Closed Form Solution of Qualitative Differential Equations

Numerical simulation, phase-space analysis, and analytic techniques are three methods used to solve quantitative differential equations . Most work in Qualitative Reasoning has dealt with analogs of the first two techniques, producing capabilities applicable to a wide range of systems . Although potentially of benefit, little has been done to provide closedform, analytic solution techniques for qualitative differential equations (QDEs) . This paper presents one such technique for the solution of a class of ordinary linear and nonlinear differential equations . The technique is capable of deriving closed-form descriptions of the qualitative temporal behavior represented by such equations . A language QFL for describing qualitative temporal behaviors is presented, and procedures and an implementation QDIFF that solves equations in this form are demonstrated. I . Introduction Various techniques have been described in the literature for inferring qualitative behavior of physical systems . The first techniques were based on simulation [De Kleer & Brown 84, Forbus 84, Kuipers 86] . Analogous to numerical simulation, these techniques compute the progression of qualitative values over time . More recently, qualitative phase-space approaches have been introduced [Lee&Kuipers; 88, Struss 88, Sacks 87] . Augmenting simulation, these techniques explore trajectories in phase space, showing how the qualitative values in a system will change from any point in the space. Similar to the phase-space methods used in quantitative analysis [Thompson & Stewart 86], these techniques are strong at indicating convergence, stability, etc ., but weaker at explicitly describing the temporal behavior of the values. Closed-form, analytic solution of differential equations is a well-known technique in mathematics [Boyce & DiPrima 77] . Rather than using point-by-point simulation, this methodology describes entire temporal behaviors in terms of a set offunctions. The set of these functions includes tn, exp(t), sin(t), log(t), etc . Manipulation of these symbols according to the laws of mathematics is used to find behaviors in closed form. Although familiar in quantitative mathematics, closed-form analysis of differential equations has seen little attention in qualitative reasoning, although closed-form algebraic analysis has been described by various authors [e.g., Williams 88] . For differential equations, however, techniques such as aggregation [Weld 86], abduction [Williams 90], and dynamical systems theory [Struss 88] have been used to infer properties of behaviors computed in other ways. To perform qualitative, closed-form analysis, qualitative reasoning needs a set of symbolic descriptions of qualitative behavior analogous to the i(t), log(t), etc ., of quantitative mathematics, and rules to manipulate and transform :se functional descriptions . Such qualitative solutions to differential equations are desirable for several reasons. First, if an exact solution to an equation is not known, a qualitative solution can indicate the types of behavior that are possible, augmenting numerical simulation results . Also, for complex equations where an exact solution is known, it may be so complex as to not be comprehensible to a person examining it . A simpler, qualitative solution may be preferable for obtaining an intuitive understanding of system behavior . The advantages of qualitative descriptions of behavior are covered further in [Yip 88 and Williams 90] . This paper discusses a preliminary set of such analytic tools . Section II presents a framework, QFL, in which to represent functions qualitatively. Section III describes how derivatives of QFL qualitative functions are computed. Section IV defines the effects of applying nonlinear functions to qualitative behaviors . Finally, Section V presents an implementation QDIFF, and some examples outlining the solution of QDEs. We close with a brief evaluation of the approach and some ideas for how it can be extended II. Describing Qualitative Functions Various techniques currently exist for describing qualitative values . These include the (+,0,-) representation of [DeKleer & Brown 84], values defined in terms of a quantity space [Forbus 84], and dynamically-defined values represented in terms of landmarks [Kuipers 86] . For qualitative analytic solution, a representation for behavior over time, similar to the quantitative functions such as sin(t), exp(t), and tn, is needed . One way to do this is to define a generic quantitative template that describes a wide set of