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22c145-Fall’02: Knowledge Representation and Reasoning Knowledge Representation and Reasoning Readings: Chapter 6 of Russell & Norvig. Hantao Zhang 1 22c145-Fall’02: Knowledge Representation and Reasoning Reasoning Agents Remember our goal-based agent. Sensors State What the world is like now What my actions do What it will be like if I do action A Goals What action I should do now Agent Hantao Zhang Environment How the world evolves Effectors 2 22c145-Fall’02: Knowledge Representation and Reasoning (Knowledge-based) Reasoning Agents Know about the world They maintain a collection of facts (sentences) about the world, the Knowledge Base, expressed in some formal language. Reason about the world They are able to derive new facts from those in the KB using some inference mechanism. Act upon the world They map percepts to actions by querying and updating the KB. Hantao Zhang 3 22c145-Fall’02: Knowledge Representation and Reasoning Automated Reasoning Main Assumption (or the “Church Thesis” of AI) 1. Facts about the world can be represented as particular configurations of symbols 1. 2. Reasoning about the world can be achieved by mere symbol manipulation. AI researchers believe that reasoning is symbol manipulation, nothing else. (After all, the human brain is a physical system itself!) 1 ie physical entities such as marks on a piece of paper, states in a computer’s memory, and so on. Hantao Zhang 4 22c145-Fall’02: Knowledge Representation and Reasoning Abstraction Levels It looks like we can describe every reasoning agent (natural or not) at two different abstraction levels. • Knowledge level: what the agent knows and what the agent’s goals are. • Symbol level: what symbols the agent is manipulating and how. Agent’s Knowledge and Goals Internal configuration of symbols reasoning symbol manipulation Agent’s Knowledge and Goals Internal configuration of symbols Knowledge Level Symbol Level At least for artificial agents, we may say that the knowledge level is the metaphor by which we explain the behavior of the agent, which is really at the symbol level. Hantao Zhang 5 22c145-Fall’02: Knowledge Representation and Reasoning A Motivating Example: The Wumpus World! 4 Breeze Stench Breeze 3 Stench PIT Breeze PIT Actions Gold 2 Breeze 1 Move Forward, Turn Left, Turn Right, Grab, Shoot, Climb Breeze Stench Breeze PIT START 1 Hantao Zhang 2 3 Percepts Stench Breeze Glitter Bump Scream y/n y/n y/n y/n y/n 4 Rewards/Punishments 1,000pts for climbing out with gold -1pt per action taken -10,000pts for getting killed 6 22c145-Fall’02: Knowledge Representation and Reasoning The Wumpus World (cont’t) Perceipts = ( Stench, Breeze, Glitter, Bump, Screm ) • A stench is perceived in the squares containing or adjacent to the wumpus. • A breeze is perceived in the squares adjacent to a pit. • A glitter is perceived in the square containing the gold. • A bump is perceived if the agent walks into a wall. • A scream is perceived anywhere if the wumpus is killed. Hantao Zhang 7 22c145-Fall’02: Knowledge Representation and Reasoning 1,4 2,4 3,4 4,4 1,3 2,3 3,3 4,3 1,2 2,2 3,2 4,2 A B G OK P S V W = Agent = Breeze = Glitter, Gold = Safe square = Pit = Stench = Visited = Wumpus 1,4 2,4 3,4 4,4 1,3 2,3 3,3 4,3 1,2 2,2 3,2 4,2 OK OK 1,1 2,1 3,1 4,1 A OK P? 1,1 2,1 V OK OK (a) • What are the safe moves from (1,1)? A B OK 3,1 P? 4,1 (b) Move to (1,2), (2,1), or stay in (1,1) • Move to (2,1) then. • What are the safe moves from (2,1)? B in (2,1) ⇒ P in (2,2) or (3,1) or (1,1) • Move to (1,2) then. Hantao Zhang 8 22c145-Fall’02: Knowledge Representation and Reasoning 1,4 1,3 1,2 W! A 2,4 3,4 4,4 2,3 3,3 4,3 2,2 3,2 4,2 S OK 1,1 = Agent = Breeze = Glitter, Gold = Safe square = Pit = Stench = Visited = Wumpus 1,4 2,4 1,3 W! 1,2 OK 2,1 V OK A B G OK P S V W B V OK 3,1 P! 4,1 S V OK 1,1 3,4 4,4 2,3 3,3 P? 4,3 2,2 3,2 4,2 A S G B V OK 2,1 V OK (a) P? B V OK 3,1 P! 4,1 (b) S in (1,2) ⇒ W in (1,1) or (2,2) or (1,3) • Survived in (1,1) and no S in (2,1) ⇒ W in (1,3) No B in (1,2) ⇒ P in (3,1) • Move to (2,2), then to (2,3). • G in (2,3). • Grab G and come home ... Hantao Zhang 9 22c145-Fall’02: Knowledge Representation and Reasoning Knowledge Representation An (artificial) agent represents knowledge as a collection of sentences in some formal language, the knowledge representation language. A knowledge representation language is defined by its • syntax, which describes all the possible symbol configurations that constitute a sentence, • semantics, which maps each sentence of the language to a fact about the world. Ex: Arithmetic • x + y > 3 is a sentence; x+ > y is not. • the semantics of x + y > 3 is either the fact “true” or the