Combining Numeric and Symbolic Reasoning in SNePS

1
CombiningNumeric and Symbolic Reasoning in SNePS

• Stuart C. Shapiro
• Department of Computer Science Engineering
• Center for MultiSource Information Fusion
• Center for Cognitive Science
• University at Buffalo, The State University of
  New York

2
SNePS

• SNePS is a
• Logic-Based
• Frame-Based
• Network-Based
• knowledge representation, reasoning, and acting
  system.

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This Talk

• The logic-based view of SNePS
• The SNePSLOG user interface.
• The recently added procedural attachment to
  combine numeric with symbolic reasoning.

4
Some SNePSLOG Syntax

• P(a,b) The proposition that P is true of a and
  b.
• P(a1, , an, b1, , bm) The proposition
  that P is true of each ai and bj.
• andor(i,j)P1, , Pn The proposition that at
  least i and at most j of the Pi are true.
• andor(1,1)P1, , Pn The proposition that
  exactly 1 of the Pi are true.
• andor(0,0)P1, , Pn The proposition that none
  of the Pi are true.
• P The proposition that P
  is false, abbreviation of andor(0,0)P.
• all(x,y,)(P(x,y) gt Q(x,y)) The proposition
  that for every x and y, if P(x,y) then
  Q(x,y).
• all(x,y,)(P(x,y) gt Q1(x,y), Qn(x,y))
  The proposition that for every x and y, if
  P(x,y) then Qi(x,y), for each i.
• P1, , Pn gt Q The conjunction of the Pi
  (solved in parallel) implies Q.
• P1 gt ( (Pn gt Q)) The conjunction of the Pi
  (solved in serial) implies Q.

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Procedural Attachment

• A predicate (proposition-forming function) symbol
• may be attached to a procedure
• so instances may be computed
• in the underlying programming language.

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Example of Procedural Attachment

• Diff(7,3,?x)?
• wff24! Diff(7,3,4)
• Diff(10,?x,7)?
• wff25! Diff(10,3,7)
• Diff(?x,5,7)?
• wff26! Diff(12,5,7)
• Diff(15,8,7)?
• wff314! Diff(15,8,7)
• Diff(15,8,9)?
• wff316! Diff(15,8,9)

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Illustration

• Implementing a Bayesian Network in SNePS using
  combined symbolic and numeric reasoning

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Some Facts of Bayesian Probability in SNePSLOG

• P(x,p) The prior probability of x is p.
• Bel(x,p) The posterior probability of x is p.
• calculatedBel(x,p) The calculated posterior
  probability of x is p.
• all(x,p)(P(x,p) gt Bel(x,p)).
• all(x,p)(Bel(x,p) gt all(q)(Diff(1,p,q) gt
  Bel(x,q))).
• all(x,y)(andor(1,1)x,y
• gt all(p)(Bel(x,p)
• gt all(q)(Diff(1,p,q) gt Bel(y,q)))).
• all(x,p)(calculatedBel(x,p) gt Bel(x,p)).

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An Example Bayesian Network
Pollution
Smoker
P(PL)
0.90
P(ST)
Joe 0.30 Jane 0.50
P S P(CTP,S)
H T 0.05
H F 0.02
L T 0.03
L F 0.001
Cancer
C P(XposC)
T 0.90
F 0.20
C P(DTC)
T 0.65
F 0.30
Dyspnoea
XRay
From Kevin B. Korb Ann E. Nicholson, Bayesian
Artificial Intelligence, Chapman Hall/CRC,
2004, p. 31 ff.
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The Patient-Independent CPTs

• CP(x,y,p) The conditional probability of x
  given y is p.
• JCP(x,y,z,p) The conditional probability of x
  given y and z is p.
• ExposedTo(x,v,f) x has been exposed to a v
  level of factor f.
• Does(x,a) x engages in the activity a.
• Has(x,d) x has the disease d.
• Positive(x) The procedure x gave a
  positive result
• all(x)(andor(1,1)ExposedTo(x,low,pollution),
  ExposedTo(x,high,pollution)).
• all(x)(Patient(x)
• gt JCP(Has(x,cancer), ExposedTo(x,high,pollut
  ion), Does(x,smoke), 0.05),
• JCP(Has(x,cancer), ExposedTo(x,high,polluti
  on), Does(x,smoke), 0.02),
• JCP(Has(x,cancer), ExposedTo(x,low,pollutio
  n), Does(x,smoke), 0.03),
• JCP(Has(x,cancer), ExposedTo(x,low,pollutio
  n), Does(x,smoke), 0.001),
• CP(Positive(X-ray(x)), Has(x,cancer),
  0.90),
• CP(Positive(X-ray(x)), Has(x,cancer),
  0.20),
• CP(Has(x,dyspnoea), Has(x,cancer), 0.65),
• CP(Has(x,dyspnoea), Has(x,cancer),
  0.30)).

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Algorithm for calculatedBel

• Given an X, To find the p such that
  calculatedBel(X,p)
• Infer from the KB all Ei, pi s.t. CP(X,Ei,pi).
• Infer from the KB the qi s.t. Bel(Ei,qi).
• Set pcp to Si (pi qi).
• Infer from the KB all E1i, E2i, pi s.t.
  JCP(X,E1i,E2i,pi).
• Infer from the KB the q1i s.t. Bel(E1i,q1i).
• and the q2i s.t. Bel(E2i,q2i).
• Set pjcp to Si (pi q1i q2i).
• Set p to pcp pjcp.

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The Patients and Their Priors

• Patient(x) x is a patient.
• Patient(Joe,Jane).
• P(ExposedTo(Joe,low,pollution), 0.90).
• P(Does(Joe,smoke), 0.30).
• P(ExposedTo(Jane,low,pollution), 0.90).
• P(Does(Jane,smoke), 0.50).

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Inferring Posteriors

• Bel(ExposedTo(Joe,low,pollution), ?p)?
  should be 0.9
• wff12! Bel(ExposedTo(Joe,low,pollution),0.9)
• Bel(Does(Joe,smoke), ?p)? should be 0.7
• wff32! Bel(Does(Joe,smoke),0.7)
• Bel(ExposedTo(Joe,high,pollution), ?p)?
  should be 0.1
• wff46! Bel(ExposedTo(Joe,high,pollution),0.1)
• Bel(Has(Joe,cancer), ?p)? should be 0.011
• wff90! Bel(Has(Joe,cancer),0.011)
• Bel(Positive(X-ray(Joe)), ?p)? should be
  0.208
• wff104! Bel(Positive(X-ray(Joe)),0.208)
• Bel(Has(Joe, dyspnoea), ?p)? should be
  0.304
• wff126! Bel(Has(Joe,dyspnoea),0.304)
• Bel(Has(Jane,cancer), ?p)? should be 0.017
• wff280! Bel(Has(Jane,cancer),0.017)
• Bel(Positive(X-ray(Jane)), ?p)?
  should be 0.212
• wff294! Bel(Positive(X-ray(Jane)),0.212)
• Bel(Has(Jane, dyspnoea), ?p)? should be
  0.306
• wff377! Bel(Has(Jane,dyspnoea),0.306)

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Conclusions

• SNePS uses procedural attachment to combine
  numeric with symbolic reasoning.
• New attached procedures may be added by the KE.
• Strengths of symbolic reasoning
• Representation of, and reasoning about general
  cases.
• Instantiating multiple specific cases.
• Clarity of declarative statements.
• Strengths of numerical reasoning
• Complicated numerical calculations.
• Especially sums and products of series.
• Sometimes faster than logical deduction.