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The Nuances of Non-monotonic Reasoning in Commonsense Logic

Non-monotonic reasoning reflects how our conclusions can change with new information. While first-order logic traditionally assumes monotonicity, real-world commonsense reasoning often involves retracting previous conclusions when faced with additional data. This principle underlies concepts like the Closed World Assumption, negation as failure, and domain-closure assumptions. Default rules and probabilistic reasoning further reveal complexities in knowledge representation, especially in uncertain domains like medical diagnosis, where exceptions and incomplete information challenge rigid logical constructs.

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The Nuances of Non-monotonic Reasoning in Commonsense Logic

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  1. Non-monotonic Reasoning • Are we having a pop quiz today? • You assume not. • But can you prove it? • In commonsense reasoning, • we often jump to conclusions, • can’t always list the assumptions we made, • need to retract conclusions, when we get more information. • In first-order logic, our conclusion set is monotonically growing.

  2. The Closed World Assumption • KB contains: Student(Joe), Student(Mary) • Query: Student(Fred)? • Intuitively, no; but can’t prove it. • Solution: when appropriate, close the predicate Student. • X Student(X) <=>X=Joe v X=Mary • Closing can be subtle when multiple predicates are involved: • X In(X) <=> Out(X)

  3. More on CWA • Negation as failure: x,y,z edge(x,z)  path(z,y)  path(x,y) x,y edge(x,y)  path(x,y) edge(A,B), edge(B,C), edge(A,D) • Conclude: path(C,D). • Domain-closure assumption: the only named constants in the KB exist in the universe. • Unique-names assumption: every constant is mapped to a different object in the universe. (already assumed in Description Logics and Databases).

  4. Default Rules Bird(X) C(Flies(X)) : Flies(X) is consistent. Flies(X) • Application of default rules: the order matters! Liberal(X) Hunter(X) C(Dem(X)) C(Rep(X)) Dem(X) Rep(X)  X  (Dem(X)  Rep(X)) Liberal(Tom), Hunter(Tom)

  5. Minimal Models: Circumscription • Consider only models in which the extension of some predicates is minimized. •  X (Bird(X) abnormal(X))  Flies(X) • Some predicates are distinguished as “abnormal”. • An interpretation I1 is preferred to I2 if: • I1 and I2 agree on the extensions of all objects, functions and non-abnormal predicates. • The extension of abnormal in I1 is a strict subset of its extension in I2. • KB |= S, if S is satisfied in every minimal model of KB (I is minimal if no I2 is preferred to it).

  6. But Uncertainty is Everywhere • Medical knowledge in logic? • Toothache <=> Cavity • Problems • Too many exceptions to any logical rule • Hard to code accurate rules, hard to use them. • Doctors have no complete theory for the domain • Don’t know the state of a given patient state • Uncertainty is ubiquitous in any problem-solving domain (except maybe puzzles) • Agent has degree of belief, not certain knowledge

  7. Ways to Represent Uncertainty • Disjunction • If information is correct but complete, your knowledge might be of the form • I am in either s3, or s19, or s55 • If I am in s3 and execute a15 I will transition either to s92 or s63 • What we can’t represent • There is very unlikely to be a full fuel drum at the depot this time of day • When I execute pickup(?Obj) I am almost always holding the object afterwards • The smoke alarm tells me there’s a fire in my kitchen, but sometimes it’s wrong

  8. Numerical Repr of Uncertainty • Interval-based methods • .4 <= prob(p) <= .6 • Fuzzy methods • D(tall(john)) = 0.8 • Certainty Factors • Used in MYCIN expert system • Probability Theory • Where do numeric probabilities come from? • Two interpretations of probabilistic statements: • Frequentist: based on observing a set of similar events. • Subjective probabilities: a person’s degree of belief in a proposition.

