lines: initial q-values; columns: final qvalue. This matrix is calculated from equation (8) and QTE0+(qi,qj), given in Table 4. final → initial ↓ -∞ -00 0+ + ++ +∞

--

-∞

0.901 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.012

-

0.012 0.587 0.111 0.074 0.074 0.044 0.037 0.032 0.012

0.012 0.032 0.481 0.074 0.074 0.044 0.037 0.032 0.012

0-

0

0.012 0.032 0.037 0.378 0.185 0.081 0.037 0.032 0.012

0.012 0.032 0.037 0.081 0.111 0.081 0.037 0.032 0.012

0+

0.012 0.032 0.037 0.081 0.185 0.378 0.037 0.032 0.012

+

0.012 0.032 0.037 0.044 0.074 0.074 0.481 0.032 0.012

++

0.012 0.032 0.037 0.044 0.074 0.074 0.111 0.587 0.012

+∞

0.012 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.901

Asymptotic stationary distribution of a variable X. This distribution is unique for each homogeneous Markovian process. It is the only probability distribution for X that remains constant, i.e. that is not altered at the following step. It is the solution of the following vector equation: (9)

P* = M P*

where P* is the vector representing the probabilities of each q-value, and M is the probability transition matrix. Determination of P*. This can be done as follows. 1) Find the eigenvalues of M, 2) find the eigenvector V corresponding to the eigenvalue “1", 3) divide V by the sum of its values (in order to get a probability distribution, which sum is 1). VII. CALCULUS OF THE ASYMPTOTIC STATIONARY DISTRIBUTION OF A FREE VARIABLE With the foregoing simplifications, the transition probability matrix and the corresponding asymptotic stationary distribution have been calculated for the three possible time steps (infinitesimal, finite, unbounded), and for ten qualitative probability distributions, namely the four symmetrical uniform distributions and the four symmetrical bipolar distributions presented in the previous section, plus two random distributions, presented below. Table 10. Random qualitative probability distributions used for the experiment. pi -∞ -00 0+ + ++ +∞

R1 0.148 0.230 0.138 0.027 0.070 0.071 0.250 0.060 0.006

R2 0.166 0.137 0.127 0.208 0.020 0.033 0.059 0.103 0.147

18

R1

R2

Figure 6. Random qualitative probability distributions used for the experiments.

The 30 transition probability matrices were calculated in excel (r) from Tables 4, 5, 6 and equation (8). The eigenvalues and eigenvectors for each probability transition matrix M were calculated in Matlab (r). In each case, the eigenvalue closest from "1" (unitary) has been selected, and the probability distribution P* obtained from the corresponding eigenvector. In all cases, the unitary eigenvalue was always in the range 1±0.1 (the difference results numerical approximations in the matrices M and in the algorithms of Matlab). Results. The stationary distributions of the q-values of the free variable are bipolar in all the cases. Moreover, the following points can be observed. Range of q-values. The range of allowed q-values has a strong incidence on the shape of the stationary distribution (see diagrams below). Uncertain q-values. The stationary distribution is never strictly bipolar, because of the presence of uncertain q-values (no q-value has a null probability at any step). Symmetry of the stationary distribution. The stationary distribution is symmetrical whenever the probability distribution on the q-values is; else it is asymmetrical (as in the case of the random distributions). Time step. The time step has almost no effect on the stationary distribution. The differences may even be due to numerical roundings. The results are now presented in the following tables and the following figures for each time interval. Table 11. Stationary probability distributions of the q-values of a free variable resulting from each qualitative probability distribution and an infinitesimal time step. δt (0+) -∞ -00 0+ + ++ +∞

Psu∞ 0.362 0.053 0.038 0.036 0.023 0.036 0.038 0.053 0.362

Psu++ 0.000 0.230 0.126 0.112 0.064 0.112 0.126 0.230 0.000

Psu+ 0.000 0.000 0.285 0.175 0.080 0.175 0.285 0.000 0.000

Psu0+ 0.000 0.000 0.000 0.429 0.143 0.429 0.000 0.000 0.000

Psb∞ 0.360 0.044 0.040 0.040 0.033 0.040 0.040 0.044 0.360

19

Psb++ 0.000 0.231 0.126 0.108 0.070 0.108 0.126 0.231 0.000

Psb+ 0.000 0.000 0.285 0.175 0.080 0.175 0.285 0.000 0.000

Psb0+ 0.000 0.000 0.000 0.429 0.143 0.429 0.000 0.000 0.000

R1 0.942 0.011 0.007 0.006 0.004 0.006 0.006 0.009 0.009

R2 0.401 0.063 0.042 0.044 0.024 0.028 0.034 0.041 0.323

Psu∞

Psu+

Psu++

Psu0+

Figure 7. Stationary probability distributions of the q-values of a free variable (down) resulting from symmetrical uniform qualitative probability distributions (up) and an infinitesimal time step. Psb∞

Psb+

Psb++

Psb0+

Figure 8. Stationary probability distributions of the q-values of a free variable (down) resulting from strictly bipolar qualitative probability distributions (up) and an infinitesimal time step. R1

R2

Figure 9. Stationary probability distributions of the q-values of a free variable (down) resulting from random qualitative probability distributions, and an infinitesimal time step (up).

