1. Logical Agents Chapter 7 How do we reason about reasonable decisions
Fall 2009
Copyright, 1996 © Dale Carnegie & Associates, Inc.

2. A knowledge-based agent
Accepting new tasks in explicit goals
Knowing about its world
current state of the world, unseen properties from percepts, how the world evolves
help deal with partially observable environments
help understand “John threw the brick thru the window and broke it.” – natural language understanding
Reasoning about its possible course of actions
Achieving competency quickly by being told or learning new knowledge
Adapting to changes by updating the relevant knowledge
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3. Knowledge Base
A knowledge base (KB) is a set of representations (sentences) of facts about the world.
TELL and ASK - two basic operations
to add new knowledge to the KB
to query what is known to the KB
Infer - what should follow after the KB has been TELLed.
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4. A generic KB agent (Fig 7.1)
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5. Three levels of A KB Agent
Knowledge level (the most abstract)
Logical level (knowledge is of sentences)
Implementation level
Building a knowledge base
A declarative approach - telling a KB agent what it needs to know
A procedural approach – encoding desired behaviors directly as program code
A learning approach - making it autonomous
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6. Specifying the environment
The Wumpus world (Fig 7.2) in PEAS
Performance: for getting the gold, for being dead, -1 for each action taken, -10 for using up the arrow
Goal: bring back gold as quickly as possible
Environment: 4X4, start at (1,1) ...
Actions: Turn, Grab, Shoot, Climb, Die
Sensors: (Stench, Breeze, Glitter, Bump, Scream)
It’s possible that the gold is in a pit or surrounded by pits -> try not to risk life, just go home empty-handed
The variants of the Wumpus world – they can be very difficult
Multiple agents
Mobile wumpus
Multiple wumpuses
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7. 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 a 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
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8. Acting & reasoning Let’s play the wumpus game!
The conclusion: “what a fun game!”
Another conclusion: If the available information is correct, the conclusion is guaranteed to be correct.
Figs 7.3 and 7.4
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9. Logic The primary vehicle for representing knowledge
Simple
Concise
Precise
Can be manipulated following rules
It cannot represent uncertain knowledge well (so it’s where new research is about)
We will learn Logic first and other techniques later
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10. Logics A logic consists of the following: Some examples of logics ...
A formal system for describing states of affairs, consisting of syntax (how to make sentences) and semantics (to relate sentences to states of affairs).
A proof theory - a set of rules for deducing the entailments of a set of sentences.
Some examples of logics ...
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11. Propositional Logic e.g., D means “the wumpus is dead”
In this logic, symbols represent whole propositions (facts)
e.g., D means “the wumpus is dead”
W1,1 Wumpus is in square (1,1)
S1,1 there is stench in square (1,1).
Propositional logic can be connected using Boolean connectives to generate sentences with more complex meanings, but does not specify how objects are represented.
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12. Other logics
First order logic represents worlds using objects and predicates on objects with connectives and quantifiers.
Temporal logic assumes that the world is ordered by a set of time points or intervals and includes mechanisms for reasoning about time.
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13. Other logics (2)
Probability theory allows the specification of any degree of belief.
Fuzzy logic allows degrees of belief in a sentence and degrees of truth.
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14. Propositional logic Syntax Semantics
A set of rules to construct sentences:
and, or, imply, equivalent, not
literals, atomic or complex sentences
BNF grammar (Fig 7.7, P205)
Semantics
Specifies how to compute the truth value of any sentence
Truth table for 5 logical connectives (Fig 7.8)
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15. Knowledge Representation
Syntax - the possible configurations that can constitute sentences
Semantics - the meaning of the sentences
x > y is a sentence about numbers; or x+y=4;
A sentence can be true or false
Defines the truth of each sentence w.r.t. each possible world
What are possible worlds for x+y = 4
Entailment: one sentence logically follows another
 |= , iff in every model in which  is true,  is also true
`Sentences’ entails `sentence’ w.r.t. `aspects’ follows `aspect’ (Fig 7.6)
Sentence is x+y=4; one model (a possible world) is x = 2 and y = 2
Sentence is true in some models and false in other models
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16. Reasoning KB entails sentence  if KB is true,  is true
Model checking (Fig 7.5) for two sentences/models
Asking whether KB entails s given KB?
1 = “There is no pit in [1,2]” -> yes or no?
2 = “There is no pit in [2,2]” -> yes or no?
P[1,2], P[2,2], and P[3,1], a total of 2^3 = 8 models, KB is in read, sentences (alpha1 and alpha2)
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17. An inference procedure
can generate new valid sentences or verify if a sentence is valid given KB
is sound if it generates only entailed sentences
A proof is the record of operation of a sound inference procedure
An inference procedure is complete if it can find a proof for any sentence that is entailed.
Sound reasoning is called logical inference or deduction.
