Knowledge Representation in Intelligent Agents

1. Knowledge Representation Lecture # 17, 18 & 19

2. Motivation (1) • Up to now, we concentrated on search methods in worlds that can be relatively easily represented by states and actions on them • a few objects, rules, relatively simple states • problem-specific heuristics to guide the search • complete knowledge: know all what’s needed • no new knowledge is deduced or added • well-defined start and goal states • Appropriate for accessible, static, discrete problems.

3. Motivation (2) • What about other types of problems? • More objects, more complex relations • Not all knowledge is explicitly stated • Dynamic environments: the rules change! • Agents change their knowledge • Deduction: how to derive new conclusions • Examples 1. queries on family relations 2. credit approval 3. diagnosis of circuits

4. Knowledge Representation • A complete intelligent agent needs to be able to perform several tasks: – Perception: what is my state? – Deliberation: what action should I take? – Action: how do I execute the action? • State recognition implies some form of knowledge representation • Figuring out the right action implies some form of inference Two levels to think about: • Knowledge level; what does the agent know? • Implementation level: how is the knowledge represented? Key question: What is a good representation?

5. Knowledge Bases The golden dream: • Tell the agent what it needs to know • The agent uses rules of inference to deduce consequences This is the declarative approach to building agents. Agents have two different parts: • A knowledge base, which contains a set of facts expressed in some formal, standard language • An inference engine, with general rules for deducing new facts

6. Knowledge Base Knowledge Base: set of sentences represented in a knowledge representation language and represents assertions about the world. Inference rule: when one ASKs questions of the KB, the answer should follow from what has been TELLed to the KB previously. ask tell

7. CONTD… • The Knowledge Base is a set of sentences. • Syntactically well-formed • Semantically meaningful • A user can perform two actions to the KB: • Tell the KB a new fact • Ask the KB a question

8. Syntax of Sentences • English acceptable an one is sentence This vs. • This English sentence is an acceptable one. • V P – ^Q R vs. • P V –Q ^ R

9. Semantics of Sentences This hungry classroom is a jobless moon. • Why is this syntactically correct sentence not meaningful? P V –Q ^ R • Represents a world where either P is true, or Q is not true and R is true.

10. Logical Agents • Reflex agents find their goal state by dumb luck • Logic (Knowledge-Based) agents combine general knowledge with current percepts to infer hidden aspects of current state prior to selecting actions • Crucial in partially observable environments

11. sensors environment ? agent Domain-specific content actuators Knowledge base Inference Engine Domain-independent algorithms Knowledge-Based Agent

12. A Knowledge-Based Agent • A knowledge-based agent consists of a knowledge base (KB) and an inference engine (IE). • A knowledge-base is a set of representations of what one knows about the world (objects and classes of objects, the fact about objects, relationships among objects, etc.) • Each individual representation is called a sentence.

13. Abilities KB agent • Agent must be able to: • Represent states and actions, • Incorporate new percepts • Update internal representation of the world • Deduce hidden properties of the world • Deduce appropriate actions

14. Knowledgebase Agents • The sentences are expressed in a knowledge representation language. • Examples of sentences • The moon is made of green cheese • If A is true then B is true • A is false • All humans are mortal • Confucius is a human

15. Inference Engine Input from environment Output (actions) Learning (KB update) Knowledge Base • The Inference engine derives new sentences from the input and KB • The inference mechanism depends on representation in KB • The agent operates as follows: • 1. It receives percepts from environment • 2. It computes what action it should perform (by IE and KB) • 3. It performs the chosen action (some actions are simply inserting inferred new facts into KB).

16. The Wumpus World The Wumpus computer game • The agent explores a cave consisting of rooms connected by passageways. • Lurking somewhere in the cave is the Wumpus, a beast that eats any agent that enters its room. • Some rooms contain bottomless pits that trap any agent that wanders into the room. • Occasionally, there is a heap of gold in a room. • The goal is to collect the gold and exit the world without being eaten

17. Wumpus PEAS description • Performance measure: gold +1000, death -1000, -1 per step, -10 use arrow • Environment: • Squares adjacent to wumpus are smelly • Squares adjacent to pit are breezy • Glitter iff gold is in the same square • Bump iff move into a wall • Woeful scream iff the wumpus is killed • Shooting kills wumpus if you are facing it • Shooting uses up the only arrow • Grabbing picks up gold if in same square • Releasing drops the gold in same square • Sensors:Stench, Breeze, Glitter, Bump, Scream • Actuators:Let turn, Right turn, Forward, Grab, Release, Shoot

20. Exploring the Wumpus World [1,1] The KB initially contains the rules of the environment. The first percept is [none, none,none,none,none], move to safe cell e.g. 2,1

21. Exploring the Wumpus World [2,1] = breeze indicates that there is a pit in [2,2] or [3,1], return to [1,1] to try next safe cell

22. Exploring the Wumpus World [1,2] Stench in cell which means that wumpus is in [1,3] or [2,2] YET … not in [1,1] YET … not in [2,2] or stench would have been detected in [2,1] (this is relatively sophisticated reasoning!)

