CS 331: Artificial Intelligence Propositional Logic 2. Review of Last Time

Transcription

1 CS 33 Artificia Inteigence Propositiona Logic 2 Review of Last Time = means ogicay foows - i means can be derived from If your inference agorithm derives ony things that foow ogicay from the KB, the inference is sound If everything that foows ogicay from the KB can be derived using your inference agorithm, the inference is compete 2

2 Outine. Propositiona Logic Syntax and Semantics 2. Reasoning Patterns in Propositiona Logic 3 Propositiona Logic Syntax and Semantics 4 2

3 Syntax Backus-Naur Form grammar of sentences in propositiona ogic Sentence AtomicSentence CompexSentence AtomicSentence True Fase Symbo Symbo P Q R CompexSentence Sentence ( Sentence Sentence ) ( Sentence Sentence ) ( Sentence Sentence ) ( Sentence Sentence ) 5 Atomic Sentences The indivisibe syntactic eements Consist of a singe propositiona symbo eg. P, Q, R that stands for a proposition that can be or eg. P=, Q= We aso ca an atomic sentence a itera 2 specia propositiona symbos True (the aways proposition) Fase (the aways proposition) 6 3

4 Compex Sentences Made up of sentences (either compex or atomic) 5 common ogica connectives (not) negates a itera (and) conjunction eg. P Q where P and Q are caed the conjuncts (or) disjunction eg. P Q where P and Q are caed the disjuncts (impies) eg. P Q where P is the premise/antecedent and Q is the concusion/consequent (if and ony if) eg. P Q is a biconditiona 7 Precedence of Connectives In order of precedence, from highest to owest,,,, Eg. P Q R S is equivaent to (( P) (Q R)) S You can rey on the precedence of the connectives or use parentheses to make the order expicit Parentheses are necessary if the meaning is ambiguous 8 4

5 Semantics (Are sentences?) Defines the rues for determining if a sentence is with respect to a particuar mode For exampe, suppose we have the foowing mode P=, Q=, R= Is (P Q R)? I want the truth! 9 Semantics For atomic sentences True is, Fase is A symbo has its vaue specified in the mode For compex sentences (for any sentence S and mode m) S is in m iff S is in m S S 2 is in m iff S is in m and S 2 is in m S S 2 is in m iff S is in m or S 2 is in m S S 2 is in m iff S is in m or S 2 is in m i.e., can transate it as S S 2 S S 2 is iff S S 2 is in m and S 2 S is in m 0 5

6 Note on impication P Q seems weird doesn t fit intuitive understanding of if P then Q Propositiona ogic does not require causation or reevance between P and Q Impication is whenever the antecedent is (remember P Q can be transated as P Q ) Impication says if P is, then I am caiming that Q is. Otherwise I am making no caim The ony way for this to be is if P is but Q is Truth Tabes for the Connectives P P P Q P Q P Q P Q P Q With the truth tabes, we can compute the truth vaue of any sentence with a recursive evauation eg. Suppose the mode is P=, Q=, R= P (Q R) = ( ) = = 2 6

7 The Wumpus Word KB (ony deaing with knowedge about pits) For each i, j Let P i,j be if there is a pit in [i, j] Let B i,j be if there is a breeze in [i, j] The KB contains the foowing sentences. There is no pit in [,] R P, 2. A square is breezy iff there is a pit in a neighboring square (not a sentences are shown) R 2 B, P,2 P 2, R 3 B 2, (P, P 2,2 P 3, ) 3 The Wumpus Word KB 3. We add the percepts for the first two squares ([,] and [2,]) visited in the Wumpus Word exampe R 4 B, R 5 B 2, The KB is now a conjunction of sentences R R 2 R 3 R 4 R 5 because a of these sentences are asserted to be. 4 7

8 Inference How do we decide if KB = α? Enumerate the modes, check that α is in every mode in which KB is B, B 2, P, P,2 P 2, P 2,2 P 3, R R 2 R 3 R 4 R 5 KB Inference Suppose we want to know if KB = P,2? In the 3 modes in which KB is, P,2 is aso B, B 2, P, P,2 P 2, P 2,2 P 3, R R 2 R 3 R 4 R 5 KB 6 8

9 Compexity If the KB and α contain n symbos in tota, what is the time compexity of the truth tabe enumeration agorithm? Space compexity is O(n) because the actua agorithm uses DFS 7 The reay depressing news Every known inference agorithm for propositiona ogic has a worse-case compexity that is exponentia in the size of the input You can t hande the truth! But some agorithms are more efficient in practice 8 9

