1	Logical Agents Chapter 7 (continued)
2	Outline: Inference Resolution in CNF Sound and Complete Forward and Backward Chaining using Modus Ponens in Horn Form Sound and Complete
3	Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm Typically require transformation of sentences into a normal form Model checking truth table enumeration (always exponential in n) improved backtracking, e.g., Davis-Putnam-Logemann-Loveland (DPLL) heuristic search in model space (sound but incomplete) e.g., min-conflicts like hill-climbing algorithms
4	Inference by enumeration Depth-first enumeration of all models is sound and complete For n symbols, time complexity is O(2ⁿ), space complexity is O(n)
5	Wumpus world sentences Let P_i,j be true if there is a pit in [i, j]. Let B_i,j be true if there is a breeze in [i, j]. ØP_1,1 ØB_1,1 B_2,1 "Pits cause breezes in adjacent squares" B_1,1Û(P_1,2 Ú P_2,1) B_2,1Û (P_1,1 Ú P_2,2Ú P_3,1)
6	Truth tables for inference
7	Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm Typically require transformation of sentences into a normal form Model checking truth table enumeration (always exponential in n) improved backtracking, e.g., Davis-Putnam-Logemann-Loveland (DPLL) heuristic search in model space (sound but incomplete) e.g., min-conflicts like hill-climbing algorithms
8	Reasoning Patterns in Prop Logic Given(s) Conclusion A Þ B, A B B Ù A A Rules that allow us to introduce new propositions while preserving truth values: logically equivalent Two Examples: Modus Ponens And Elimination
9	Logical equivalence Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α
10	Resolution Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses E.g., (A Ú ØB) Ù (B Ú ØC Ú ØD) Resolution inference rule (for CNF): l_i Ú… Ú l_k, m₁ Ú … Ú m_n l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k Ú m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n where l_i and m_j are complementary literals. E.g., P_1,3 Ú P_2,2, ØP_2,2 P_1,3 Resolution is sound and complete for propositional logic
11	Resolution example KB = (B_1,1 Û (P_1,2Ú P_2,1)) ÙØ B_1,1 α = ØP_1,2(negate the premise for proof by refutation)
12	The power of false Given: (P) Ù (ØP) Prove: Z Can we prove ØZ using the givens above?
13	Applying inference rules Equivalent to a search problem KB state = node Inference rule application = edge
14	Inference Define: KB ├_iα = sentence α can be derived from KB by procedure i Soundness: i is sound if whenever KB ├_iα, it is also true that KB╞ α Completeness: i is complete if whenever KB╞ α, it is also true that KB ├_iα Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure. That is, the procedure will answer any question whose answer follows from what is known by the KB.
15	Completeness Completeness: i is complete if whenever KB╞ α, it is also true that KB ├_i α An incomplete inference algorithm cannot reach all possible conclusions Equivalent to completeness in search (chapter 3)
16	Resolution Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses E.g., (A Ú ØB) Ù (B Ú ØC Ú ØD) Resolution inference rule (for CNF): l_i Ú… Ú l_k, m₁ Ú … Ú m_n l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k Ú m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n where l_i and m_j are complementary literals. E.g., P_1,3 Ú P_2,2, ØP_2,2 P_1,3 Resolution is sound and complete for propositional logic
17	Resolution Soundness of resolution inference rule: Ø(l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k) Þ l_i Øm_j Þ (m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n) Ø(l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k) Þ (m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n) where l_i and m_j are complementary literals. What if l_i and Øm_jare false? What if l_i and Øm_j are true?
18	Completeness of Resolution That is, that resolution can decide the truth value of S S = set of clauses RC(S) = Resolution closure of S = Set of all clauses that can be derived from S by the resolution inference rule. RC(S) has finite cardinality (finite number of symbols P₁, P₂, … P_k), thus resolution refutation must terminate.
19	Completeness of Resolution (cont) Ground resolution theorem = if S unsatisfiable, RC(S) contains empty clause. Prove by proving contrapositive: i.e., if RC(S) doesn’t contain empty clause, S is satisfiable Do this by constructing a model: For each P_i, if there is a clause in RC(S) containing ØP_iand all other literals in the clause are false, assign P_i = false Otherwise P_i = true This assignment of P_iis a model for S.
20	Forward and backward chaining Horn Form (restricted) KB = conjunction of Horn clauses Horn clause = proposition symbol; or (conjunction of symbols) Þ symbol E.g., C Ù (B Þ A) Ù (C Ù D Þ B) Modus Ponens (for Horn Form): complete for Horn KBs α₁, … ,α_n, α₁ Ù … Ù α_n Þ β β Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time
21	Forward chaining example
22	Forward chaining example
23	Forward chaining example
24	Proof of completeness FC derives every atomic sentence that is entailed by KB (only for clauses in Horn form) FC reaches a fixed point (the deductive closure) where no new atomic sentences are derived Consider the final state as a model m, assigning true/false to symbols Every clause in the original KB is true in m a₁Ù … Ù a_k_Þb Hence m is a model of KB If KB╞ q, q is true in every model of KB, including m
25	Backward chaining example
26	Backward chaining example
27	Backward chaining example
28	Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm Typically require transformation of sentences into a normal form Model checking truth table enumeration (always exponential in n) improved backtracking, e.g., Davis-Putnam-Logemann-Loveland (DPLL) heuristic search in model space (sound but incomplete) e.g., min-conflicts like hill-climbing algorithms
29	Efficient propositional inference Two families of efficient algorithms for propositional inference: Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms WalkSAT algorithm
30	The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: Early termination A clause is true if any literal is true. A sentence is false if any clause is false. Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A Ú ØB), (ØB Ú ØC), (C Ú A), A and B are pure, C is impure. Make a pure symbol literal true. Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.
31	The DPLL algorithm
32	The WalkSAT algorithm Incomplete, local search algorithm Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses Balance between greediness and randomness
33	The WalkSAT algorithm
34	Hard satisfiability problems Consider random 3-CNF sentences. e.g., (ØD Ú ØB Ú C) Ù (B Ú ØA Ú ØC) Ù (ØC Ú ØB Ú E) Ù (E Ú ØD Ú B) Ù (B Ú E Ú ØC) m = number of clauses n = number of symbols Hard problems seem to cluster near m/n = 4.3 (critical point)
35	Hard satisfiability problems
36	Hard satisfiability problems Median runtime for 100 satisfiable random 3-CNF sentences, n = 50
37	Inference-based agents in the wumpus world A wumpus-world agent using propositional logic: ØP_1,1 ØW_1,1 B_x,y Û (P_x,y+1 Ú P_x,y-1 Ú P_x+1,y Ú P_x-1,y) S_x,y Û (W_x,y+1 Ú W_x,y-1 Ú W_x+1,y Ú W_x-1,y) W_1,1 Ú W_1,2 Ú … Ú W_4,4 ØW_1,1 Ú ØW_1,2 ØW_1,1 Ú ØW_1,3 … Þ 64 distinct proposition symbols, 155 sentences
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39	Expressiveness limitation of propositional logic We didn’t keep track of location and time in the KB. To do this we need more variables: L_1,1to show that agent in L_1,1.Does this work? KB contains "physics" sentences for every single square For every time t and every location [x,y], L _x,y Ù FacingRight ^t Ù Forward ^t Þ L _x+1,y Rapid proliferation of clauses
40	Summary Logical agents apply inference to a knowledge base to derive new information and make decisions Basic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences Wumpus world requires the ability to represent partial and negated information, reason by cases, etc. Resolution is complete for propositional logic Forward, backward chaining are linear-time, complete for Horn clauses Propositional logic lacks expressive power