Logical Agents Logic Propositional Logic Summary Knowledge-Based Agents Logic Propositional Logic Summary

Knowledge-Based Agents Central component is a knowledge-base Knowledge is made of sentences Sentences are expressed in a language We wish to derive new sentences through inference.

Knowledge-Based Algorthim Function KB (percept) returns action TELL (KB; PERCEPT-SENTENCE(percept,t)) action = ASK(KB,MAKE-ACTION-QUERY(t)) TELL (KB;ACTION-SENTENCE(action,t)) T = t + 1 Return action

Declarative vs Procedural Two approaches to store knowledge: Declarative. Sentences representing knowledge Procedural. Encode behavior directly as code.

An Example Moving on a square to reach a goal state while avoiding risks.

Logical Agents Logic Propositional Logic Summary Knowledge-Based Agents Logic Propositional Logic Summary

Logic Important terms: Syntax .- rules to represent well-formed sentences Semantics.- define the truth of each sentence. Model.- Possible world (m is a model of a) Entailment: a |= b a entails b means: In every model where a is true, b is also true.

Logic Sentence a is derived from KB by algorithm i: KB |=i a Algorithm i is called sound (truth preserving) if it derives only entailed sentences. Algorithm i is called complete if it derives all entailed sentences.

Physical configuration Sentence Sentence entails Representation semantics semantics Real World Real World follows

Allen Newell Born in 1927 – died in 1992 1982 “The Knowledge Level” Rational agents can be described by the knowledge they contain With Herbert Simon he developed the Logic Theorist (first real AI program). Winner of the Turing Award

Logical Agents Logic Propositional Logic Summary Knowledge-Based Agents Logic Propositional Logic Summary

Syntax Atomic sentences: propositional symbols Complex sentences use logical connectives Negation,conjunction,disjunction Implication and biconditional.

Syntax Sentence  Atom | Complex Structure Atom  True | False | Symbol Symbol  P | Q | R Complex S  - sentence | ( sentence ^ sentence ) | ( sentence V sentence ) | ( sentence  sentence) | ( sentence  sentence)

Semantics How to define the truth value of statements? Connectives associate to truth tables. The knowledge base of an agent grows by telling it of new statements: TELL(KB,S1), … TELL(KB,Sn) KB = S1 ^ … ^ Sn

Inference We want to know if KB entails a: KB |= a Approach 1: Try all models KB: {b , b  a} b a Models consistent with KB t t YES t f NO f t NO f f NO

Check All Models Method is characterized as: sound (directly implements entailment) complete (it works for any KB and a and it always terminates. Problem: With n symbols there are 2n models. Time complexity is O(2n) Space complexity is O(n)

Some concepts Equivalence: a = b  a |= b and b |= a Validity A sentence is valid if it is true in all models Example p V -p Satisfiability A sentence is satisfiable if it is true in some model

Reasoning Patterns Modus Ponens a  b, a b And-Elimination a ^ b a Commutativity of ^ and V de Morgan’s Laws, etc.

Proofs Applying a sequence of rules is called a proof. Equivalent to searching for a solution Monotonicity: if KB |= a then KB ^ b |= a {b, b a} a Modus ponens

Resolution Complete and sound inference algorithm a v b ~a v c b v c Cannot derive all conclusions but can tell if they are true or false (refutation completeness).

Applying Resolution Resolution only works on disjunction of literals. How can we apply resolution to all clauses? Solution: convert to conjunctive normal form (CNF) CNF : (L11 V .. V L1k) ^ … ^ (Ln1 V … V Lnk)

Resolution How does it work? To show that KB |= a We show that (KB ^ ~a) is unsatisfiable. Every pair that contains complementary literals is resolved. Continue until there are no new clauses (KB |= a) Or we derive the empty clause (a V ~a) (KB |= a)

Example a v b v ~c d v c KB a v b v d ~b a v d ~d a ~a KB |= b

Horn Clauses In practice we restrict our clauses to Horn Clauses. (base of logic programming) Horn Clause: disjunction of literals with at most one of them being positive. Example: (~a v ~b v c) (definite clause) Equivalent to ( a ^ b  c )

Forward Chaining Begins from known facts in the knowledge base. If premises of implication are known derive conclusion. Continue until query is added or no more inferences can be made. a ^ b ^ c  d a,b,c conclude d

Example Q P  Q L ^ M  P P B ^ L  M A ^ P  L M A ^ B  L A B L A B AND-OR Graph

Backward Chaining Works backwards from the query q Find implications that conclude q If all premises can be proved true then q is true a ^ b ^ c  q How do I prove a? and b? and c?

Logical Agents Logic Propositional Logic Summary Knowledge-Based Agents Logic Propositional Logic Summary

Summary Rational agents need to work with knowledge bases. A language is defined by syntax and semantics KB |= a if a is true in all models where KB is true Inference techniques can be characterized based on soundness and completeness. Simplest logic is propositional logic A strong inference rule is called “resolution” When using Horn clauses, reasoning methods are forward and backward chaining.