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Unformatted text preview: Logic Intro CPS 170 Ron Parr Historical Perspec8ve I • Logic was one of the classical founda8ons of AI • Dream: A Knowledge
Based agent – Tell the agent facts – Agent uses rules of inference to deduce consequences – Example: prolog • Dis8nc8on between data and program • Embodied in ﬁeld of “Expert Systems” 1 Example: Minesweeper • How do you play minesweeper? • How would you program a machine to do it? – Hacking – Search – Logic • Logic approach – Tell the system of rules of minesweeper – System uses logic to make the best moves What is logic, really? • Syntax: Rules for construc8ng valid sentences • Seman8cs: Relate syntax to the real world 2 Entailment • Aim: Rule for genera8ng (or tes8ng) new sentences that are necessarily true • The truth of sentence may depend upon the interpreta,on of the sentence Interpreta8ons • An interpreta8on is a way of matching up objects in the universe with symbols in a sentence (or database). • A sentence may be true in one interpreta8on, but false in another • A necessarily true sentence is true in all interpreta8ons (perhaps given some premises in our KB) 3 Examples • Premises (facts in our database): – (X or Y) – Not X – Conclude: Y is necessarily true • Premises – If P then Q – Q – Conclude: P is not necessarily true (though might be true in some interpreta8ons) Soundness & Completeness • A (set of) rule(s) of inference is sound if it generates only sentences that are entailed by the knowledge base, i.e., only necessary truths • A (set of) rule(s) of inference is complete if it can generate all necessary truths • Can we have one w/o the other? 4 Historical Perspec8ve II • Things that are not true necessarily but s8ll true are some8mes said to be “con8ngent,” “accidental,” or “synthe8c,” truths. • A deep understanding of this dis8nc8on evolved through thousands of years of philosophy and mathema8cs • Arguably one of the most important intellectual accomplishments of mankind – Basis of mathema8c proofs – Provides a rigorous procedure for verifying statements Proposi8onal Logic • Proposi8onal logic is the simplest logic • All sentences are composed of – Atoms – Nega8on – Disjunc8on, conjunc8on (or, and) – Condi8onal, bicondi8onals • Atoms can map to any proposi,on about the universe (depending upon the interpreta8on) 5 Checking Validity • Classic method for checking validity: truth table • Enumerate all possible values (t/f) of atomic elements of a sentence (P ∨ H )
¬H
P Horizontal
line separates
premises from
conclusion • Enumerate all 4 (or more) combina8ons € Inference Rules • Inference rules are (typically) sound methods of genera8ng new sentences given a set of previous sentences • Inference rules save us the trouble of genera8ng truth tables all of the 8me 6 Inference Rules I • Modus Ponens " # $, "
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• And
Elimina8on ! "" # "# # … # " "
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!! ! Inference Rules II • And
Introduc8on " " , " # ,…, " "
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• Or
Introduc8on ! ""
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7 Inference Rules III • Double Nega8on Elimina8on ¬¬"
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• Unit Resolu8on ! " # $, ¬$
" ! Resolu8on α ∨ β, ¬β ∨ γ
α ∨γ
Resolu8on is perhaps the most important inference rule! € Why? Resolu8on is both sound and complete! 8 Complexity of Inference • What is the complexity of exhaustsively verifying the validity of a sentence with n literals (variables)? ""
!! • Special Case: Horn Logic – Horn clauses are disjunc8ons with at most one posi8ve literal !
– Equivalent to " " " " … " "# # $
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!! ! Remember De Morgan’s Law? • not(P and Q) = (not P) or (not Q) • not(P or Q) = (not P) and (not Q) • Surprisingly, no rela8onship to Captain Morgan 9 Implica8ons and Horn Clauses • If P then Q – Same as: (not (P and (not Q)) – Same as: (not P) or Q – …and this is horn! • If (P1 and P2 and … Pn) then Q –
–
–
– Same as: (not ((P1 and P2 and … Pn) and (not Q)) Same as: not (P1 and P2 and … Pn) or Q Same as: ((not P1) or (not P2) or … (not Pn) or Q) …and this is horn! Horn Clause Inference • Horn clause inference is polynomial – Why? – Every sentence establishes exactly one new fact – Can add every possible new fact implied by our KB in n passes over our database • What types of things are easy to represent with horn clauses? – Diagnos8c rules – “Expert Systems” 10 Shortcomings of Horn Clauses • Suppose you want to say, “If you have a runny nose and fever, then you have a cold or the ﬂu.” • If (runny_nose and fever) then (cold or ﬂu) • But this isn’t a horn clause: (not runny_nose) or (not fever) or (cold) or (ﬂu) • Does adding two separate horn clauses work? – (not runny_nose) or (not fever) or (cold) – (not runny_nose) or (not fever) or (ﬂu) Proposi8onal Logic Conclusion • Logic gives formal rules for reasoning • Necessarily true = true in all interpreta8ons • Contrast with CSPs: Sa8sﬁable = true in some, but not necessarily all interpreta8ons • Sound inference rules generate only necessary truths • Resolu8on is a sound and complete inference rule • Inference with a horn KB is poly 8me 11 ...
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 Spring '11
 Parr
 Artificial Intelligence

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