23. Exploring the Wumpus World [1,2] Stench in cell which means that wumpus is in [1,3] or [2,2] YET … not in [1,1] YET … not in [2,2] or stench would have been detected in [2,1] (this is relatively sophisticated reasoning!) THUS … wumpus is in [1,3] THUS [2,2] is safe because of lack of breeze in [1,2] THUS pit in [1,3] (again a clever inference) move to next safe cell [2,2]

24. Exploring the Wumpus World [2,2] move to [2,3] [2,3] detect glitter , smell, breeze THUS pick up gold THUS pit in [3,3] or [2,4]

26. Representation, Reasoning, and Logic • The objective of knowledge representation is to express knowledge in a computer-tractable form, so that agents can perform well. • A knowledge representation language is defined by: • Its syntax which defines all possible sequences of symbols that constitute sentences of the language (grammar to form sentences)

27. Representation, Reasoning, and Logic • Its semantics determines the facts in the world to which the sentences refer (meaning of sentences) • Each sentence makes a claim about the world. • Its proof theory (inference rules and proof procedures)

28. Logic in general • Logics are formal languages for representing information such that conclusions can be drawn • Syntax defines the sentences in the language • Semantics define the "meaning" of sentences; • i.e., define truth of a sentence in a world • E.g., the language of arithmetic • x+2 ≥ y is a sentence; x2+y > {} is not a sentence • x+2 ≥ y is true iff the number x+2 is no less than the number y • x+2 ≥ y is true in a world where x = 7, y = 1 • x+2 ≥ y is false in a world where x = 0, y = 6

29. Types of Logic • Logics are characterized by what they commit to as “primitives” • Ontological commitment: what exists—facts? objects? time? • beliefs? • Epistemological commitment: what states of knowledge?

30. Interpretations • We want to have a rule for generating (or testing) new sentences that are always true • But the truth of a sentence may depend on its interpretation! • Formally, an interpretationis a way of matching objects in the world with symbols in the sentence (or in the knowledge database) • A sentence may be true in one interpretation and false in another • Terminology: • A sentence is validif it is true in all interpretations • A sentence is satisfiableif it is true in at least one interpretation • A sentence is unsatisfiableif it is false in all interpretations

31. Entailment • Entailment means that one thing follows from another: KB ╞α • Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true • For example (P ^ Q) ⊨ (P V R) • Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

32. M() M(KB) Models • Models are formal definitions of possible states of the world • We say m is a model of a sentence  if  is true in m • M() is the set of all models of  • Then KB  if and only if M(KB)  M()

33. Inference • KB |-i : sentence  can be derived from KB by inference procedure I • For example -(AVB) ⊢ (-A ^ -B) • Soundness: KB ⊢ f → KB ⊨ f i.e. all conclusions arrived at via the proof procedure are correct: they are logical consequences. • Completeness: KB ⊨ f → KB ⊢ f i.e. every logical consequence can be generated by the proof procedure

34. Propositional Logic: Syntax

35. Propositional Logic: Semantics

36. Truth Tables

37. Propositional Inference: Truth Table Method

38. Logical equivalence • Two sentences are logically equivalent iff both are true in same models: α ≡ ß iff α╞ βand β╞ α

39. Wumpus world sentences • Let Pi,j be true if there is a pit in [i,j] • Let Bi,j be true if there is a breeze in [i,j] • R1: ¬P1,1 • “Pits cause breezes in adjacent squares” • R2: B11  P12 v P21 • R3: B2,1 (P1,1 P2,2 P3,1) • Include breeze precepts for the first 2 moves • R4: ¬B1,1 • R5: B2,1

40. Normal Forms

41. Horn Sentences

42. Example: Conversion to CNF Any KB can be converted into CNF B1,1 (P1,2 P2,1) • Eliminate , replacing α  β with (α  β)(β  α). (B1,1 (P1,2 P2,1))  ((P1,2 P2,1)  B1,1) 2. Eliminate , replacing α  β with α β. (B1,1 P1,2 P2,1)  ((P1,2 P2,1)  B1,1) 3. Move  inwards using de Morgan's rules and double-negation: (B1,1  P1,2 P2,1)  ((P1,2 P2,1)  B1,1) 4. Apply distributive law ( over ) and flatten: (B1,1 P1,2 P2,1)  (P1,2  B1,1)  (P2,1 B1,1)

43. Validity and Satisfiability

44. Complementary Literals • A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P, P • Two literals are complementary if one is the negation of the other, e.g.: P and P

45. Reasoning Patterns/Rules of Inference

46. Proofs using Inference in Wumpus World

47. Proof Methods Proof methods divide into (roughly) two kinds: • Model checking: • Truth table enumeration (sound and complete for propositional logic) • Heuristic search in model space (sound but incomplete) • Application of inference rules: • Legitimate (sound) generation of new sentences from old • A proof is a sequence of inference rule applications • Inference rules can be used as operators in a standard search algorithm!

48. PL is Too Weak a Representational Language • Consider the problem of representing the following information: • Every person is mortal. (S1) • Confucius is a person. (S2) • Confucius is mortal. (S3) • S3 is clearly a logical consequence of S1 and S2. But how can these sentences be represented using PL so that we can infer the third sentence from the first two? • We can use symbols P, Q, and R to denote the three propositions, but this leads us to nowhere because knowledge important to infer R from P and Q (i.e., relationship between being a human and mortality, and the membership relation between Confucius and human class) is not expressed in a way that can be used by inference rules

49. Weakness of PL

50. Weakness of PL