10 Logica equivaence Intuitivey two sentences α and β are ogicay equivaent (ie. α β) if they are in the same set of modes Formay α β if and ony if α = β and β = α Can prove this with truth tabes 9 Standard Logic Equivaences In the above, α, β, and γ are arbitrary sentences of propositiona ogic 20 0

11 Vaidity A sentence is vaid if it is in a modes eg. P Pis vaid Vaid sentences = Tautoogies Tautoogies are vacuous Deduction theorem For any sentences α and β, α = β iff the sentence (α β) is vaid 2 Satisfiabiity A sentence is satisfiabe if it is in some mode. A sentence is unsatisfiabe if it is in no modes Determining the satisfiabiity of sentences in propositiona ogic was the first probem proved to be NP-compete Satisfiabiity is connected to vaidity α is vaid iff α is unsatisfiabe Satisfiabiity is connected to entaiment α = β iff the sentence (α β) is unsatisfiabe (proof by contradiction) 22

12 Reasoning Patterns in Propositiona Logic 23 Proof methods How do we prove that α can be entaied from the KB?. Mode checking eg. check that α is in a modes in which KB is 2. Inference rues 24 2

13 . Modus Ponens α β, α β 2. And-Eimination α β α Inference Rues These are both sound inference rues. You don t need to enumerate modes now 25 Other Inference Rues A of the ogica equivaences can be turned into inference rues eg. α β ( α β ) ( β α) 3

14 Exampe Given the foowing KB, can we prove R? KB P (Q R) P Proof (Q R) by Modus Ponens Q R by De Morgan s Law R by And-Eimination 27 Proofs A sequence of appications of inference rues is caed a proof Instead of enumerating modes, we can search for proofs Proofs ignore irreevant propositions 2 methods Go forward from initia KB, appying inference rues to get to the goa sentence Go backward from goa sentence to get to the KB 28 4

15 Monotonicity Proofs ony work because of monotonicity Monotonicity the set of entaied sentences can ony increase as information is added to the knowedge base For any sentences α and β, if KB = α then KB β = α 29 Resoution An inference rue that is sound and compete Forms the basis for a famiy of compete inference procedures Here, compete means refutation competeness resoution can refute or confirm the truth of any sentence with respect to the KB 30 5

16 6 Resoution Here s how resoution works ( 2 and 2 are caed compementary iteras) Note that you need to remove mutipe copies of iteras (caed factoring) ie. If i and m j are compementary iteras, the fu resoution rue ooks ike n j j k i i n k m m m m m m + + L L L L L L, , 2 2, 32 Conjunctive Norma Form Resoution ony appies to sentences of the form 2 k This is caed a disjunction of iteras It turns out that every sentence of propositiona ogic is ogicay equivaent to a conjunction of disjunction of iteras Caed Conjunctive Norma Form or CNF eg. ( ) ( ) k-cnf sentences have exacty k iteras per cause eg. A 3-CNF sentence woud be ( 2 3 ) ( ) ( )

17 Recipe for Converting to CNF. Eiminate, repacing α βwith (α β) (β α) 2. Eiminate, repacing α βwith α β 3. Move inwards using ( α) α(doube-negation eimination) (α β) α β (De Morgan s Law) (α β) α β (De Morgan s Law) 4. Appy distributive aw (α(β γ)) ((α β) (αγ)) 33 A resoution agorithm To prove KB = α, we show that (KB α) is unsatisfiabe (Remember that α = β iff the sentence (α β) is unsatisfiabe) The agorithm. Convert (KB α) to CNF 2. Appy resoution rue to resuting causes. Each pair with compementary iteras is resoved to produce a new cause which is added to the KB 3. Keep going unti There are no new causes that can be added ( meaning KB α) Two causes resove to yied the empty cause ( meaning KB = α ) The empty cause is equivaent to because a disjunction is ony if one of its disjuncts is 34 7

18 Resoution Pseudocode 35 Things you shoud know Understand the syntax and semantics of propositiona ogic Know how to do a proof in propositiona ogic using inference rues Know how resoution works Know how to convert arbitrary sentences to CNF 36 8

19 In-cass Exercise KB Can we show that Person Morta Socrates Person KB = (Socrates Morta)? 37 If it is October, there wi not be a footba game at OSU If it is October and it is Saturday, I wi be in Corvais If it doesn t rain or if there is a footba game, I wi ride my bike to OSU Today is Saturday and it is October If I am in Corvais, it wi not rain In-cass Exercise Can you prove that I wi ride my bike to OSU? 38 9