functions, using qualitative values for its parameters to represent particular functions . A desirable starting-point template would describe constant, increasing, and decreasing behavior, as well as a wide variety of periodic and non-periodic oscillations . One such template is : N f(t) = 1: A,(t) sin(kB(t) + 0(k)) (eq, 1) where 0(k) = n/2 if k even and zero otherwise. Intuitively, the set Ak(t) describes the envelope of the waveform of f(t) and B(t) describes the behavior of the period of oscillation of the waveform (or that there is no oscillation, if dB(t)/dt = 0) . A great many functions can be described in this form. The variation of A(t) with k allows for dynamically-varying harmonic content of the waveform, and the use of B(t), rather than a constant times t allows the time scale to be varied with time . These variations from the familiar Fourier expansions [Gabel & Roberts 80] allow a wider variety of behaviors than might initially be expected . We define a language QFL (Qualitative Function Language) in which functions are described in terms of the attributes of the sets of functions Ak(t) and dBk(t)/dt, the sets henceforth referred to as A(t) and dB(t) . In QFL, A(t) and dB(t) fall into one of the following categories : 1 . inc : Ak(t) is non-negative, and for all nonzero Ak(t), monotonically increases as t approaches infinity 2. dec: Ak(t) is non-negative, and for all nonzero Ak(t), monotonically decreases asymptotically as t approaches infinity . 3 . static: for every k, Ak(t) is non-negative constant . In addition, QFL allows subcategories of each of the above: 1 . incC : inc, starting at a positive value, increasing without bound. 2. inc0 : inc, starting at zero, increasing without bound. 3 . incA: inc, increasing asymptotically toward a bounding final value . 4 . decF : dec, decreasing toward a nonzero final value . 5 . deco : dec, decreasing toward a final value of zero . 6 . con: static, for some k, Ak is nonzero . 7. 0 : static, for all k, Ak is zero . A QFL function is represented by the expression (, ) If aB(t) is zero, the second argument is omitted . Figure 1 shows an example of the function FI(inc0,decF) . Figure 1 . Example of Fl(inc0,decF) Figure 2. Example of (sharp Fl)(inc0,decF) In addition to specifying the types of A(t) and DB(t), QFL allows functions to be specified relative to other QFL functions, by use of a set of qualitative shape operators . Shape operators express relationships between the amplitude envelopes of different functions . The shape operators supported by QFL include : 1 . (sharp f) : to scale the range of function f by a positive, nonlinear scaling function which increases with distance from the origin . 2 . (flat f) : to scale the range of function f by a positive, nonlinear scaling function which decreases with distance from the origin. III . Derivatives of Qualitative Functions Assume that we wish to solve a nonlinear differential equation of the form i (fk(f(t)) d`f(t) ) + f o(f(t)) = 0 (eq. 2) -, dt df(t) = 7_ (a A(t) sin(k b(t) + 0(k)) + k A(t)a b(t) cos(k b(t) + 0(k))) Table I. Derivative Effect on A(t) A(t) of nth derivative dA(t) (t) A(t) = (I . dt 3 . (invert f) : to nonlinearly reverse the scale of the range of function f, hence changing the type of f. Figure 2 shows an example of the function (sharp FI)(inc0,decF) . for the behavior f(t), where the fk(x) are nonlinear functions of x . To process the terms of an equation in this form, we need to compute the derivatives of qualitative functions, as well as compute the results of applying nonlinear functions to qualitative behaviors . We can elucidate the mapping between function and derivative by differentiating the template of Equation 1 and determining the implied qualitative transformations . Operator tables for functions, analogous to the operator transforms described for values [De Kleer & Brown 84, Forbus 84, Kuipers 86], can then be constructed . In the following, aB will be considered equivalent to dB(t)/dt, aDB to d2B(t)/dt2 , etc . This equation contains a component lagging f(t) in phase by n/2 and a component in phase with f(t) . The oscillation characteristics of f(t) (the argument to the sin terms) are preserved. The results for derivatives zero through two are tabulated below : It would be desirable to express the entries in this table in algebraic terms, free of the a operators, so that the solution of the differential equations could be found algebraically . This is achieved by the following process, which converts the expresion dA(t)/dt into a product. Let d(t) be the function such that where d(t) is one of the qualitative