mathematical foundations of qualitative reasoning


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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 - PowerPoint PPT Presentation



    Presented by Nuri Tademir

  • OverviewDifferent formalisms for modeling physical systemsMathematical aspects of processes, potential and limitationsBenefits of QR in system identificationOpen research issues

  • QR as a good alternative for modeling

    cope with uncertain and incomplete knowledgequalitative output corresponds to infinitely many quantitative outputqualitative predictions provide qualitative distinction in systems behaviourmore intuitive interpretation

  • QRCombine discrete states-continous dynamicsFinite no. of states transitions obeying continuity constraintsBehaviour: sequence of statesDomain abstractionFunction abstraction

  • Domain Abstraction and Computation of Qualitative StatesReal numbers finite no. of ordered symbolsquantity space: totally ordered set of all possible qualitative valuesQualititativization of quantitave operatorsa 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 CQ(C): set of qualitative constraints obtained from CSoundness: C, Q(Sol(C)) Q-Sol(Q(C))Completeness: Q-C, Q-Sol(Q-C) Q(Sol(C))

  • Reasoning about SignsDirection of changeS={-,0,+,?}Qualitative equality ()a,b S, (a b iff (a = b or a = ? or b = ?))

  • Reasoning about SignsQuasi-transitivity: If a b and b c and b ? then a cCompatibility of addition:a + b c iff a c - bQualitative resolution rule: If x + y a and x + z b and x ? then y + z a + b

  • Absolute Orders of MagnitudeS1 = { NL,NM,NS,0,PS,PM,PL }S = S1 {[X,Y] S1-{0} and X
  • Semi-Lattice Structure

  • Relative Order of MagnitudeInvariant by translationInvariant 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 xx Co y y Co xx Co y, y Vo z x Co zx Ne y (x + y) Vo y

  • Qualitative SimulationThree 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

  • Q-SIMVariables in form transitions obtained by MVT and IVTP-transitions: one time point time interval I-transitions:time interval one time pointTemporal branchingAllens algebra does not fit to qualitative simulation

  • Allens AlgebraThe Allen Calculus specifies the results of combining intervals. There are precisely 13 possible combinations including symmetries (6 * 2 + 1)

  • Time RepresentationShould time be abstracted qualitatively?State-based approach(Struss): sensors give information at sampled time pointsUse continuity and differentiability to constrain variablesUse 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

  • System IdentificationAim: deriving quantitative model looking at input and outputinvolves experimental data and a model spaceunderlying 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)

  • Gray-Box SytemsRHEOLO specific domain behaviour of viscoelastic materialsinstantaneous and delayed elasticity is modeled with same ODEEither:(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 materialCompare QB(S) with Q(S)QRA:qualitative response abstraction

  • Gray-Box Sytems

  • Black-Box Sytemsgiven input and output find fdifficult when inadequate inputAlternative to NNs, multi-variate splines, fuzzy systemsused successfully in construction of fuzzy rule base

  • Conclusion and Open IssuesQR as a significant modeling methodologylimitations due to weakness of qualitative informationOpen issues:- Automation of modeling process- determining landmarks- Compositional Modeling


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