Logical Agents Copyright, 1996 © Dale Carnegie & Associates, Inc. Chapter 7 Spring 2005

CS 471/598, CBS598 by H. Liu2 A knowledge-based agent Accepting new tasks in explicit goals Knowing about its world current state of the world, unseen properties from percepts, how the world evolves help deal with partially observable environments help understand “John threw the brick thru the window and broke it.” – natural language understanding Reasoning about its possible course of actions Achieving competency quickly by being told or learning new knowledge

CS 471/598, CBS598 by H. Liu3 Knowledge Base A knowledge base (KB) is a set of representations (sentences) of facts about the world. TELL and ASK - two basic operations to add new knowledge to the KB to query what is known to the KB Infer - what should follow after the KB has been TELLed. A generic KB agent (Fig 7.1)

CS 471/598, CBS598 by H. Liu4 Three levels of A KB Agent Knowledge level (the most abstract) Logical level (knowledge is of sentences) Implementation level Building a knowledge base A declarative approach - telling a KB agent what it needs to know A procedural approach – encoding desired behaviors directly as program code A learning approach - making it autonomous

CS 471/598, CBS598 by H. Liu5 Specifying the environment The Wumpus world (Fig 7.2) in PEAS Performance: for getting the gold, for being dead, -1 for each action taken, -10 for using up the arrow  Goal: bring back gold as quickly as possible Environment: 4X4, start at (1,1)... Actions: Turn, Grab, Shoot, Climb, Die Sensors: (Stench, Breeze, Glitter, Bump, Scream) It’s possible that the gold is in a pit or surrounded by pits -> try not to risk life, just go home empty-handed The variants of the Wumpus world – they can be very difficult Multiple agents Mobile wumpus Multiple wumpuses

CS 471/598, CBS598 by H. Liu6 Acting & reasoning Let’s play the wumpus game! The conclusion: “what a fun game!” Another conclusion: If the available information is correct, the conclusion is guaranteed to be correct.

CS 471/598, CBS598 by H. Liu7 Representation Knowledge representation Syntax - the possible configurations that can constitute sentences Semantics - the meaning of the sentences  x > y is a sentence about numbers; or x+y=4;  A sentence can be true or false  Defines the truth of each sentence w.r.t. each possible world What are possible worlds for x+y = 4 Entailment: one sentence logically follows another  |= , iff  is true,  is also true Sentences entails sentence w.r.t. aspects follows aspect (Fig 7.6)

CS 471/598, CBS598 by H. Liu8 Reasoning KB entails sentence s if KB is true, s is true Model checking (Fig 7.5) for two sentences/models  Asking whether KB entails s? S1 = “There is no pit in [1,2]” -> yes or no? S2 = “There is no pit in [2,2]” -> yes or no? An inference procedure can generate new valid sentences or verify if a sentence is valid given KB is sound if it generates only entailed sentences A proof is the record of operation of a sound inference procedure An inference procedure is complete if it can find a proof for any sentence that is entailed.

CS 471/598, CBS598 by H. Liu9 Inference Sound reasoning is called logical inference or deduction. A sentence is valid or necessarily true iff it is true under all possible interpretations in all possible worlds (a model is a world). Valid sentences are tautologies A sentence is satisfiable iff there is some interpretation in some world for which it is true. E.g., in Figure 7.9, there are three true models for the KB with 5 rules.

CS 471/598, CBS598 by H. Liu10 Logics A logic consists of the following: A formal system for describing states of affairs, consisting of syntax (how to make sentences) and semantics (to relate sentences to states of affairs). A proof theory - a set of rules for deducing the entailments of a set of sentences. Some examples of logics...

CS 471/598, CBS598 by H. Liu11 Propositional Logic In this logic, symbols represent whole propositions (facts) e.g., D means “the wumpus is dead” W 1,1 Wumpus is in square (1,1) S 1,1 there is stench in square (1,1). Propositional logic can be connected using Boolean connectives to generate sentences with more complex meanings, but does not specify how objects are represented.

CS 471/598, CBS598 by H. Liu12 Other logics First order logic represents worlds using objects and predicates on objects with connectives and quantifiers. Temporal logic assumes that the world is ordered by a set of time points or intervals and includes mechanisms for reasoning about time.

CS 471/598, CBS598 by H. Liu13 Other logics (2) Probability theory allows the specification of any degree of belief. Fuzzy logic allows degrees of belief in a sentence and degrees of truth.