20

Table 12. Stationary probability distributions of the q-values of a free variable resulting from each qualitative probability distribution and a finite time step. δt (+) -∞ -00 0+ + ++ +∞

Psu∞ 0.362 0.073 0.028 0.028 0.020 0.028 0.028 0.073 0.362 Psu∞

Psu++ 0.000 0.323 0.077 0.077 0.049 0.077 0.077 0.323 0.000

Psu+ 0.000 0.000 0.285 0.175 0.080 0.175 0.285 0.000 0.000

Psu0+ 0.000 0.000 0.000 0.429 0.143 0.429 0.000 0.000 0.000

Psb∞ 0.360 0.051 0.036 0.037 0.032 0.037 0.036 0.051 0.360

Psb++ 0.000 0.356 0.059 0.060 0.050 0.060 0.059 0.356 0.000 Psu+

Psu++

Psb+ 0.000 0.000 0.285 0.175 0.080 0.175 0.285 0.000 0.000

Psb0+ 0.000 0.000 0.000 0.429 0.143 0.429 0.000 0.000 0.000

R1 0.942 0.017 0.004 0.004 0.003 0.005 0.005 0.010 0.009

R2 0.401 0.085 0.030 0.035 0.021 0.023 0.026 0.057 0.323

Psu0+

Figure 10. Stationary probability distributions of the q-values of a free variable (down) resulting from symmetrical uniform qualitative probability distributions (up) and a finite time step. Psb∞

Psb+

Psb++

Psb0+

Figure 11. Stationary probability distributions of the q-values of a free variable (down) resulting from strictly bipolar qualitative probability distributions (up) and a finite time step.

21

R1

R2

Figure 12. Stationary probability distributions of the q-values of a free variable (down) resulting from random qualitative probability distributions, and a finite time step (up). Table 13. Stationary probability distributions of the q-values of a free variable resulting from each qualitative probability distribution and an unbounded time step. δt (++) -∞ -00 0+ + ++ +∞

Psu∞ 0.362 0.071 0.029 0.027 0.023 0.027 0.029 0.071 0.362 Psu∞

Psu++ 0.000 0.301 0.089 0.079 0.063 0.079 0.089 0.301 0.000

Psu+ 0.000 0.000 0.287 0.164 0.098 0.164 0.287 0.000 0.000

Psu0+ 0.000 0.000 0.000 0.429 0.143 0.429 0.000 0.000 0.000

Psb∞ 0.360 0.052 0.036 0.035 0.033 0.035 0.036 0.052 0.360

Psb++ 0.000 0.346 0.065 0.062 0.056 0.062 0.065 0.346 0.000 Psu+

Psu++

Psb+ 0.000 0.000 0.287 0.164 0.098 0.164 0.287 0.000 0.000

Psb0+ 0.000 0.000 0.000 0.429 0.143 0.429 0.000 0.000 0.000

R1 0.942 0.016 0.004 0.004 0.004 0.005 0.005 0.011 0.009

R2 0.402 0.099 0.033 0.028 0.022 0.024 0.023 0.045 0.323

Psu0+

Figure 13. Stationary probability distributions of the q-values of a free variable (down) resulting from symmetrical uniform qualitative probability distributions (up) and an unbounded time step.

22

Psb+

Psb++

Psb∞

Psb0+

Figure 14. Stationary probability distributions of the q-values of a free variable (down) resulting from strictly bipolar qualitative probability distributions (up) and an unbounded time step. R1

R2

Figure 15. Stationary probability distributions of the q-values of a free variable (down) resulting from random qualitative probability distributions and an unbounded time step (up).