function types . It can be shown, for the class of A(t) represented in QFL, that n In-phase Out-of-phase 0 A 0 1 aA A aB 2 aDA A DBaB DA aB A DDB dkA(t) d(t) . A(t) dtk where d-(t) is of the same basic qualitative type as d(t) . The same, of course, applies to the derivatives of DB(t) . Therefore, we can rewrite the terms from Table I in terms of sums and products of A(t), dB(t), a new function, D(t) (the function equivalent to the derivative of A(t)), and E(t) (the function equivalent to the derivative of DB(t)) . For example, the out-of-phase part of the second derivative from the table, aA dB + A BBB, would be rewritten as D(t) A(t) dB(t) + A(t) E(t) dB(t), or, in the shorthand we will use from now on, DAdB+AEdB. Given qualitative types for A(t) and DB(t), qualitative types for D(t) and E(t) can be computed as follows . First, the qualitative types of the derivatives of qualitative behaviors are found with Table II : Table II. Derivative of Qualitative Types f(t) af(t) deco -deco decF -deco incC incC or inc0 or incA or decF inc0 incC or inc0 or incA or decF incA deco con 0 0 0 Next, the effective product resulting from the derivative transformation can be found with Table III. This table was created by examining each pair of qualitative behaviors, determining what function multiplied by the before behavior would yield the after behavior. The values ofX and Y are variables, matched to any qualitative behavior : Table III. Equivalent Multiplication of Various Transformations Before After Effective Multiplier f(X) (flat f)(0) 0 f(X) f(X) con f(con) (Y WX) X f(inc) (invert f)(dec0) deco f(inc) (invert f)(decF) decF f(inc) (sharp f)(inc) incC f(inc) (flat f)(inc) decF f(inc0) (flat f)(incA) dec f(incC) (flat f)(incA) dec f(dec) (flat f)(dec) incA f(dec) (sharp f)(dec) decF f(decF) (Y f)(dec0) deco f(decF) (invert f)(incA) incC f(decF) (invert f)(inc0) inc0 f(decF) (invert f)(incC) incC For each of the qualitative types of A(t) and aB(t), the corresponding possible types of D(t) and E(t) have been tabulated in Table IV by use of Tables 11 and III . The table was computed by considering the possible behaviors and derivatives of each function type . Where ambiguous, all possible types were included : Table IV. Derivative Functions type of f(t) d(t) for d(t)-f(t) = af(t) decF -deco deco decF or deco or con or -incC or inc0 incC incC or inc0 or deco or con or decF inc0 incC or inc0 or deco or con or decF incA deco con zero zero X Finally, by use of multiplication, the expresions representing the derivatives of a behavior can be reduced to a sum of qualitative values, given qualitative values for A, dB, D, and E. This is achieved with the following multiplication table : Consider the previous example, in which the out-of-phase part of the second derivative from Table I is dA dB + A ddB. This was rewritten above as D A dB + A E dB. Suppose that A is of type decF and dB of type con. From Table III, we see that D, the effective multiplication of the derivative of A, must be of type -deco. Similarly, E, the effective multiplication of the derivative of dB, is of type 0. This yields the sum -deco decF con + -deco 0 con which, from Table IV, is equal to -deco. Table V. QFL Multiplication f(t) g (t) f(t) g(t) X con X X 0 0 X X X decF deco deco decF inc0 inc0 or incA or decF decF incC incF or decF or con decF incA decF or incA or con deco inc0 inc0 or incA or decF or deco deco incC incC or incA or decF or deco deco incA deco inc0 incC inc0 inc0 incA inc0 incC incA incC or inc0 The remaining analytic tool needed to solve differential equations in the form of Equation 2 is the mechanism for determining the qualitative effects of the nonlinear functions fk(t) . As is apparent from the equation, nonlinear functions will be applied directly to the unknown f(t) . We take care to consider the effects of the transformation both on the characteristic A(t) of f(t) and on the phase of the result. IV.A. Properties of fk(f(t)) IV. Nonlinear Functions Assume that any nonlinear function fk(t) of interest can be represented as a power series in t . The following characteristics will therefore occur when applying fk(t) to qualitative behavior f(t) in the form of Equation 1 : 1 . The constant term in the expansion of fk(t) will lead to the appearance of terms sin(k B(t) + phase(k)) . 2. Quadratic terms in fk(t) will lead to contributions of the form Am(t)sin(m B(t)) An(t)sin(n B(t)), when m and n are odd. Applying a trigonometric identity yields Am(t) An(t){cos((m n)B(t)) + cos((m + n)B(t))) = Am(t) An(t){sin((m n)B(t) + n/2) + sin((m + n)B(t) + n/2)) . (m n) and (m + n) are both even numbers, so the result will be in phase with the terms of Equation 1 . 