Building on Knowledge-Based Agents

In my previous post, I covered knowledge-based agents and introduced knowledge bases (KBs) and their constituent sentences. Now, let’s explore how these sentences are structured and interpreted through logic.

This is the second post in a series on knowledge-based agents and logic:

The Components of Logic

Every logical system has three components that create a framework for representing and reasoning about knowledge.

Syntax: The Rules of Formation

Syntax defines how sentences are constructed—the rules that determine if a sentence is well-formed.

Consider English: “name Hunter mine is” violates syntax rules, while “my name is Hunter” follows them. In mathematics, “1+=2 3” breaks syntax, but “1+2=3” follows it.

Knowledge bases need proper syntax because they’re built from collections of sentences.

Semantics: The Meaning Behind Sentences

While syntax governs structure, semantics determines meaning—whether a sentence is true or false in a given context.

The sentence “my name is Hunter” is true in our context, but “my name is Paul” would be false. In math, “x=5” is true only when x actually equals 5.

We use model to describe a specific world state where variables have particular values. Different models let us evaluate sentence truth. One model might have x=4, another x=5.

When a sentence is true under a model, the model satisfies the sentence. If model₅ has x=5, then model₅ satisfies x=5.

For any sentence A, M(A) represents all models that satisfy A.

Entailment: Logical Relationships

Entailment enables reasoning. When sentence A entails sentence B (written A ⊨ B), B must be true whenever A is true.

More precisely, A entails B if B is true in every model where A is true:

  • A is the premise
  • B is the consequent
  • B is a necessary consequence of A

Example: x=1 entails xy=y. If x equals 1, then xy will always equal y, regardless of y’s value.

This helps knowledge bases reason. When we’re uncertain about xy=y but add x=1 to our KB, entailment lets us assert that xy=y is now true.

To check if a KB entails sentence A, we verify that M(KB) ⊆ M(A)—every model satisfying our KB also satisfies A.

When we add sentences to our KB, we need systematic ways to discover newly entailed sentences. Inference algorithms do this work.

Inference Algorithms: Deriving New Knowledge

Inference algorithms systematically derive new sentences entailed by our knowledge base. These algorithms need two properties to be useful and trustworthy.

Soundness: Truth Preservation

An inference algorithm is sound if it only derives sentences actually entailed by the KB. If an algorithm adds unjustified sentences, it’s fabricating information—undermining the KB’s reliability.

Completeness: Finding Everything

A complete inference algorithm finds all sentences entailed by the KB, missing no valid conclusions. This ensures we don’t overlook important logical consequences.

Achieving both soundness and completeness grows challenging as environments become more complex. Small, bounded models make this manageable. Unbounded environments require sophisticated algorithms that guarantee both properties while maintaining reasonable performance.

Grounding: Connecting Logic to Reality

Beyond formal algorithm properties lies a practical concern: grounding. Is our knowledge base actually grounded in reality? Does it accurately represent what’s true?

Grounding depends on how knowledge enters the system:

  • Sensors: Agent perceptions are only as accurate as their sensors
  • Learning algorithms: Learned sentences are only as reliable as the learning process

With accurate sensors and robust learning, we can trust our KB’s grounding. This remains important when deploying knowledge-based systems in real applications.

Conclusion

Logic provides the foundation for knowledge representation in AI systems. Understanding syntax (forming valid sentences), semantics (sentence meaning), entailment (how conclusions follow from premises), and inference algorithms (deriving new knowledge) enables building robust knowledge-based agents.

The interplay between these components—governed by soundness, completeness, and grounding—determines how effectively our AI systems represent and reason about the world.