MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING Louise-Travé-Massuyès, Liliana Ironi, Philippe Dague Presented by Nuri Taşdemir

2 Overview Different formalisms for modeling physical systems Mathematical aspects of processes, potential and limitations Benefits of QR in system identification Open research issues

3 QR as a good alternative for modeling cope with uncertain and incomplete knowledge qualitative output corresponds to infinitely many quantitative output qualitative predictions provide qualitative distinction in system’s behaviour more intuitive interpretation

4 QRQR Combine discrete states-continous dynamics Finite no. of states – transitions obeying continuity constraints Behaviour: sequence of states Domain abstraction Function abstraction

5 Domain Abstraction and Computation of Qualitative States Real numbers  finite no. of ordered symbols quantity space: totally ordered set of all possible qualitative values Qualititativization of quantitave operators a Q-op b = { Q(x op y) | Q(x) = a and Q(y) = b } C: set of real valued constraints Sol(C) : real solutions to C Q(C): set of qualitative constraints obtained from C Soundness:  C, Q(Sol(C))  Q-Sol(Q(C)) Completeness:  Q-C, Q-Sol(Q-C)  Q(Sol(C))

6 Reasoning about Signs Direction of change S={-,0,+,?} Qualitative equality (≈)  a,b S, (a ≈ b iff (a = b or a = ? or b = ?))

7 Reasoning about Signs Quasi-transitivity: If a ≈ b and b ≈ c and b ≠ ? then a ≈ c Compatibility of addition: a + b ≈ c iff a ≈ c - b Qualitative resolution rule: If x + y ≈ a and –x + z ≈ b and x ≠ ? then y + z ≈ a + b

8 Absolute Orders of Magnitude S 1 = { NL,NM,NS,0,PS,PM,PL } S = S 1  {[X,Y]  S 1 -{0} and X<Y}, where X < Y means  x  X and  y  Y, x < y S is semilattice under ordering  define q-sum and q-product in lattice commutative, associative, is distributive over (S,,, ≈ ) is defined as Q-Algebra

9 Semi-Lattice Structure

10 Relative Order of Magnitude Invariant by translation Invariant by homothety (proportional transf.) A Vo B: A is close to B A Co B: A is comparable to B A Ne B: A is negligible with respect to B x Vo y → y Vo x x Co y → y Co x x Co y, y Vo z → x Co z x Ne y → (x + y) Vo y

11 Qualitative Simulation Three approaches: 1-the component-centered approach of ENVISION by de Kleer and Brown 2-the process-centered approach of QPT by Forbus 3-the constraint-centered approach of QSIM by Kuipers

12 Q-SIM Variables in form transitions obtained by MVT and IVT P-transitions: one time point  time interval I-transitions:time interval  one time point Temporal branching Allen’s algebra does not fit to qualitative simulation

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14 Allen’s Algebra The “Allen Calculus” specifies the results of combining intervals. There are precisely 13 possible combinations including symmetries (6 * 2 + 1)

15 Time Representation Should time be abstracted qualitatively? State-based approach(Struss): sensors give information at sampled time points Use continuity and differentiability to constrain variables Use linear interpolation to combine x(t), dx/dt, x(t+1) uncertainty in x causes more uncertainty in dx/dt so use sign algebra for dx/dt

16 System Identification Aim: deriving quantitative model looking at input and output involves experimental data and a model space underlying physics of system (gray box) incomplete knowledge about internal system structure ( black box) Two steps: (1) structural identification(selection within the model space of the equation form) (2) parameter estimation(evaluation of the numeric values of the equation unknown parameters from the observations)

17 Gray-Box Sytems RHEOLO  specific domain behaviour of viscoelastic materials instantaneous and delayed elasticity is modeled with same ODE Either: (1)the experimental assesment of material (high costs and poor informative content) or (2) a blind search over a possibly incomplete model space (might fail to capture material complexity andmaterial features QR  brings generality to model space M (model classes) S: structure of material Compare QB(S) with Q(S) QRA:qualitative response abstraction

18 Gray-Box Sytems

19 Black-Box Sytems given input and output find f difficult when inadequate input Alternative to NNs, multi-variate splines, fuzzy systems used successfully in construction of fuzzy rule base

20 Conclusion and Open Issues QR as a significant modeling methodology limitations due to weakness of qualitative information Open issues: - Automation of modeling process - determining landmarks - Compositional Modeling

21 THANKS FOR LISTENING!