Logical AgentsOutlineKnowledge basesA simple knowledge-based agentWumpus World PEAS descriptionWumpus world characterizationExploring a wumpus worldSlide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Logic in generalEntailmentModelsEntailment in the wumpus worldWumpus modelsSlide 20Slide 21Slide 22Slide 23InferencePropositional logic: SyntaxPropositional logic: SemanticsTruth tables for connectivesWumpus world sentencesTruth tables for inferenceInference by enumerationLogical equivalenceValidity and satisfiabilityProof methodsResolutionSlide 35Conversion to CNFResolution algorithmResolution exampleForward and backward chainingForward chainingForward chaining algorithmForward chaining exampleSlide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Proof of completenessBackward chainingBackward chaining exampleSlide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Forward vs. backward chainingEfficient propositional inferenceThe DPLL algorithmSlide 65The WalkSAT algorithmSlide 67Hard satisfiability problemsSlide 69Slide 70Inference-based agents in the wumpus worldPowerPoint PresentationExpressiveness limitation of propositional logicSummaryLogical AgentsChapter 7Outline•Knowledge-based agents•Wumpus world•Logic in general - models and entailment•Propositional (Boolean) logic•Equivalence, validity, satisfiability•Inference rules and theorem proving–forward chaining–backward chaining–resolution–Knowledge bases•Knowledge base = set of sentences in a formal language•Declarative approach to building an agent (or other system):–Tell it what it needs to know•Then it can Ask itself what to do - answers should follow from the KB•Agents can be viewed at the knowledge leveli.e., what they know, regardless of how implemented•Or at the implementation level–i.e., data structures in KB and algorithms that manipulate them––»–•A simple knowledge-based agent•The agent must be able to:–Represent states, actions, etc.–Incorporate new percepts–Update internal representations of the world–Deduce hidden properties of the world–Deduce appropriate actions–––––•Wumpus World PEAS description•Performance measure–gold +1000, death -1000–-1 per step, -10 for using the arrow•Environment–Squares adjacent to wumpus are smelly–Squares adjacent to pit are breezy–Glitter iff gold is in the same square–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: Left turn, Right turn, Forward, Grab, Release, Shoot•»–––––––»Wumpus world characterization•Fully Observable No – only local perception•Deterministic Yes – outcomes exactly specified•Episodic No – sequential at the level of actions•Static Yes – Wumpus and Pits do not move•Discrete Yes•Single-agent? Yes – Wumpus is essentially a natural feature••••••Exploring a wumpus worldExploring a wumpus worldExploring a wumpus worldExploring a wumpus worldExploring a wumpus worldExploring a wumpus worldExploring a wumpus worldExploring a wumpus worldLogic 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–––»–•••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–E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won”–E.g., x+y = 4 entails 4 = x+y–Entailment is a relationship between sentences (i.e., syntax) that is based on semantics––••Models•Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated•We say m is a model of a sentence α if α is true in m•M(α) is the set of all models of α•Then KB ╞ α iff M(KB)  M(α)–E.g. KB = Giants won and Redswon α = Giants won–•••Entailment in the wumpus worldSituation after detecting nothing in [1,1], moving right, breeze in [2,1]Consider possible models for KB assuming only pits3 Boolean choices  8 possible modelsWumpus modelsWumpus models•KB = wumpus-world rules + observations•Wumpus models•KB = wumpus-world rules + observations•α1 = "[1,2] is safe", KB ╞ α1, proved by model checking••Wumpus models•KB = wumpus-world rules + observationsWumpus models•KB = wumpus-world rules + observations•α2 = "[2,2] is safe", KB ╞ α2•Inference•KB ├i α = sentence α can be derived from KB by procedure i•Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α•Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α •Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure.•That is, the procedure will answer any question whose answer follows from what is known by the KB.•••••Propositional logic: Syntax•Propositional logic is the simplest logic – illustrates basic ideas•The proposition symbols P1, P2 etc are sentences–If S is a sentence, S is a sentence (negation)–If S1 and S2 are sentences, S1  S2 is a sentence (conjunction)–If S1 and S2 are sentences, S1  S2 is a sentence (disjunction)–If S1 and S2 are sentences, S1  S2 is a sentence (implication)–If S1 and S2 are sentences, S1  S2 is a sentence (biconditional)–––––•Propositional logic: SemanticsEach model specifies true/false for each proposition symbolE.g. P1,2 P2,2 P3,1 false true falseWith these symbols, 8 possible models, can be enumerated automatically.Rules for evaluating truth with respect to a model m:S is true iff S is false S1  S2 is true iff S1 is true and S2 is trueS1  S2 is true iff S1is true or S2 is trueS1  S2 is true iff S1 is