Chapter 6

Knowledge-Based Agent

Agents that know about their world and reason about possible courses of action.

May be able to accept new tasks in the form of explicitly described goals. Can learn new knowledge about environment. Needs to know:

• current state of world
• how to infer unseen properties from percepts
• how the world evolves over time
• what it wants to achieve
• what its actions do in various circumstances

Design of Knowledge-Based Agent

• central component is knowledge base (KB)
• set of facts about the world
• each fact is called a sentence
• expressed in knowledge representation language
• TELL adds fact
• ASK queries
• inference mechanism makes use of knowledge
• may start with initial background knowledge
• agent takes input (TELLS percept) and returns action (ASK)

3 levels of description of KB agent:

• knowledge/epistemological - say what agent knows
• logical level - knowledge encoded into sentences. Describe agent based on what logical sentences it knows.
• implementation level - runs on agent architecture; physical representations of the sentences (e.g., as string, as alist, etc.)

Declarative approach: build knowledge from sentences. Not procedural. May also include a learning approach, for acquiring new sentences. (easier than learning new procedural knowledge?)

153-157 Wumpus Example Interesting

Representation, Reasoning and Logic

Goal of knowledge representation: express knowledge in computer-tractable form so it can be used to help agent perform well. Two aspects:

• syntax = configurations of sentences
• semantics = world facts to which sentences refer

If syntax and semantics are defined precisely, call the language a logic.

Fact is part of world, representation is inside computer.

Proper reasoning ensures that representation-entails->rep only when facts -follow-> facts (i.e., correspondence between logical entailment in representation and real world).

Goal of KB is to generate new sentences that are necessarily true, given that the old sentences are true. This relation is called entailment. Two ways for inference procedure to operate:

• Given KB, it can generate new sentences that are entailed by KB
• Given KB and sentence, can report whether or not sentence is entailed by KB

Inference procedure that generates only entailed sentences is called sound or truth preserving.

In the inference procedure:

• the record of operation of a sound inference procedure is called a proof
• an inference procedure is complete if it can find a proof for any sentence that is entailed. For many KB, the consequences are infinite, completeness is an issue.
• proof theory - specifies the reasoning steps that are sound. Inference steps must respect the semantics of sentences.

Representation

Programming language

• good for algorithms and concrete data structures
• not good when don't have complete info, just possibilities

Natural language

• goal is communication rather than representation
• meaning depends on sentence and context
• often no direct representation
• ambiguous

Goals for representation

• expressive and concise
• unambiguous and independent of context
• effective inference procedure to make new inferences

First-order logic forms basis of most representation schemes in AI. - Specifics of logical notation not important, main thing is how a precise formal language represents knowledge and how mechanical procedures can operate on knowledge to perform

reasoning.

Semantics

• writer of sentence must provide interpretation. Sentence does not mean anything by itself (but meaning may have been fixed long ago).
• Languages are compositional. meaning of sentence is function of meaning of its parts
• sentence can be true or false. Depends on interpretation of sentence and state of world

Inference

• logical inference/deduction
• valid sentence/tautology (necesssarily true) iff true under all possible interpretations in all possible worlds A v ~A
• satisfiable sentence - iff some interpretation in some world for which it is true (doesn't have to be true right now)
• unsatisfiable/contradiction - not satisfiable. e.g., self-contradictory A ^ ~A

Computer inference

• doesn't know interpretation of sentences
• doesn't know about world except via sentences
• can only do symbolic manipulation of sentence representations
• we provide interpretation of conclusions (valid sentences reached via inference process)
• can handle very complex sentences

Logic

• formal system for describing state of affairs, including syntax and semantics
• proof theory or set of rules for deducing entailments

Propositional logic

• symbols represent whole propositions (facts)
• use Boolean connectives to generate more complex sentences
• little commitment about how things are represented
• not very useful

First-order logic

• contains objects and predicates on objects (i.e., properties of objects or relationships between them)
• includes connectives and quantifiers
• Can capture much of what we know about the world

Ontology - nature of reality... True or False in propositional. Properties of objects and relationships between them in first-order.

Temporal logic - set of time points, can reason about time.

Epistemology - states of knowledge. What can we know? For these logics, have three states of belief: believe true, believe false, or be unable to conclude. Some systems will have degrees of belief or assign a degree of truth (fuzzy).

Propositional Logic

Syntax:

• Symbols include True/False/prepositional symbol such as P, logical connectives, and parentheses.

A sentence consists of:

• logical constants True and False
• propositional symbol such as P or Q
• symbols inside parentheses are a sentence (P v Q)
• ^ and/conjunction
• v or/disjunction
• =>implies/implication. (P ^ Q) => R. premise or antecedent is (P ^ Q). conclusion or consequent is R. Also known as rules or if-then statements.
• <=> equivalence/biconditional
• ~ not/negation. Operates on single sentence.

• atomic sentence = single symbol, e.g., P
• complex sentence includes connectives and parentheses
• literal is atomic sentence or negated atomic sentence

• grammar is ambiguous P v Q ^ R is either (PvQ)^R or P v (Q^R)
• need precedence: ~ ^ v => <=>

Semantics

• propositional symbol can mean whatever you want. Can be satisfiable but not valid.
• true: way the world is. False: way the world is not.
• complex sentence has meaning derived from parts. Standard truth tables.
• implication doesn't follow intuition about English (false antecedent always means true statement (making no claim when false), no causal relation between antecedent and conclusion is required)

Validity and Inference

• Truth tables used to test for valid sentences
• For P => C build truth table, if valid (true in every row), then conclude C

Models

Any world in which a sentence is true under a particular interpretation is called a model of that sentence under that interpretation

May be many models for given sentence

More claims -> fewer models

• entailment: sentence s is entailed by KB if models of KB are all models of s
• P ^ Q is intersection of models of P and models of Q

Rules of inference for prepositional logic

• certain recurring patterns of inference can be shown as sound without.
• Can then use those patterns, in the form of inference rules, without having to build truth table
• a |- b says b is derived from a by inference rule

7 common inference rules (Figure 6.13)

• modus ponens: a=>b, a conclude b
• and-eliminination: a^b^c can infer any of the conjuncts
• and-introduction: a,b,c as separate true statements infers a^b^c
• or-introduction: a can infer avbvc etc.
• double-negation elimination: ~~a implies a
• unit resolution: a v B, ~B, infer a
• resolution: avB and ~Bvg infer avg

Complexity

• Checking a set of sentences for satisfiability is NP-complete
• monotonic logic - if add new sentences to KB, all sentences entailed by original KB are still entailed. Propositional and first-order logic are monotonic.
• if not monotonic, couldn't have any local rules, because rest of KB might affect soundness of inference. Probability theory is not monotonic.
• There is a useful class of sentences for which a polynomial-time inference procedure exists.
• Horn sentence/Horn clause: P1 ^ P2 ^ ... Pn => Q
• Ps and Q are non-negated atoms
• Special case 1: True=> Q is equivalent to atomic sentence Q
• Special case 2: If Q is false, get ~P1 v ~P2 v … ~Pv
• If can write KB as collection of Horn sentences, simple inference procedure: apply Modus Ponens wherever possible until no new inferences remain to be made.

How to represent problems in prepositional logic

Read Wumpus World example

Limitations of propositional logic

• too weak. 64 rules just for "don't go forward if wumpus"
• doesn't handle change (e.g., agent moving around)