11 Artificial Intelligence CS 165A Thursday, October 25, 2007  Knowledge and reasoning (Ch 7) Propositional logic 1

2 Who is this key historical AI figure? Built a calculating machine that could add and subtract (which Pascal’s couldn’t) But his dream was much grander – to reduce human reasoning to a kind of calculation and to ultimately build a machine capable of carrying out such calculations Co-inventor of the calculus “For it is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if the machine were used.” Gottfried Leibniz ( )

33 Notes HW#2 posted, due Tuesday –5x5 Tic Tac Toe game –Can work in teams of two, but different partner than in HW#1 –Can use your (or your team’s) code from HW#1 as starting point –You’ll need to use a heuristic to evaluate positions and go down N ply in your search so that the program returns an answer in a “short” amount of time  About 10 seconds maximum.... ___________ |X|O|X|X|X| | |O|O|X| | | | |O|X| | | | | |O| | | | | | |O| ¯¯¯¯¯¯¯¯¯¯¯ 3 in a row : 1 point 4 in a row : 3 points 5 in a row : 5 points X : 2 points O : 3 points

4 Knowledge Base Inference engine Domain specific content; facts ASK TELL Domain independent algorithms; can deduce new facts from the KB KB Agents True sentences

5 Thursday Quiz 1.If KB is the knowledge base and  is a logical sentence, what is the meaning of this statement? KB  2.Propositions P and Q are both true. Is this sentence true or false:  P   Q Briefly explain.

6 Syntax and semantics example The PQ system – Syntax –A correct sentence must have the form  {0 or more dashes} p {0 or more dashes} q {0 or more dashes} –For example --p---q p-q- pq --p-q- --p---q---- -p-q-- p-q --pq-- --p---q--p-- -q-p-- -p-- --p-p-q---- The PQ system – Semantics –# plus # equals # –Which of the above sentence are TRUE? Not allowable LEGAL sentences and TRUE sentences are not the same thing!

7 Semantics Representation (abstraction) World Sentences Facts Semantics Note: Facts may or may not be true If sentence P is false, then sentence  P is true --p---q pq- --p--q = = = 5

8 Inference and Entailment Given a set of (true) sentences, logical inference generates new sentences –Sentence  follows from sentences {  i } –Sentences {  i } entail sentence  –The classic example is modus ponens: P  Q and P entail what? An inference procedure i can derive  from KB KB i  KB  A knowledge base (KB) entails sentences 

9 Inference and Entailment (cont.) Representation World Fact F OLLOWS E NTAILS Facts Sentences Semantics Sentence Semantics

10 Using propositional logic: rules of inference Inference (n.): a. The act or process of deriving logical conclusions from premises known or assumed to be true. b. The act of reasoning from factual knowledge or evidence. Inference rules capture patterns of sound inference –Once established, we don’t need to show the truth table every time Examples of inference rules: –((P  H)   H) P  I.e., if ((P  H)   H) is in the KB, then we can conclude P –((P  Q)  P) Q  I.e., if ((P  Q)  P) is in the KB, then we can conclude Q

11 Inference engine An inference engine is a program that applies inference rules to knowledge –Goal: To infer new (and useful) knowledge Separation of –Knowledge –Rules –Control Which rules should we apply when? Inference engine

12 Inference procedures An inference procedure –Generates new sentences  that purport to be entailed by the knowledge base …or... –Reports whether or not a sentence  is entailed by the knowledge base Not every inference procedure can derive all sentences that are entailed by the KB A sound or truth-preserving inference procedure generates only entailed sentences Inference derives valid conclusions independent of the semantics (i.e., independent of the interpretation)

13 Inference procedures (cont.) Soundness of an inference procedure –i is sound if whenever KB i , it is also true that KB  –I.e., the procedure only generates entailed sentences Completeness of an inference procedure –i is complete if whenever KB , it is also true that KB i  –I.e., the procedure can find a proof for any sentence that is entailed The derivation of a sentence by a sound inference procedure is called a proof –Hence, the proof theory of a logical language specifies the reasoning steps that are sound