fact “false”. • x + y > 3 is “true” iff the number x + y is greater than the number three. Hantao Zhang 10 22c145-Fall’02: Knowledge Representation and Reasoning Knowledge Representation and Reasoning At the semantical level, reasoning is the process of deriving new facts from previous ones. At the syntactical level, this process is mirrored by that of producing new sentences from previous ones. The production of sentences from previous ones should not be arbitray. Only entailed sentences should be derivable. Hantao Zhang 11 22c145-Fall’02: Knowledge Representation and Reasoning Entailment Informally, a sentence ϕ is entailed by a set of sentences Γ iff the fact denoted by ϕ follows from the facts denoted by Γ. Sentences Facts Hantao Zhang Semantics World Semantics Representation Sentence Entails Follows Fact 12 22c145-Fall’02: Knowledge Representation and Reasoning Entailment Notation Γ |= ϕ if the set of sentences Γ entail the sentence ϕ. Intuitive reading of Γ |= ϕ: Whenever Γ is true in the world, ϕ is also true. Examples Let Γ consist of the axioms of arithmetic. {x = y, y = z} |= x = z Γ ∪ {x + y ≥ 0} |= x ≥ −y Γ ∪ {x + y = 3, x − y = 1} |= x = 2 Γ ∪ {x + y = 3} Hantao Zhang 6|= x = 2 13 22c145-Fall’02: Knowledge Representation and Reasoning Inference Systems At the knowledge representation level, reasoning is achieved by an inference system I, a computational device able to derive new sentences from previous ones. Notation Γ `I ϕ if I can derive the sentence ϕ from the set Γ. To be useful at all, an inference system must be sound: if Γ `I ϕ then Γ |= ϕ holds as well. An ideal inference system is also complete: if Γ |= ϕ then Γ `I ϕ holds as well. Hantao Zhang 14 22c145-Fall’02: Knowledge Representation and Reasoning Inference Rules An inference system is typically described as a set of inference (or derivation) rules. Each derivation rule has the form: P1, . . . , Pn ←− premises ←− conclusion C Hantao Zhang 15 22c145-Fall’02: Knowledge Representation and Reasoning Derivation Rules and Soundness A derivation rule is sound if it derives true conclusions from true premises. All men are mortal Aristotle is a man Aristotle is mortal All men are mortal Aristotle is mortal 2 All men are Aristotle 2 Sound Inference Unsound Inference! Woody Allen in Love and Death, 1975. Hantao Zhang 16 22c145-Fall’02: Knowledge Representation and Reasoning Knowledge Representation Languages Why don’t we use natural language (eg, English) to represent knowledge? • Natural language is certainly expressive enough! • But it is also too ambiguous for automated reasoning. Ex: I saw the boy on the hill with the telescope. Why don’t we use programming languages? • They are certainly well-defined and unambiguous. • But they are not expressive enough. Hantao Zhang 17 22c145-Fall’02: Knowledge Representation and Reasoning Knowledge Representation and Logic The field of Mathematical Logic provides powerful, formal knowledge representation languages and inference systems to build reasoning agents. Inference and Logic Soundness Indispensable but easily achievable: an inference system is sound if all of its rules are sound. Completeness Generally hard or even impossible to achieve. But most of the times we can live without it . . . Hantao Zhang 18 22c145-Fall’02: Knowledge Representation and Reasoning Logics A logic is a triple hL, S, Ri where • L, the logic’s language, is a class of sentences described by a formal grammar. • S, the logic’s semantics is a formal specification of how to assign meaning in the “real world” to the elements of L. • R, the logic’s inference system, is a set of formal derivation rules over L. There are several logics: propositional, first-order, higher-order, modal, temporal, intuitionistic, linear, equational, non-monotonic, fuzzy, . . . We will concentrate on propositional logic and first-order logic. Hantao Zhang 19 22c145-Fall’02: Knowledge Representation and Reasoning Propositional Logic Each sentence is made of • propositional variables (A, B, . . . , P, Q, . . . ) • logical constants (True, False). • logical connectives (∧, ∨, ⇒, . . . ). Every propositional variable stands for a basic fact. Ex: I’m hungry, Apples are red, Bill and Hillary are married. Hantao Zhang 20 22c145-Fall’02: Knowledge Representation and Reasoning Propositional Logic The Language • Each propositional variable (A, B, . . . , P, Q, . . . ) is a sentence. • Each logical constant (True, False) is a sentence. • If ϕ and ψ are sentences, all of the following are also sentences. (ϕ) ¬ϕ ϕ∧ψ ϕ∨ψ ϕ⇒ψ ϕ⇔ψ • Nothing else is a sentence. Hantao Zhang 21 22c145-Fall’02: Knowledge Representation and Reasoning The Language of Propositional Logic Formally, it is the language generated