1	Logical Agents Chapter 7
2	Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution
3	Knowledge bases Knowledge base = set of sentences in a formal language Declarative approach to building an agent (or other system): Tell it what it needs to know Then it can Ask itself what to do - answers should follow from the KB Agents can be viewed at the knowledge level i.e., what they know, regardless of how implemented Or at the implementation level i.e., data structures in KB and algorithms that manipulate them
4	A simple knowledge-based agent The agent must be able to: Represent states, actions, etc. Incorporate new percepts Update internal representations of the world Deduce hidden properties of the world Deduce appropriate actions
5	Wumpus World PEAS description Performance measure gold +1000, death -1000 -1 per step, -10 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square Sensors: Stench, Breeze, Glitter, Bump, Scream Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot
6	Wumpus world characterization Fully Observable No – only local perception Deterministic Yes – outcomes exactly specified Episodic No – sequential at the level of actions Static Yes – Wumpus and Pits do not move Discrete Yes Single-agent? Yes – Wumpus is essentially a natural feature
7	Exploring a wumpus world
8	Exploring a wumpus world
9	Exploring a wumpus world
10	Exploring a wumpus world
11	Exploring a wumpus world
12	Exploring a wumpus world
13	Exploring a wumpus world
14	Exploring a wumpus world
15	Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the "meaning" of sentences; i.e., define truth of a sentence in a world E.g., the language of arithmetic x+2 ≥ y is a sentence; x2+y > {} is not a sentence x+2 ≥ y is true iff the number x+2 is no less than the number y x+2 ≥ y is true in a world where x = 7, y = 1 x+2 ≥ y is false in a world where x = 0, y = 6
16	Entailment Entailment means that one thing follows from another: KB ╞ α Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won” E.g., x+y = 4 entails 4 = x+y Entailment is a relationship between sentences (i.e., syntax) that is based on semantics
17	Models Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated We say m is a model of a sentence α if α is true in m M(α) is the set of all models of α Then KB ╞ α iff M(KB) Í M(α) E.g. KB = Giants won and Reds won α = Giants won
18	Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices Þ 8 possible models
19	Wumpus models
20	Wumpus models KB = wumpus-world rules + observations
21	Wumpus models KB = wumpus-world rules + observations α₁ = "[1,2] is safe", KB ╞ α₁, proved by model checking
22	Wumpus models KB = wumpus-world rules + observations
23	Wumpus models KB = wumpus-world rules + observations α₂ = "[2,2] is safe", KB ╞ α₂
24	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.
25	Propositional logic: Syntax Propositional logic is the simplest logic – illustrates basic ideas The proposition symbols P₁, P₂ etc are sentences If S is a sentence, ØS is a sentence (negation) If S₁ and S₂ are sentences, S₁ Ù S₂ is a sentence (conjunction) If S₁ and S₂ are sentences, S₁ Ú S₂ is a sentence (disjunction) If S₁ and S₂ are sentences, S₁ Þ S₂ is a sentence (implication) If S₁ and S₂ are sentences, S₁ Û S₂ is a sentence (biconditional)
26	Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P_1,2 P_2,2 P_3,1 false true false With these symbols, 8 possible models, can be enumerated automatically. Rules for evaluating truth with respect to a model m: ØS is true iff S is false S₁ Ù S₂ is true iff S₁ is true and S₂ is true S₁ Ú S₂ is true iff S₁is true or S₂ is true S₁ Þ S₂ is true iff S₁ is false or S₂ is true i.e., is false iff S₁ is true and S₂is false S₁ Û S₂is true iff S₁ÞS₂ is true andS₂ÞS₁ is true Simple recursive process evaluates an arbitrary sentence, e.g., ØP_1,2Ù (P_2,2ÚP_3,1) = true Ù (true Ú false) = true Ù true = true
27	Truth tables for connectives
28	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)
29	Truth tables for inference
30	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)
31	Logical equivalence Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α
32	Validity and satisfiability A sentence is valid if it is true in all models, e.g., True, A ÚØA, A Þ A, (A Ù (A Þ B)) Þ B Validity is connected to inference via the Deduction Theorem: KB ╞ α if and only if (KB Þ α) is valid A sentence is satisfiable if it is true in some model e.g., AÚ B, C A sentence is unsatisfiable if it is true in no models e.g., AÙØA Satisfiability is connected to inference via the following: KB ╞ α if and only if (KB ÙØα) is unsatisfiable
33	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
34	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
35	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.
36	Conversion to CNF B_1,1 Û (P_1,2 Ú P_2,1) Eliminate Û, replacing α Û β with (α Þ β)Ù(β Þ α). (B_1,1 Þ (P_1,2 Ú P_2,1)) Ù ((P_1,2 Ú P_2,1) Þ B_1,1) 2. Eliminate Þ, replacing α Þ β with ØαÚ β. (ØB_1,1 Ú P_1,2 Ú P_2,1) Ù (Ø(P_1,2 Ú P_2,1) Ú B_1,1) 3. Move Ø inwards using de Morgan's rules and double-negation: (ØB_1,1Ú P_1,2 Ú P_2,1) Ù ((ØP_1,2Ú ØP_2,1) Ú B_1,1) 4. Apply distributivity law (Ù over Ú) and flatten: (ØB_1,1 Ú P_1,2 Ú P_2,1) Ù (ØP_1,2Ú B_1,1) Ù (ØP_2,1 Ú B_1,1)
37	Resolution algorithm Proof by contradiction, i.e., show KBÙØα unsatisfiable
38	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)
39	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
40	Forward chaining Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found
41	Forward chaining algorithm Forward chaining is sound and complete for Horn KB
42	Forward chaining example
43	Forward chaining example
44	Forward chaining example
45	Forward chaining example
46	Forward chaining example
47	Forward chaining example
48	Forward chaining example
49	Forward chaining example
50	Proof of completeness FC derives every atomic sentence that is entailed by KB FC reaches a fixed point 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
51	Backward chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q Avoid loops: check if new subgoal is already on the goal stack Avoid repeated work: check if new subgoal has already been proved true, or has already failed
52	Backward chaining example
53	Backward chaining example
54	Backward chaining example
55	Backward chaining example
56	Backward chaining example
57	Backward chaining example
58	Backward chaining example
59	Backward chaining example
60	Backward chaining example
61	Backward chaining example
62	Forward vs. backward chaining FC is data-driven, automatic, unconscious processing, e.g., object recognition, routine decisions May do lots of work that is irrelevant to the goal BC is goal-driven, appropriate for problem-solving, e.g., Where are my keys? How do I get into a PhD program? Complexity of BC can be much less than linear in size of KB
63	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
64	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.
65	The DPLL algorithm
66	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
67	The WalkSAT algorithm
68	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)
69	Hard satisfiability problems
70	Hard satisfiability problems Median runtime for 100 satisfiable random 3-CNF sentences, n = 50
71	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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73	Expressiveness limitation of propositional logic 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
74	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