14 Logics We will soon define a logic which is expressive enough to say most things of interest, and for which there exists a sound and complete inference procedure –I.e., the procedure will be able to derive anything that is derivable from the KB –This is first-order logic, a.k.a. first-order predicate calculus –But first, we need to define propositional logic

15 Propositional (Boolean) Logic Symbols represent propositions (statements of fact, sentences) –P means “San Francisco is the capital of California” –Q means “It is raining in Seattle” Sentences are generated by combining proposition symbols with Boolean (logical) connectives

16 Propositional Logic Syntax –True, false, propositional symbols –( ),  (not),  (and),  (or),  (implies),  (equivalent) Examples of sentences in propositional logic P 1, P 2, etc. (propositions) ( S 1 )  S 1 S 1  S 2 S 1  S 2 S 1  S 2 S 1  S 2 true P 1  true   ( P 2  false ) P  Q  Q  P

17 Propositional (Boolean) Logic (cont.) Semantics –Defined by clearly interpreted symbols and straightforward application of truth tables –Rules for evaluating truth: Boolean algebra –Simple method: truth tables 2 N rows for N propositions

18 Propositional (Boolean) Logic (cont.) Make sure you know simple Boolean logic Associative, commutative, and distributive laws P  Q  Q  P (P  Q)  R  P  (Q  R) P  (Q  R)  (P  Q)  (P  R)  P  P  (P  Q)   P   Q  (P  Q)   P   Q P  Q   P  Q DeMorgan’s Laws Important!

19 Basic logical equivalences Equivalent Sentences  P  QP  QP  Q P True  P P  QP  QTrue  P  Q P  False P PPP  False

20 Satisfiability and Validity Is this true: ( P  Q ) ? –It depends on the values of P and Q –This is a satisfiable sentence – there are some interpretations for which it is true –In other words, it depends: it could be true or false Is this true: ( P   P ) ? –No, it is never true –This is an unsatisfiable sentence (self-contradictory) – there is no interpretation for which it is true Is this true: ( ((P  Q)   Q)  P ) ? –Yes, independent of the values of P and Q –This is a valid sentence – it is true under all possible interpretations (a.k.a. a tautology) –Truth tables can test for validity

21 Propositional logic (cont.) A  C –The premise implies the conclusion... or...  The antecedent implies the consequent –What if A  C is always true? Example: Then we can say that the antecedent entails the consequent. In other words, P and H can be seen as variables – this is true for any statements P and Q.

22 One can say that the premise entails the conclusion –((P  H)   H) entails P –((P  H)   H) P Have we also shown that ((P  H)   H) P ? Well, yes and no.... We still don’t have a method to derive this. But if we add this inference rule explicitly to our system, with P and H as variables representing any two propositions, then we do have such a procedure.

23 Using propositional logic: rules of inference Inference rules capture patterns of sound inference –Once established, don’t need to show the truth table every time –E.g., we can define an inference rule: ((P  H)   H) P for variables P and H  “If we know , then we can conclude  ” Alternate notation for inference rule   : (where  and  are propositional logic sentences)

24 Inference KB  1 KB,  1  2 KB,  1,  2  3 …  Inference steps KB   1,  2, …  or We’re particularly interested in So we need a mechanism to do this! Inference rules that can be applied to sentences in our KB

25 Important Inference Rules for Propositional Logic

26 Resolution Rule: one rule for all inferences Propositional calculus resolution or Remember: p  q   p  q, so let’s rewrite it as: Resolution is really the “chaining” of implications. Would like to show that resolution is sound and (essentially) complete (use Deduction Rule for proof)

27 Inference in Propositional Logic Three ways to answer: “Is Mary the grandparent of Ann?” –One way:  Find sentence P (“Mary is the grandparent of Ann”) in KB –Another way:  Put sentence  P in the KB, and use inference rules to prove a contradiction (false) –Yet another way:  Find sentence Q (“Mary is parent of Frank”), sentence R (“Frank is the parent of Ann”), and sentence Q  R  S, where S is the proposition “Mary is grandparent of Ann”  Then apply modus ponens: Q  R  S, Q  R to conclude S (“Mary is the grandparent of Ann”) “If Mary is the parent of Frank and Frank is the parent of Ann, then Mary is the grandparent of Ann”