by the following grammar. • Symbols: – Propositional variables: A, B, . . . , P, Q, . . . – Logical constants: True (true) False (false) ∧ (and) ∨ (or) ⇒ (implies) ⇔ (equivalent) ¬ (not) • Grammar Rules: Sentence AtomicS ComplexS Connective Hantao Zhang ::= ::= ::= ::= AtomicS | ComplexS True | False | A | B | . . . | P | Q | . . . (Sentence) | Sentence Connective Sentence | ¬Sentence ∧| ∨ | ⇒ | ⇔ 22 22c145-Fall’02: Knowledge Representation and Reasoning Propositional Logic Ontological Commitments Propositional Logic is about facts in the world that are either true or false, nothing else. Semantics of Propositional Logic Since each propositional variable stands for a fact about the world, its meaning ranges over the Boolean values {True, False}. Note: Do note confuse, as the textbook does, True, False, which are values (ie semantical entities) with True, False which are logical constants (ie symbols). Hantao Zhang 23 22c145-Fall’02: Knowledge Representation and Reasoning Semantics of Propositional Logic • The meaning (value) of True is always True. The meaning of False is always False. • The meaning of the other sentences depends on the meaning of the propositional variables. – Base cases: Truth Tables True True False False Q False False False True P Q False True True True P Q True True False True P False True False True P False False True True P Q P Q True False False True – Non-base Cases: Given by reduction to the base cases. Ex: the meaning of (P ∨ Q) ∧ R is the same as the meaning of A ∧ R where A has the same meaning as P ∨ Q. Hantao Zhang 24 22c145-Fall’02: Knowledge Representation and Reasoning The Meaning of Logical Connectives: A Warning Disjunction • A ∨ B is true when A or B or or both are true (inclusive or). • A ⊕ B is sometimes used to mean “either A or B but not both” (exclusive or). Implication • A ⇒ B does not require a causal connection between A and B. Ex: Sky-is-blue ⇒ Snow-is-white • When A is false, A ⇒ B is always true regardless of the value of B. Ex: Two-equals-four ⇒ Apples-are-red • Beware of negations in implications. Ex: (¬Has-blue-seal) ⇒ (¬Chiquita) Hantao Zhang 25 22c145-Fall’02: Knowledge Representation and Reasoning Semantics of Propositional Logic • An assignment of Boolean values to the propositional variables of a sentence is an interpretation of the sentence. H) False False True False H ((P H) H) False True True True (P False True False True H False False True True P H P P True True True True • The semantics of Propositional logic is compositional: – The meaning of a sentence is given recursively in terms of the meaning of the sentence’s components (all the way down to its propositional variables). Hantao Zhang 26 22c145-Fall’02: Knowledge Representation and Reasoning Semantics of Propositional Logic The meaning of a sentence in general depends on its interpretation. Some sentences, however, have always the same meaning. H) H False False True False ((P H) H) False True True True (P False True False True H False False True True P H P P True True True True A sentence is • satisfiable if it is true in some interpretation, • valid if it is true in every possible interpretation. Hantao Zhang 27 22c145-Fall’02: Knowledge Representation and Reasoning Entailment in Propositional Logic Given a set Γ of sentences and a sentence ϕ, we write Γ |= ϕ iff every interpretation that makes all sentences in Γ true makes ϕ also true. Γ |= ϕ is read as “Γ entails ϕ” or “ϕ logically follows from Γ.” Hantao Zhang 28 22c145-Fall’02: Knowledge Representation and Reasoning Entailment in Propositional Logic: Examples {A, A ⇒ B} {A} {A, B} {} {A} {A ∨ ¬A} 1. 2. 3. 4. Hantao Zhang A False False True True B A⇒B False True True True False False True True |= |= |= |= 6|= 6|= B A∨B A∧B A ∨ ¬A A∧B A A∨B False True True True A ∧ B A ∨ ¬A False True False True False True True True 29 22c145-Fall’02: Knowledge Representation and Reasoning Entailment in Propositional Logic Note: • Γ |= ϕ, for all ϕ ∈ Γ (inclusion property of PL) • if Γ |= ϕ, then Γ0 |= ϕ for all Γ0 ⊇ Γ (monotonicity of PL) • ϕ is valid iff True |= ϕ (also written as |= ϕ) • ϕ is unsatisfiable iff ϕ |= False • Γ |= ϕ iff the set Γ ∪ {¬ϕ} is unsatisfiable Hantao Zhang 30 22c145-Fall’02: Knowledge Representation and Reasoning Logical Equivalence Two sentences ϕ1 and ϕ2 are logically equivalent ϕ1 ≡ ϕ2 if ϕ1 |= ϕ2 and ϕ2 |= ϕ1. Note: • If ϕ1 ≡ ϕ2, every interpretation of their propositional variables will assign the same Boolean value to ϕ1 and ϕ2. • Implication and equivalence (⇒, ⇔), which are syntactical entities, are intimately related to entailment and logical equivalence (|=, ≡), which are semantical notions: ϕ1 |= ϕ2 iff |= ϕ1 ⇒ ϕ2 ϕ1 ≡ ϕ2 iff |= ϕ1 ⇔ ϕ2 Hantao Zhang 31