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Lecture of 11/13 Resolution theorem proving (end) Propositional Probabilistic Logic (start) Announcements: 1. Homework 4 socket closed; Due next week 2. Homework 5 and Project 5 made available for perusal (these will be the only other assignments) Rest of the semester: ~3 lectures on Uncertainty (project 5) ~3 lectures on learning 1 last class (loose ends+ interactive review) Example of FOPC Resolution.. Everyone is loved by someone xy loves( y, x) loves(SK ( x' ), x' ) If x loves y, x will give a valentine card to y xy loves( y, x) GV ( y, x) loves( y, x) GV ( y, x) Will anyone give Rao a valentine card? zGV ( z, Rao) zGV ( z, Rao) zGV ( z, Rao) GV ( z, Rao) y/z;x/Rao ~loves(z,Rao) z/SK(rao);x’/rao Finding where you left your key.. Atkey(Home) V Atkey(Office) 1 Where is the key? Ex Atkey(x) Negate Forall x ~Atkey(x) CNF ~Atkey(x) 2 Resolve 2 and 1 with x/home You get Atkey(office) 3 Resolve 3 and 2 with x/office You get empty clause So resolution refutation “found” that there does exist a place where the key is… Where is it? what is x bound to? x is bound to office once and home once. so x is either home or office Existential proofs.. • The previous example shows that resolution refutation is powerful enough to model existential proofs. In contrast, generalized modus ponens is only able to model constructive proofs.. • (We also discussed a cute example of existential proof—is it possible for an irrational number power another irrational number to be a rational number—we proved it is possible, without actually giving an example). GMP vs. Resolution Refutation • While resolution refutation is a complete inference for FOPC, it is computationally semi-decidable, which is a far cry from polynomial property of GMP inferences. • So, most common uses of FOPC involve doing GMP-style reasoning rather than the full theorem-proving.. • There is a controversy in the community as to whether the right way to handle the computational complexity is to – a. Develop ―tractable subclasses‖ of languages and require the expert to write all their knowlede in the procrustean beds of those sub-classes (so we can claim ―complete and tractable inference‖ for that class) OR – Let users write their knowledge in the fully expressive FOPC, but just do incomplete (but sound) inference. – See Doyle & Patil’s ―Two Theses of Knowledge Representation‖ Probabilistic Logic Need for modeling uncertainity • Consider a simple scenario: You know that rain makes grass wet. Sprinklers also make grass wet. Wet grass means wet news paper. You woke up one morning and found that your newspaper (brought in by your trusted dog/kid/spouse) was wet – Can we say something about whether it rained the previous day? – Will logic allow you to do it? • You hear the whooshing sound of the sprinklers outside the window – Does your belief in rain-the-previous-night change? – Will logic capture this? • No—our ―belief‖ in rain has reduced… that makes it ―non-monotonic‖ change • Standard logic is MONOTONIC • (By the way, this is a form of inference called ―explaining away‖—increased belief in one explanation for a cause reduces the belief in the competing explanations). ―Monotonic Logics‖ • Standard logic is monotonic – Given a database D and fact f, such that D|=f • Adding new knowledge to D doesn’t reverse the entailment – D+d |= f if D|=f • Plausible reasoning doesn’t have this property – Told that Tweety is a bird, we believe it will fly. Told that it is an ostrich, we believe it doesn’t. Told that it is a magical ostrich, we believe it does… – Probabilistic reasoning (effectively) allows non-monotonicity – (So does a class of logics called ―default logics‖—Chitta Baral is the Big Cheese in the default logic community). Qualification problem --impossible to enumerate all preconditions Ramification problem --impossible to enumerate all effects Frame problem --impossible to enumerate all that stays unchanged Prob. Prop logic: The Gameplan • We will review elementary ―discrete variable‖ probability • We will recall that joint probability distribution is all we need to answer any probabilistic query over a set of discrete variables. • We will recognize that the hardest part here is not the cost of inference (which is really only O(2n) –no worse than the (deterministic) prop logic • The real problem is assessing probabilities. – You could need as many as 2n numbers (if all variables are dependent on all other variables); or just n numbers if each variable is independent of all other variables. Generally, you are likely to need somewhere between these two extremes. – The challenge is to • Recognize the ―conditional independences‖ between the variables, and exploit them to get by with as few input probabilities as possible and • Use the assessed probabilities to compute the probabilities of the user queries efficiently.
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