A reasoning system should be able to draw conclusions that follow from the premises, regardless of the world to which the sentences are intended to refer.
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18. Equivalence, validity, and satisfiability
Logical equivalence requires  |= and  |= 
Validity: a sentence  is true in all models
Valid sentences are tautologies (P v !P)
“deduction theorem”: for any  and ,  |= iff the sentence ( ) is valid
Satisfiability: a sentence  is satisfiable if it is true in some models
E.g., A v B, P
 |= iff the sentence ( ^ !) is unsatisfiable or !( ^ !) is valid .
Connecting validity and satisfiability:
 is valid iff ! is unstatisfiable;
contrapositively,  is satisfiable iff ! is not valid.
alpha /= beta iff in every model in which alpha is true, beta is also true
! (a =>b), !(!a v b), a ^ !b
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19. Inference
Truth tables can be used not only to define the connectives, but also to test for validity:
If a sentence is true in every row, it is valid.
What is a truth table for “Premises imply Conclusion”
A simple knowledge base for Wumpus
A simple KB with five rules (P208)
What if we write R2 as B1,1 => (P1,2 v P2,1)
Think about the definition of =>
KB |= . Let’s check its validity (Fig 7.9)
E.g., in Figure 7.9, there are three true models for the KB with 5 rules.
A truth-table enumeration algorithm (Fig 7.10)
There are only finitely many models to examine, but it is exponential in size of the input (n)
Can we prove this?
The proof – what’s the size of the truth table?
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20. Reasoning Patterns in Prop Logic
 |= iff the sentence ( ^ !) is unstatisfiable
 are known axioms, thus true (T)
Proof by refutation (or contradiction): assuming  is F, !  is T, we now need to prove !(^T) is valid, …
Inference rules
Modus Ponens, AND-elimination, Bicond-elimination
All the logical equivalences in Fig 7.11
A proof is a sequence of applications of inference rules
An example to conclude neither [1,2] nor [2,1] contains a pit
Start with R2
Monotonicity (consistency): the set of entailed sentences can only increase as information is added to KB
For  and , if KB |=  then KB^ |= 
Propositional logic and first-order logic are monotonic
The first bullet follows from the deduction theorem.
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21. Resolution – an inference rule
An example of resolution
R11, R12 (new facts added), R13, R14 (derived from R11, and R12), R15 from R3 and R5, R16, R17 – P3,1 (there is a pit in [3,1]) (P213)
Unit resolution: l1 v l2 …v lk, m = !li
We have seen examples earlier
Full resolution: l1 v l2 …v lk, m1 v…v mn where li = mj
An example: (P1,1vP3,1, !P1,1v!P2,2)/P3,1v!P2,2
Soundness of resolution
Considering literal li,
If it’s true, mj is false, then …
If it’s false, …
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22. Refutation completeness
Resolution can always be used to either confirm or refute a sentence
Conjunctive normal form (CNF)
A conjunction of disjunctions of literals
A sentence in k-CNF has exactly k literals per clause (l1,1 v … v l1,k) ^…^ (ln,1 v …v ln,k)
A simple conversion procedure (turn R2 to CNF, next slide or see P.215)
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23. Conversion to CNF B1,1  (P1,2  P2,1)
Eliminate , replacing α  β with (α  β)(β  α).
(B1,1  (P1,2  P2,1))  ((P1,2  P2,1)  B1,1)
2. Eliminate , replacing α  β with α β.
(B1,1  P1,2  P2,1)  ((P1,2  P2,1)  B1,1)
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)
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24. A resolution algorithm (Fig 7.12)
An example (KB= R2^R4, to prove !P1,2, Fig. 7.13)
Completeness of resolution
Ground resolution theorem
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25. Horn cluases
A Horn clause is a disjunction of literals of which at most one is positive
An example: (!L1,1 v !Breeze V B1,1)
An Horn sentence can be written in the form P1^P2^…^Pn=>Q, where Pi and Q are nonnegated atoms
Deciding entailment with Horn clauses can be done in linear time in size of KB
Inference with Horn clauses can be done thru forward and backward chaining
Forward chaining is data driven
Backward chaining works backwards from the query, goal-directed reasoning
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26. An Agent for Wumpus The knowledge base (an example on p208)
Bx,y  …, Sx,y …
There is exactly one W: (1) there is at least one W, and (2) there is at most one W
Finding pits and wumpus using logical inference
Keeping track of location and orientation
Translating knowledge into action
A1,1^EastA^W2,1=>!Forward
Problems with the propositional agent
too many propositions to handle (“Don’t go forward if…”)
hard to deal with change (time dependent propositions)
(p225) has the details for the first bullet
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27. Summary Knowledge is important for intelligent agents
Sentences, knowledge base
Propositional logic and other logics
Inference: sound, complete; valid sentences
Propositional logic is impractical for even very small worlds
Therefore, we need to continue our AI class ...
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