  9. KR with Probabilities • Our knowledge about the world is a distribution of the form prob(s), for sS. (S is the set of all states) • s S,0  prob(s)  1 • sSprob(s) = 1 • For subsets S1 and S2, prob(S1S2) = prob(S1) + prob(S2) - prob(S1S2) • Note we can equivalently talk about propositions:prob(p  q) = prob(p) + prob(q) - prob(p  q) • where prob(p) means sS | p holds in s prob(s) • prob(TRUE) = 1 • prob(FALSE) = 0

  10. Probability As “Softened Logic” • “Statements of fact” • Prob(TB) = .06 • Soft rules • TB  cough • Prob(cough | TB) = 0.9 • (Causative versus diagnostic rules) • Prob(cough | TB) = 0.9 • Prob(TB | cough) = 0.05 • Probabilities allow us to reason about • Possibly inaccurate observations • Omitted qualifications to our rules that are (either epistemological or practically) necessary

  11. Probabilistic Knowledge Representation and Updating • Prior probabilities: • Prob(TB) (probability that population as a whole, or population under observation, has the disease) • Conditional probabilities: • Prob(TB | cough) • updated belief in TB given a symptom • Prob(TB | test=neg) • updated belief based on possibly imperfect sensor • Prob(“TB tomorrow” | “treatment today”) • reasoning about a treatment (action) • The basic update: • Prob(H)  Prob(H|E1)  Prob(H|E1, E2)  ...

  12. Ache Ache Cavity 0.04 0.06 0.01 0.89 Cavity Basics • Random variable takes values • Cavity: yes or no • Joint Probability Distribution • Unconditional probability (“prior probability”) • P(A) • P(Cavity) = 0.1 • Conditional Probability • P(A|B) • P(Cavity | Toothache) = 0.8

  13. Bayes Rule • P(B|A) = P(A|B)P(B) ----------------- P(A) A = red spots B = measles We know P(A|B), but want P(B|A).

  14. C A P Prob F F F 0.534 F F T 0.356 F T F 0.006 F T T 0.004 T F F 0.048 T F T 0.012 T T F 0.032 T T T 0.008 Conditional Independence • “A and P are independent” • P(A) = P(A | P) and P(P) = P(P | A) • Can determine directly from JPD • Powerful, but rare(I.e. not true here) • “A and P are independent given C” • P(A|P,C) = P(A|C) and P(P|C) = P(P|A,C) • Still powerful, and also common • E.g. suppose • Cavities causes aches • Cavities causes probe to catch Ache Cavity Probe

  15. C A P Prob F F F 0.534 F F T 0.356 F T F 0.006 F T T 0.004 T F F 0.012 T F T 0.048 T T F 0.008 T T T 0.032 Conditional Independence • “A and P are independent given C” • P(A | P,C) = P(A | C) and also P(P | A,C) = P(P | C)

  16. Suppose C=True P(A|P,C) = 0.032/(0.032+0.048) = 0.032/0.080 = 0.4

  17. P(A|C) = 0.032+0.008/ (0.048+0.012+0.032+0.008) = 0.04 / 0.1 = 0.4

  18. Summary so Far • Bayesian updating • Probabilities as degree of belief (subjective) • Belief updating by conditioning • Prob(H)  Prob(H|E1)  Prob(H|E1, E2)  ... • Basic form of Bayes’ rule • Prob(H | E) = Prob(E | H) P(H) / Prob(E) • Conditional independence • Knowing the value of Cavity renders Probe Catching probabilistically independent of Ache • General form of this relationship: knowing the values of all the variables in some separator set S renders the variables in set A independent of the variables in B. Prob(A|B,S) = Prob(A|S) • Graphical Representation...

  19. Computational Models for Probabilistic Reasoning • What we want • a “probabilistic knowledge base” where domain knowledge is represented by propositions, unconditional, and conditional probabilities • an inference engine that will computeProb(formula | “all evidence collected so far”) • Problems • elicitation: what parameters do we need to ensure a complete and consistent knowledge base? • computation: how do we compute the probabilities efficiently? • Belief nets (“Bayes nets”) = Answer (to both problems) • a representation that makes structure (dependencies and independencies) explicit

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