Logical CS-480 Agents and Inference
AIMA, Chapter 7
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
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
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
Wumpus World PEAS
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
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
Exploring a wumpus world
Exploring a wumpus world
Exploring a wumpus world
Exploring a wumpus world
Exploring a wumpus world
Exploring a wumpus world
Exploring a wumpus world
Exploring a wumpus world
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
Entailment
Entailment means that one thing follows from another: KB Knowledge base KB entails sentence if and only if 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
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 if is true in m M() is the set of all models of Then KB iff M(KB) M()
E.g. KB = Giants won and Reds won = Giants won
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 8 possible models
Wumpus models
Wumpus models
KB = wumpus-world rules + observations
Wumpus models
KB = wumpus-world rules + observations 1 = "[1,2] is safe", KB 1, proved by model checking
Wumpus models
KB = wumpus-world rules + observations
Wumpus models
KB = wumpus-world rules + observations 2 = "[2,2] is safe", KB 2
Inference
KB i = sentence can be derived from KB by procedure i Soundness: i is sound if whenever KB i , it is also true that KB Completeness: i is complete if whenever KB , it is also true that KB i Preview: We will (soon) define a logic (first-order logic) which is expressive enough to say most things 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.
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, S is a sentence (negation) If S1 and S2 are sentences, S1 S2 is a sentence (conjunction) If S1 and S2 are sentences, S1 S2 is a sentence (disjunction) If S1 and S2 are sentences, S1 S2 is a sentence (implication) If S1 and S2 are sentences, S1 S2 is a sentence (biconditional)
Propositional logic: Semantics
A model specifies a true/false value for each proposition symbol
E.g. P1,2 false P2,2 true P3,1 false
With these symbols, 8 possible models can be enumerated automatically. Rules for evaluating truth with respect to a model m: S is true iff S is false S1 S2 is true iff S1 is true and S2 is true S1 S2 is true iff S1is true or S2 is true S1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is false S1 S2 is true iff S1S2 true and S2S1 is true Simple recursive process evaluates an arbitrary sentence, e.g., P1,2 (P2,2 P3,1) = true (true false) = true true = true
Truth tables for connectives
Wumpus world sentences
Let Pi,j be true iff there is a pit in [i, j]; Let Bi,j be true iff there is a breeze in [i, j]:
P1,1 B1,1 B2,1
"Pits cause breezes in adjacent squares:
B1,1 B2,1
(P1,2 P2,1) (P1,1 P2,2 P3,1)
Truth tables for inference
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)
Logical equivalence
Two sentences are logically equivalent iff they true in exactly the same models: iff and
Validity and satisfiability
A sentence is valid if it is true in all models,
e.g., True, A A, A A, (A (A B)) B
Validity is connected to inference via the Deduction Theorem:
KB if and only if (KB ) is valid
A sentence is satisfiable if it is true in some model
e.g., AB, C
A sentence is unsatisfiable if it is true in no models
e.g., A A
Satisfiability is connected to inference via the following:
KB if and only if (KB is unsatisfiable )
Proof methods
Proof methods divide into (roughly) kinds:
Application two 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
Inference by resolution
Conjunctive Normal Form (CNF)
conjunction of disjunctions of literals
E.g.,
(A B) (B C D)
Resolution inference rule (for CNF): li lk, m1 mn li li-1 li+1 lk m1 mj-1 mj+1 mn ... where li and mj are complementary literals. E.g., P1,3 P2,2, P1,3 P2,2
Resolution is sound and complete for propositional logic
}
clauses
Resolution
Soundness of resolution inference rule:
(li li-1 li+1 lk) li mj (m1 mj-1 mj+1 mn) ... (li li-1 li+1 lk) (m1 mj-1 mj+1 mn) ...
Conversion to CNF
B1,1 (P1,2 P2,1)
1.
Eliminate , replacing with ( ) ). (
(B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1) (B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)
2. Eliminate , replacing with . 3. Move inwards using de Morgan's rules and double-negation:
(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)
4. Apply distributivity law (over and flatten: )
(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)
Resolution proof algorithm
Proof by contradiction, i.e., show KB unsatisfiable
Resolution example
KB = (B1,1 (P1,2P2,1)) B1,1 = P1,2
Forward and backward chaining
Horn Form (restricted)
KB = conjunction of Horn clauses Horn clause =
E.g., C (B A) (C D B)
proposition symbol; or (conjunction of symbols) symbol
Modus Ponens (for Horn Form): complete for Horn KBs 1, ,n, 1 n Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time
Forward chaining
Idea: fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found
Forward chaining algorithm
Forward chaining is sound and complete for Horn KB
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Proof of completeness
FC derives every atomic sentence that is entailed by KB
1.
2.
3.
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
a1 ak b
1. 2.
Hence m is a model of KB If KB q, q is true in every model of KB, including m
3.
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 1. has already been proved true, or 2. has already failed
3.
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
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
Efficient propositional inference
Two families of efficient algorithms for propositional i...