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CIS 675 Knowledge and Reasoning c P´ eter Moln´ Inference in First-Order Logic (Chapter 9)First-Order Logic (Chapter 7)Agents That Reason Logically (Chaper 6) ar, 2001 Slide 2
CIS 675 I. Agents That Reason Logically c P´ eter Moln´ Propositional LogicRepresentation, Reasoning, and LogicA Knowledge-Based Agent ar, 2001 Slide 3
CIS 675 Agents That Reason Logically A Knowledge-Based Agent c P´ eter Moln´that have been added to theInferenceAdd new sentences, query knowledge: functions Arepresentation languageindividual representation id called aKnowledge base returnt T actionT staticfunction ELL ELL ar, 2001 t+ 1( ( KB KB : KB-A action t, a counter, initially 0, indicating timeKB , MA , M , a knowledge base mechanism: determine what follows from the sentencesAKESK AKE GENT (KB -A, M-P CTIONERCEPT (percept AKE (KB -S -A -S ) ENTENCECTIONENTENCE returns ): set of representations of facts. Each . -Q (actionUERY( an KB percept, t action (t)) . ,t )) )) sentence of a SK knowledge and T ELL . Slide 4
CIS 675 Agents That Reason Logically A Knowledge-Based Agent (cont’d) c P´ eter Moln´given a series of percepts.Implementation ofdeclarativeRepresentation of knowledge in sentences simplifies the designDescription of knowledge-based agents in three levels: 3. 2. 1. pointers, or entry in 3-dim. table indexed by road links and location pairs.representation of sentences on logical level. E.g. string, objects withimplementation levelLinks(GGBridge,logical levelGate Bridge links San Francisco with Marin County”.by saying what it knows. E.g. “An automated taxi knows that the Goldenknowledge level ar, 2001 approach to system building. : knowledge is encoded into (formal) sentences. E.g. learning or SF, Marin) epistemological level : level that runs on the agent architecture; physical mechanisms that produce general knowledge : most abstract, describe agent Slide 5
CIS 675 Agents That Reason Logically Representation, Reasoning, and Logic c P´ eter Moln´exists.refer. AgentsSemanticsconfiguration or physical property of some part of the agent.constitute a sentence. Each sentence is implemented by a physicalSyntax World Representation ar, 2001 : of a language describes the possible configurations that can : determines the facts in the world to which the sentence believes Sentences Facts Semantics a sentence it the corresponding configuration Follows Entails Sentence Fact Semantics Slide 6
CIS 675 Agents That Reason Logically Representation, Reasoning, and Logic (cont’d) c P´ eter Moln´ 2. The 1. A formal system for describing states of affairs, consisting ofLogicsProof TheorycalledAn inference procedure that generates only entailed sentences is2. report whether or not1. produceInference (b) the(a) the of a set of sentences. ar, 2001 affairs). sound proof-theory consist of the following: semanticssyntax procedure or α of a language: reasoning steps that are sound.for given of the language (how to make sentences), andtruth-preserving of the language (how sentence relates to states ofKB a set of rules for deducing the entailmentsα KB is entailed byi α . KB . Slide 7
CIS 675 Agents That Reason Logically validity, methods, and scope.EpistemologyOntology Representation, Reasoning, and Logic (cont’d) c Fuzzy logicProbability theoryTemporal logicFirst-order logicPropositional logicLanguage P´ eter Moln´ ar, 2001 , branch of metaphysics that deals with the nature of being. , the theory of knowledge, esp. the critical study of its degree of truthfactsfacts, objects, relations, timesfacts, objects, relationsfacts(What exists in the world)Ontological Commitment degree of belief 0…1degree of belief 0…1true/false/unknowntrue/false/unknowntrue/false/unknown(What an agent believes about facts)Epistemological Commitment Slide 8
CIS 675 Agents That Reason Logically Propositional Logic c P´ eter Moln´ Grammar in BNF (Backus-Naur Form)andSymbols: logical constants ComplexSentence Q ar, 2001 AtomicSentence, logical connectives Connective Sentence Sentence Connective SentenceSentenceTrueAtomicSentence, True Sentence , and False , False , , parentheses, ()., propositional symbols ComplexSentence P Slide 9
