AI Advanced 2

Translating natural language into logical form is challenging because:

• Natural language lacks complexity, making logic unnecessary

• Ambiguities, context dependence, and implicit assumptions in language require careful interpretation into formal predicates and quantifiers

• Logic cannot handle any ambiguity

• Natural language and logic are identical in structure

Axioms serve as foundational truths in a knowledge base. Their role is:

• To be proven by other statements, never assumed

• To anchor reasoning processes, as all theorems and inferences must trace back to these foundational truths

• To be replaced by heuristics in reasoning

• To serve as arbitrary placeholders with no logical consequence

Consider the equivalence: ¬∀x P(x) ≡ ∃x ¬P(x). This relationship demonstrates that:

• Universal and existential quantifiers cannot be interchanged logically

• Universal quantification can be expressed in terms of existential quantification and negation, highlighting their logical interdependence

• Negation does not affect quantifiers

• This equivalence is not relevant in AI reasoning

Entailment means that if a knowledge base entails a sentence S:

• S must be logically derivable from the knowledge base, ensuring S is true whenever the knowledge base is true

• S is just a guess without logical consequence

• The knowledge base and S are unrelated

• Entailment requires no inference rules

Higher-Order Logic (HOL) is more expressive than First-Order Logic because:

• HOL can quantify over predicates and functions themselves, not just over object variables

• HOL reduces the complexity of inference

• HOL restricts what can be said about domains

• HOL eliminates the need for quantification

In reasoning systems, why might one select a less expressive formalism, like propositional logic, over a more expressive one like HOL?

• Less expressive formalisms are always better, regardless of problem complexity

• More expressive formalisms can be harder to reason with computationally; propositional logic often offers more efficient inference methods

• HOL cannot represent any meaningful knowledge

• Propositional logic eliminates all complexity from reasoning

Uncertainty arises in AI because:

• The world is always deterministic and fully known

• Agents often lack complete information, and outcomes may be probabilistic, requiring modeling degrees of belief

• Probabilities have no computational interpretation

• Symbolic logic can solve all problems deterministically

Bayesian networks represent joint probability distributions more compactly by:

• Representing each variable as independent of all others

• Exploiting conditional independencies, factoring the joint distribution into smaller conditional probability tables

• Eliminating the need for probabilities entirely

• Replacing probabilities with deterministic rules

Consider a Bayesian network where Weather influences Traffic, and Traffic influences TravelTime. This structure is A → B → C. If you know Traffic, what does conditional independence imply about Weather and TravelTime?

• TravelTime is independent of Weather given Traffic

• Weather and TravelTime remain dependent regardless of observing Traffic

• Observing Traffic has no influence on the relationship

• Conditional independence cannot hold in chain-structured Bayesian networks

Bayes’ Rule allows updating beliefs based on new evidence. If you have a prior belief P(Hypothesis) and new evidence E, Bayes’ Rule:

• Directly gives P(E|Hypothesis) without further calculation

• Provides a way to compute P(Hypothesis|E) by adjusting the prior belief using the likelihood of E given the hypothesis and the overall probability of E

• Ignores the prior entirely

• Forces you to assume independence among variables