VIII. EMPIRICAL VERIFICATION WITH AN EXPERIMENTAL SIMULATION SOFTWARE With the foregoing software, 100 simulations of 1024 steps have been performed on a simple model containing only a free variable and its derivatives. The q-values of the variable and all its derivatives were progressively constrained: 1) totally free, i.e. in [-∞, +∞], 2) unbounded, i.e. in [--, ++], 3) finite, i.e. in [-, +], 4) infinitesimal, i.e. in [0-, 0+]. Notice that each level of constraint corresponds to a uniform qualitative probability distribution from Psu∞ to Psu0+. The initial conditions were totally free, which means that any variable started with the universal q-value. Table 14. The model and the constraints. (..), (.), (0.) are notations for qualitative comparators, i.e. respectively “in [--,++]”, “in [-, +]”, “in [0-, 0+]”. Variables X0 X1 X2

Equations

X1 = d/dt X0 X2 = d/dt X1

Constraints 1

Constraints 2

[-∞, +∞]

[--, ++] (..) X0 (..) X1 (..) X2 (..) d/dt X2

23

Constraints 3 [-, +] (.) X0 (.) X1 (.) X2 (.) d/dt X2

Constraints 4 [0-, 0+] (0.) X0 (0.) X1 (0.) X2 (0.) d/dt X2

In each case, the overall frequencies of the q-values have been calculated for the variable and its derivatives up to the third. Results. The results are presented in the following tables and they are shown graphically in Figure 16. Briefly stated, they are compatible with the hypothesis (bipolar distribution of q-values). Let us see separately each of the variables and derivatives. Variable X0. For all the sets of constraints, the variable shows clearly bipolar distributions of q-values. First derivative X1. It also shows a bipolar distribution. It is worth noting that, as it is declared in the model, it is considered a variable, and its q-value is determined by means of the QTE. Second derivative X2. It does not show a bipolar distribution. Rather, it follows distributions where null and infinitesimal q-values are more frequent than the extremes. These conservative distributions are the effect of the neighbor-state heuristic, which, in this case, tries to avoid discontinuous changes in the q-values of X0 and X1. As X1, X2 is considered a variable and its q-value is determined by means of the QTE. Third derivative dX2/dt. It is not declared in the model, hence it is considered a free derivative and its q-value is randomly chosen. However, it shows distributions where the null and the extreme q-values are slightly less frequent than the others. This corresponds to the progression heuristic, which tends to avoid steady states. These results lead to a practical rule for model writing: always declare the derivative of a variable in the model, even if it is free. For instance, if Xi is a significant variable, the model must contain a variable X’i such that X’i = dXi/dt. This rule avoids any side effect of the heuristics on the distribution of q-values of the variable Xi.

Table 15. Relative frequencies of the q-values of a free variable and its derivatives during simulations. All variables in [-∞, +∞]. -∞ -00 0+ + ++ +∞

X0 0.502 0.008 0.002 0.002 0.001 0.001 0.001 0.008 0.474

X1 0.475 0.019 0.007 0.006 0.003 0.005 0.009 0.022 0.453

X2 0.061 0.128 0.126 0.126 0.127 0.125 0.125 0.124 0.058

d/dt X2 0.044 0.137 0.122 0.137 0.122 0.137 0.121 0.137 0.044

24

Table 16.Relative frequencies of the q-values of a free variable and its derivatives during simulations. All variables in [--, ++]. -∞ -00 0+ + ++ +∞

X0 0.000 0.477 0.015 0.007 0.003 0.007 0.015 0.476 0.000

X1 0.000 0.409 0.049 0.034 0.016 0.032 0.048 0.412 0.000

X2 0.000 0.088 0.161 0.160 0.184 0.158 0.161 0.089 0.000

d/dt X2 0.000 0.159 0.125 0.153 0.124 0.153 0.126 0.160 0.000

Table 17. Relative frequencies of the q-values of a free variable and its derivatives during simulations. All variables in [-, +]. -∞ -00 0+ + ++ +∞

X0 0.000 0.000 0.485 0.016 0.008 0.015 0.476 0.000 0.000

X1 0.000 0.000 0.364 0.120 0.048 0.110 0.358 0.000 0.000

X2 0.000 0.000 0.131 0.212 0.321 0.207 0.128 0.000 0.000

d/dt X2 0.000 0.000 0.187 0.225 0.176 0.225 0.187 0.000 0.000

Table 18. Relative frequencies of the q-values of a free variable and its derivatives during simulations. All variables in [0-, 0+]. -∞ -00 0+ + ++ +∞

X0 0.000 0.000 0.000 0.466 0.079 0.455 0.000 0.000 0.000

X1 0.000 0.000 0.000 0.419 0.169 0.412 0.000 0.000 0.000

X2 0.000 0.000 0.000 0.257 0.487 0.256 0.000 0.000 0.000

d/dt X2 0.000 0.000 0.000 0.370 0.260 0.370 0.000 0.000 0.000

25

Constraints 1

Constraints 2

[-∞, +∞]

[--, ++]

Constraints 3 [-, +]

Constraints 4 [0-, 0+]

X0

X1 = dX0 /dt

X2 = d2X0 2 /dt

d/dt X2 = d3X0 /dt3

Figure 16 . Relative frequencies of a free variable and its derivatives during repeated simulations with different sets of constraints.

