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MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING. Louise-Travé-Massuyès, Liliana Ironi, Philippe Dague Presented by Nur i Taşdemir. Overview. Different formalisms for modeling physical systems Mathematical aspects of processes, potential and limitations

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MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

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  1. 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’sbehaviour • more intuitive interpretation

  4. QR • 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-opb = { 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 • S1 = { NL,NM,NS,0,PS,PM,PL } • S = S1 {[X,Y]  S1-{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 ACo 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 <x,dx/dt> • transitions obtained by MVT and IVT • P-transitions: one time point  time interval I-transitions:time interval  one time point • Temporal branching • Allen’s algebradoes not fit to qualitative simulation

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

  14. 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

  15. 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)

  16. 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

  17. Gray-Box Sytems

  18. 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

  19. 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

  20. THANKS FOR LISTENING!

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