CS 471/598, CBS598 by H. Liu14 Propositional logic Syntax A set of rules to construct sentences:  and, or, imply, equivalent, not  literals, atomic or complex sentences  BNF grammar (Fig 7.7, P 205 ) Semantics Specifies how to compute the truth value of any sentence Truth table for 5 logical connectives (Fig 7.8)

CS 471/598, CBS598 by H. Liu15 Inference Truth tables can be used not only to define the connectives, but also to test for validity: If a sentence is true in every row, it is valid.  What if a truth table for “Premises imply Conclusion” A simple knowledge base for Wumpus  Five rules (P208)  What if we write R2 as B1,1 => (P1,2 v P2,1) KB |= . Let’s check its validity (Fig 7.9) A truth-table enumeration algorithm (Fig 7.10)  There are only finitely many models to examine, but it is exponential in size of the input (n) A reasoning system should be able to draw conclusions that follow from the premises, regardless of the world to which the sentences are intended to refer.

CS 471/598, CBS598 by H. Liu16 Equivalence, validity, and satisfiability Logical equivalence requires  |=  and  |=  Validity: a sentence  is true in all models Valid sentences are tautologies (P v !P) Deduction theorem: for any  and ,  |=  iff the sentence (   ) is valid Satisfiability: a sentence is satisfiable if it is true in some models If  is true in a model m, then m satisfies  Validity and satisfiability:  is valid iff !  is unstatisfiable; contrapositively,  is satisfiable iff !  is not valid

CS 471/598, CBS598 by H. Liu17 Reasoning Patterns in Prop Logic  |=  iff the sentence (  ^ !  ) is unstatisfiable  are known axioms Proof by refutation (or contradiction): assuming  is F, we now need to prove !(  ^T) is valid, … Inference rules Modus Ponens, AND-elimination, Bicond-elimination All the logical equivalences in Fig 7.11 A proof is a sequence of applications of inference rules An example to conclude neither [1,2] nor [2,1] contains a pit Monotonicity (consistency): the set of entailed sentences can only increase as information is added to KB For  and , if KB |=  then KB^  |=  Propositional logic and first-order logic are monotonic

CS 471/598, CBS598 by H. Liu18 Resolution – an inference rule An example of resolution R11, R12 (new facts added), R13, R14 (derived from R11, and R12), R15 from R3, R16, R17 – P3,1 (there is a pit in [3,1]) (P213) Unit resolution: l1 v l2 …v l k, m = !l i We have seen examples earlier Full resolution: l1 v l2 …v l k, m 1 v…v m n where l i = m j An example: (P1,1vP3,1, !P1,1v!P2,2)/P3,1v!P2,2 Soundness of resolution Considering literal l i,  If it’s true, m j is false, then …  If it’s false, …

CS 471/598, CBS598 by H. Liu19 Refutation completeness Resolution can always be used to either confirm or refute a sentence Conjunctive normal form (CNF) A conjunction of disjunctions of literals A sentence in k-CNF has exactly k literals per clause (l 1,1 v … v l 1,k ) ^…^ (l n,1 v …v l n,k ) A simple conversion procedure (turn R2 to CNF,P. 215) A resolution algorithm (Fig 7.12) An example (KB= R2^R4, to prove !P1,2, Fig. 7.13) Completeness of resolution Ground resolution theorem

CS 471/598, CBS598 by H. Liu20 Horn cluases A Horn clause is a disjunction of literals of which at most one is positive An example: (!L1,1 v !Breeze V B1,1) An Horn sentence can be written in the form P1^P2^…^Pn=>Q, where Pi and Q are nonnegated atoms Deciding entailment with Horn clauses can be done in linear time in size of KB Inference with Horn clauses can be done thru forward and backward chaining  Forward chaining is data driven  Backward chaining works backwards from the query, goal- directed reasoning

CS 471/598, CBS598 by H. Liu21 An Agent for Wumpus The knowledge base (p208) Finding pits and wumpus using logical inference Keeping track of location and orientation Translating knowledge into action A1,1^East A ^W2,1=>!Forward Problems with the propositional agent too many propositions to handle (“Don’t go forward if…”) hard to deal with change (time dependent propositions)

CS 471/598, CBS598 by H. Liu22 Summary Knowledge is important for intelligent agents Sentences, knowledge base Propositional logic and other logics Inference: sound, complete; valid sentences Propositional logic is impractical for even very small worlds Therefore, we need to continue our AI class...