Slide 1 Logic: Domain Modeling /Proofs + Top-Down Proofs Jim Little UBC CS 322 – CSP October 22, 2014

Slide 2 Lecture Overview Recap Using Logic to Model a Domain (Electrical System) Reasoning/Proofs (in the Electrical Domain) Top-Down Proof Procedure

Slide 3 Soundness & completeness of proof procedures A proof procedure X is sound … A proof procedure X is complete…. BottomUp for PDCL is sound and complete. We proved this in general even for domains represented by thousands of propositions and corresponding KB with millions of definite clauses ! (KB ⊦ P g) → (KB ⊧ g) (KB ⊧ g) ← (KB ⊦ P g)

Slide 4 Lecture Overview Recap Using PDCL Logic to Model a Domain (Electrical System) Reasoning/Proofs (in the Electrical Domain) Top-Down Proof Procedure

Slide 5 Electrical Environment / up /down

Slide 6 Let’s define relevant propositions / up /down For each wire w For each circuit breaker cb For each switch s For each light l For each outlet p x 2 2 How many interpretations?

Slide 7 Let’s now tell system knowledge about how the domain works / up /down live_l 1 ← live_w 0 ← live_w 1 ← live_w_0 live_w_1 ∧ up_s_2. live_w_2 ∧ down_s_2. live_w_3 ∧ up_s_1.

Slide 8 More on how the domain works…. / up /down live_w 2 ← live_w 3 ∧ down_s 1. live_l 2 ← live_w 4. live_w 4 ← live_w 3 ∧ up_s 3. live_p 1 ← live_w 3..

Slide 9 More on how the domain works…. / up /down live_w 3 ← live_w 5 ∧ ok_cb 1. live_p 2 ← live_w 6. live_w 6 ← live_w 5 ∧ ok_cb 2. live_w 5 ← live_outside.

Slide 10 What else we may know about this domain? That some simple propositions are true / up /down

Slide 11 What else we may know about this domain? That some additional simple propositions are true down_s 1. up_s 2. up_s 3. ok_cb 1. ok_cb 2. live_outside. / up /down

Slide 12 All our knowledge….. down_s 1. up_s 2. up_s 3. ok_cb 1. ok_cb 2. live_outside / up /down live_l 1 ← live_w 0 live_w 0 ← live_w 1 ∧ up_s 2. live_w 0 ← live_w 2 ∧ down_s 2. live_w 1 ← live_w 3 ∧ up_s 1. live_w 2 ← live_w 3 ∧ down_s 1. live_l 2 ← live_w 4. live_w 4 ← live_w 3 ∧ up_s 3. live_p 1 ← live_w 3. live_w 3 ← live_w 5 ∧ ok_cb 1. live_p 2 ← live_w 6. live_w 6 ← live_w 5 ∧ ok_cb 2. live_w 5 ← live_outside.

Slide 13 Lecture Overview Recap Using Logic to Model a Domain (Electrical System) Reasoning/Proofs (in the Electrical Domain) Top-Down Proof Procedure

Slide 14 What Semantics is telling us Our KB (all we know about this domain) is going to be true only in a subset of all possible 2^19 interpretations What is logically entailed by our KB are all the propositions that are true in all those models This is what we should be able to derive given a sound and complete proof procedure

Slide 15 If we apply the bottom-up (BU) proof procedure down_s 1. up_s 2. up_s 3. ok_cb 1. ok_cb 2. live_outside live_l 1 ← live_w 0 live_w 0 ← live_w 1 ∧ up_s 2. live_w 0 ← live_w 2 ∧ down_s 2. live_w 1 ← live_w 3 ∧ up_s 1. live_w 2 ← live_w 3 ∧ down_s 1. live_l 2 ← live_w 4. live_w 4 ← live_w 3 ∧ up_s 3. live_p 1 ← live_w 3.. live_w 3 ← live_w 5 ∧ ok_cb 1. live_p 2 ← live_w 6. live_w 6 ← live_w 5 ∧ ok_cb 2. live_w 5 ← live_outside. live_l 1 ? No live_w 2 ? Yes Green are in C

