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Unit 2 — Computable Functions and Predicates — Artificial Intelligence Notes

Computable Functions and Predicates

Predicate logic becomes genuinely useful to a computer program only when its predicates and functions can be evaluated or proven — this is where the ideas of computable functions and computable predicates come in.

Functions vs Predicates

	Function	Predicate
Returns	An object (a value)	A truth value (true/false)
Example	FatherOf(Ram) returns Dashrath	Student(Ram) returns true
Notation	f(x)	P(x)

A predicate can be thought of as a special case of a function whose output is restricted to {True, False}.

Computable Functions

A function is computable if there exists an effective procedure (an algorithm that always terminates) to compute its output for any valid input. Most functions used in AI knowledge bases — arithmetic functions, string operations, list operations — are computable in this sense.

Plus(2, 3) = 5
FatherOf(Ram) = Dashrath
Length([a, b, c]) = 3

Computable Predicates

A predicate is computable (also called decidable) if there is an algorithm that, given any input, always halts and correctly outputs True or False.

Predicate	Computable?	Why
Even(n) — is n even?	Yes	Simple modulus check, always halts
GreaterThan(x, y)	Yes	Direct numeric comparison
Provable(phi) in full first-order logic — is formula phi a theorem?	No (only semi-decidable)	Validity of first-order logic is undecidable in general; a prover may run forever on an unprovable formula

Why This Matters for AI Systems

An inference engine reasoning over a knowledge base needs to evaluate predicates and compute functions as part of proving goals. If a predicate is not computable, no algorithm can guarantee an answer in finite time — this is a fundamental limitation, not an engineering shortcoming.

Consequence	Explanation
Restricted logic subsets are used in practice	Systems like Prolog restrict to Horn clauses, a decidable-in-practice subset of first-order logic, to guarantee usable inference
Some queries may not terminate	A general theorem prover asked "is this formula true?" for an unprovable formula may search forever
Function evaluation must be side-effect-free and well-defined	So the same input always yields the same, predictable output — essential for consistent reasoning

Connecting Back

Computable functions and predicates are what let the facts written in the previous two lessons (Marcus the Pompeian, Tweety the canary) actually be used by a program — not merely stored as inert text, but queried, combined, and reasoned over automatically.