3 . Quadratic terms in fk(t), when m and n are both even or for m odd and n even similarly will yield results in phase with f(t) . 4 . Higher-order terms in will also result in terms in phase with the original terms in Equation 1 . This can be shown inductively, using the results of 2) and 3) . These results indicate that applying a nonlinear function to the unknown f(t) in Equation 2 will yield another function that is in phase with f(t) . By definition, f(t) has no out-of-phase components (per Table I) . Therefore, fk(f(t)) also will have no out-of-phase components . Recall that in Equation 2, the nonlinear functions of f(t) are multiplied by the various derivatives of f(t) . A derivation nearly identical to that carried out above yields the following conclusion about how those products are formed: When multiplying fk(f(t)) = g(t) by any order derivative of f(t), the in-phase part of the product will be g(t) times the in-phase part of the derivative. Similarly, the out-of phase part of the product will be g(t) times the out-of phase part of the derivative. The significance of this result is that the in-phase and out-of-phase parts of the products fk(f(t))-dnf(t)/dtn can be found by multiplying the effect of fk(x) on the envelope function A(t) by the resulting envelope function of the derivative operators found in Table I . IV.B. Characterization of Nonlinear Functions Let us now discuss how the nonlinear functions fk(f(t)) can be defined . The basic QFL facility for representing nonlinear transformations is the set of qualitative shape operators flat, sharp, and invert. . In Section IV.A, it was shown that the effect of the nonlinear functions on A(t), the envelope function of f(t), is the effect of interest. Therefore, it is adequate to define the behavior of each nonlinear function fk(f(t)) as a qualitative shape operator operating on A(t) . For example, let fk(x) be sin(x), for -n/2 < x < n/2 . Suppose that we wish to find fk(f(t)), where A(t) is of type inc. In this case, sin(A(t)) will be "flattened" more and more as A(t) gets larger. Therefore, we would use (flat A) as the factor by which the corresponding derivative in the QDE would be multiplied. In addition to the qualitative shape operator caused by nonlinear functions, the sign of the effect is important. Consider the nonlinear function, fk(x) = (1 x2 ) . Proceeding as above, we find that the corresponding factor in terns of A(t) is (con /Sharp A b. In this case, however, differing values of A(t) will lead to differing qualitative effects : when [A(t) < 1, fk(A) will be positive, and negative when [A(t) I > 1 . To avoid excessive ambiguity, therefore, consideration of the behavior is divided into distinct regions . In each region, the behavior of this equation is given by con /Sharp A /. However, when A(t) > 1, the qualitative relationship Icon I < sharp A 1, is imposed, and where A(t) < 1, Icon I > sharp A I is imposed. In Section V, this technique will be demonstrated in an example. V. Solving QDEs The results outlined above lead to a technique for solving qualitative differential equations. A program called QDIFF has been implemented for just this purpose. In this section, we describe the solution method used by QDIFF and show examples of various equations and their solution. QDIFF solves differential equations by finding values for A(t) and DB(t) that allow the sum of the in-phase and out-of phase contributions of the terms in the equations to add to zero . The problem can be broken down in this way because the in-phase and out-of phase parts are linearly independent (although not necessarily orthogonal) . The solution is achieved with the following procedure : 1 . From Table I, gather the in-phase and out-of phase expressions for the envelope function A(t) for each derivative of f(t) that appears in the QDE. 2 . For terms multiplied by a nonlinear function, obtain the expresion, in terms of A(t), that describes that function, and multiply the corresponding in-phase and out-of phase expresions from step 1) by that function . 3 . Replace a operators in the resulting in-phase and out-of phase sums with D(t) and E(t) terms, according to the translation process of Section III . 4. Using Table IV, constrain the values of D(t) and E(t) with respect to potential values of A(t) and DB(t) . Using multiplication via Table V, find all combinations of A(t) and DB(t) within these constraints that allow both sums to be zero . (If the equation is linear, the only values of DB(t) that need be tried are con and 0.)