# 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 PresentationTRANSCRIPT

MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONINGLouise-Trav-Massuys, Liliana Ironi, Philippe Dague

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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