IX. DISCUSSION In both ideal and real simulations, the q-values of the free variables follow bipolar distributions, provided that the model is written correctly, i.e. that the derivatives of all the significant variables Xi are declared in the model by means of equations X’i=dXi/dt. Henceforth we can assume that any deviation from a bipolar qualitative probability distribution results from the initial conditions and the constraints of the model. In the case of elementary properties using q-values e.g., X is positive, X is infinitesimal, etc., it is possible to discard immediately the non-significant properties by comparing their probability in constrained simulations (CSP) and in free simulations (FSP): only the properties whose CSP is above their FSP are significant. However, this result is useful but limited, and several issues remain open for interpreting repeated simulations. The first issue is the extension of this result to more complex properties such as patterns of behavior. Assuming that any property P can be expressed by means of the qvalues of variables and derivatives, the constrained simulation probability of P can be compared to its free simulation probability. Depending on the property, the FSP can be determined in two ways: 1) stationary distributions in ideal simulations assuming bipolar distributions for the the q-values of the variables or 2) empirical determination of the frequency during repeated simulations. However, this method implies a duplication of the total number of simulations (free and constrained).

26

The second issue is the interpretation of the statistical properties of repeated simulations. The previous result only allows discarding non-significant properties, but it does not give any justification for further analysis. Recall that repeated qualitative simulations cannot not provide 1) numerical probabilities 2) qualitative probabilities, i.e. a ranking of the events by order of decreasing probability, and 3) qualitative expectations, i.e. a ranking of the quantifiable properties (such as degrees) by order of decreasing expected value. These limitations lead to weak interpretations of the statistical properties. For instance, the properties may be characterized by means of their diffential frequency (DF), i.e. the difference of their frequencies in constrained and free simulations. Depending upon some threshold p, the properties may be classified in 3 categories: 1) non significant when their DF is negative, 2) possibly spurious when the DF is below p, 3) observable when their is above p. Any classification of the property must obviously be compatible with their ranking by order of DF. However, this weak approach still relies on an arbitrary assumption, namely that the order among the DF is significant. As a consequence, the issue of the interpretation of the statistical properties remains open. The third issue is the optimality of the overall approach, namely the interpretation of statistical properties of repeated random simulations. Is this approach capable of exploiting all the predictive power of a model? We do not pretend to give a definitive answer to this question, but we will defend a rather paradoxical point of view: because it allows the existence of spurious behaviors, the random simulation approach may provide a good tradeoff between significance (or relevance) and precision, as stated by Zadeh's principle of incompatibility (1973). Spurious behaviors may be seen as precise hypothesis issued from an imprecise overall prediction, namely the set of possible behaviors produced by repeated simulations. For instance, suppose that a model predicts that some variable X will be in the range [1,10]. Individual behaviors such that X=1±0.5, X=2±0.5, etc. are precise, but most of them are spurious. However, they are precise enough to be experimentally verified (or falsified), and they can be discarded either in the model itself, or by means of filters during the simulation process. The previous remarks lead to the conclusion that because the predictions of a model cannot overcome its initial uncertainty (or imprecision), spurious behaviors necessarily appear as soon as the predictions of an uncertain (or imprecise) model are reformulated in precise terms. As a consequence, repeated random simulations could provide a good way of exploiting the predictive power of a qualitative model, inasmuch as spurious behaviors are reasonably unlikely, and do not interfere with the statistical properties of repeated simulations.

27

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Plahte, E., Mestl, T., Omholt, S. (1998) "A methodological basis for the 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. 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. Thomas, R., d’Ari, R. (1990) "Biological Feedback". CRC Press. Thomas, R. and 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. Vogel, R.S. (1995). “Approximate Qualitative Probability”. Journal of Mathematical Psychology 39 pp.125-128. Zadeh, L. (1973). "Outline of a New Approach to the Analysis of Complex Systems and Decision Processes". IEEE trans. on systems, man and cybernetics smc-3(1) pp. Zhao, F. (1994) "Extracting and Representing Behaviors of Complex Systems in Phase Space". Artificial Intelligence 69(1,2) pp.51-92.

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