CIS 675 Agents That Reason Logically SemanticsPropositional Logic (cont’d) c P´ eter Moln´ Connectives can be defined by a truth table:interpretation,Logical constants:arbitrary fact.Proposition symbols can carry any meaning, interpretation is an TrueTrueFalseFalse P ar, 2001 TrueFalseTrueFalse Q False FalseFalseTrueTrue True P means the fact is false. means the fact is always true, under any TrueFalseFalseFalseP Q TrueTrueTrueFalseP Q TrueFalseTrueTrueP Q TrueFalseFalseTrueP Q Slide 10
CIS 675 Agents That Reason Logically Inference RulesPropositional Logic (cont’d) c P´ eter Moln´ conjunctions.And-Introductionconjuncts.And-Eliminationimplication, you can infer the conclusion.Modus Ponens ar, 2001 : from an implication and the premise of the : from a sequence of sentences, you can infer their: from a conjunction, you can infer any of the α α α 1 1 1 α α α α 2 2 α 2 β β i α α α α n n n Slide 11
CIS 675 Agents That Reason Logically More Inference RulesPropositional Logic (cont’d) c P´ eter Moln´ ResolutionUnit ResolutionDouble-Negation Eliminationanything.Or-Introduction ar, 2001 α : α β : : from a sentence, you can infer its disjunction with γ β γ or equivalently α 1 α : α α β α 2 α α i β α α n α β γ β γ Slide 12
CIS 675 II. First-Order Logic c P´ eter Moln´ also called Examples:for a given input.relations areobjects. Among the objects, variousindividual identities andOntological commitment: world consists of – – square; Property: smelly; Relation: neighboring.“Squares neighboring the wumpus are smelly” Objects: wumpus,two; Relations: equals; Function: plus.“One plus two equals three” Objects: one, two, three, one plus ar, 2001 First-Order Predicate Calculus functions – relations in which there is only one valueproperties relationsthat distinguish them from other . objects hold. Some of the , i.e., things with Slide 13
CIS 675 First-Order Logic c P´ eter Moln´ Prolog ExamplesUsing First-Order Logic(Extension and Notational Variations)Syntax and Semantics ar, 2001 Slide 14
CIS 675 First-Order Logic Syntax and Semantics c P´ eter Moln´ AtomicSentence ar, 2001 Sentence Term VariableConstantFunctionPredicateSentenceQuanti f ierVariableSentenceConnectiveSentenceAtomicSentence Sentence Term Term Term Sentence Term Slide 15
CIS 675 First-Order Logic Syntax and Semantics (cont’d) c P´ eter Moln´ ar, 2001 Quanti f ierConnective FunctionPredicateVariableConstant motherbe f oreA a X x 1 Namejohn le f tLegO fhasColor raining Slide 16
CIS 675 First-Order Logic Syntax and Semantics (cont’d) P´eter Moln´c objects, rather than having to enumerate the objects by name.QuantifiersComplex SentenceAtomic sentenceTerm state facts.: logical expression that refers to an object. ar, 2001 : allows to express properties of entire collections ofmarried : using logical connectives. : terms f atherO f brother objects, predicate symbols richardrichard motherO fjohn john relations Slide 17
CIS 675 First-Order Logic Syntax and Semantics: Quantifiers c P´ eter Moln´true for at least one value.“There exists an animal that is the sister of Spot and it is also a cat” isExistential quantificationfor all possible values of“For all animals that are cats we can imply they are mammals” is trueUniversal quantificationbinding variablesSubstituting the name ar, 2001 ANIMAL ANIMAL john sister cat john ANIMALANIMAL PERSON ANIMAL of an object for a variable . . spot mammal cat ANIMAL ANIMAL PERSON , Slide 18
CIS 675 First-Order Logic Using First-Order Logic c P´ eter Moln´ Example:facts of the domain.some knowledge; use ofDomain X G ar, 2001 Y C : a section of the world about which we wish to express siblinggrand parent W M H C X husbandmother Y P C X G parentmaleH M axioms X C W C Y X , and P P C f emale P male parent f emale definitions parent child H M P G C X spouse P P parent to capture basic X parentparent H W M C P P Y C Slide 19
CIS 675 First-Order Logic Prolog Examples c P´ eter Moln´ No equivalence Exceptions:Syntax: similar to first-order logic (predicate calculus). ar, 2001 α α α β β β . alpha; betaalpha, beta.beta :- alpha. alpha. beta. Slide 20
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