Slide 16 Lecture Overview Recap Using Logic to Model a Domain (Electrical System) Reasoning/Proofs (in the Electrical Domain) Top-Down Proof Procedure

Bottom-up vs. Top-down KB G is proved if G  C When does BU look at the query G ? Bottom-up D. Only at the beginning A. In every loop iterationB. Never C. Only at the end C Slide 17

Bottom-up vs. Top-down Key Idea of top-down: search backward from a query G to determine if it can be derived from KB. KB C G is proved if G  C When does BU look at the query G? At the end Bottom-up Top-down TD performs a backward search starting at G KB answer Query G Slide 18

Slide 19 Top-down Ground Proof Procedure Key Idea: search backward from a query G to determine if it can be derived from KB.

Slide 20 Top-down Proof Procedure: Basic elements Notation : An answer clause is of the form: yes ← a 1 ∧ a 2 ∧ … ∧ a m Rule of inference (called SLD Resolution) Given an answer clause of the form: yes ← a 1 ∧ a 2 ∧ … ∧ a m and the clause: a i ← b 1 ∧ b 2 ∧ … ∧ b p You can generate the answer clause yes ← a 1 ∧ … ∧ a i-1 ∧ b 1 ∧ b 2 ∧ … ∧ b p ∧ a i+1 ∧ … ∧ a m i.e., resolving a i with a i ← b 1 ∧ b 2 ∧ … ∧ b p Express query as an answer clause (e.g., query a 1 ∧ a 2 ∧ … ∧ a m ) yes ← a 1 ∧ a 2 ∧ … ∧ a m

Slide 21 Rule of inference: Examples Rule of inference (called SLD Resolution) Given an answer clause of the form: yes ← a 1 ∧ a 2 ∧ … ∧ a m and the KB clause: a i ← b 1 ∧ b 2 ∧ … ∧ b p You can generate the answer clause yes ← a 1 ∧ … ∧ a i-1 ∧ b 1 ∧ b 2 ∧ … ∧ b p ∧ a i+1 ∧ … ∧ a m yes ← b ∧ c.b ← k ∧ f. yes ← e ∧ f.e. KB clause -> yes ← k ∧ f ∧ c. -> yes ← f.

Slide 22 (successful) Derivations An answer is an answer clause with m = 0. That is, it is the answer clause yes ←. A (successful) derivation of query “?q 1 ∧ … ∧ q k “ from KB is a sequence of answer clauses γ 0, γ 1,…,γ n such that γ 0 is the answer clause yes ← q 1 ∧ … ∧ q k γ i is obtained by resolving γ i-1 with a clause in KB, and γ n is an answer. An unsuccessful derivation….. yes ←. yes ← a ∧ b.

Slide 23 Example: derivations a ← e ∧ f. a ← b ∧ c.b ← k ∧ f. c ← e.d ← k. e. f ← j ∧ e. f ← c. j ← c. Query: a (two ways) a ← b ∧ c. a ← e ∧ f. a ← k ∧ f ∧ c. a ← f. FAIL since a ← c. k cannot be eliminated a ← e. yes ←. yes ← a. yes ← a. KB

Slide 24 Example: derivations k ← e. a ← b ∧ c.b ← k ∧ f. c ← e.d ← k. e. f ← j ∧ e. f ← c. j ← c. Query: b ∧ e A. Provable by TDB. It depends C. Not Provable by TD KB

Slide 25 Course Big Picture Environment Problem Inference Planning Deterministic Stochastic Search Arc Consistency Search Value Iteration Var. Elimination Constraint Satisfaction Logics STRIPS Belief Nets Vars + Constraints Decision Nets Markov Processes Var. Elimination Static Sequential Representation Reasoning Technique SLS

Slide 26 Learning Goals for today’s class You can: Model a relatively simple domain with propositional definite clause logic (PDCL) Trace query derivation using SLD resolution rule of inference