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

2. 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. 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. 4 QRQR Combine discrete states-continous dynamics Finite no. of states – transitions obeying continuity constraints Behaviour: sequence of states Domain abstraction Function abstraction

5. 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. 6 Reasoning about Signs Direction of change S={-,0,+,?} Qualitative equality (≈)  a,b S, (a ≈ b iff (a = b or a = ? or b = ?))

7. 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. 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. 9 Semi-Lattice Structure

10. 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. 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. 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

13. 13

14. 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. 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. 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. 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. 18 Gray-Box Sytems

19. 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. 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. 21 THANKS FOR LISTENING!