Explorations in Artificial Intelligence

1
Explorations in Artificial Intelligence

• Prof. Carla P. Gomes
• gomes_at_cs.cornell.edu
• Module 3-1
• Logic Based Reasoning

2
Knowledge and Reasoning

• Knowledge and Reasoning ? humans are very good at
  acquiring new
• information by combining raw knowledge,
  experience, with reasoning
• Examples
• ?diagnosis e.g., a physician diagnoses a
  patient, i.e., it infers what disease he/she has,
  based on the knowledge he/she acquired as a
  student, textbooks, prior cases and also some
  reasoning process (patterns of association, or
  other process) that he/she may not be able to
  describe.
• ?car repair diagnosis
• Common sense reasoning
• Inventions, new ideas

Key issues Representation of knowledge Reasonin
g processes
3
Knowledge-base Agents

• Key issues
• Representation of knowledge ? knowledge base
• Reasoning processes ? inference/reasoning
  mechanisms to query what is known, to derive new
  information, to make decisions.

• Knowledge base set of sentences in a formal
  language representing facts about the world()

() called knowledge representation language
4
Knowledge bases

• Key aspects
• How to add sentences to the knowledge base
• How to query the knowledge base
• Both tasks may involve inference i.e. how to
  derive new sentences from old sentences
• Logical agents inference must obey the
  fundamental requirement that when one asks a
  question to the knowledge base, the answer should
  follow from what has been told to the knowledge
  base previously. (In other words the inference
  process should not make things up)

5
Logic Outline

• General principles of logic main vehicle for
  representing knowledge
• Wumpus World - a toy world how a knowledge
  based agent operates
• Propositional logic
• Predicate logic
• Proof Methods
• Inference Rules (including induction)
• Model Checking
• Algorithmic approaches
• Satisfiability as an Encoding language

6
Logic in general

• Logics are formal languages for formalizing
  reasoning, in particular for representing
  information such that conclusions can be drawn
• A logic involves
• A language with a syntax for specifying what is a
  legal expression in the language syntax defines
  well formed sentences in the language
• Semantics for associating elements of the
  language with elements of some subject matter.
  Semantics defines the "meaning" of sentences
  (link to the world) i.e., semantics defines the
  truth of a sentence with respect to each possible
  world
• Inference rules for manipulating sentences in the
  language

Original motivation Early Greeks, settle
arguments based on purely rigorous
(symbolic/syntactic) reasoning starting from a
given set of premises.
7
Example of a formal language Arithmetic

• E.g., the language of arithmetic
• x2 y is a sentence ( x2y is not a
  sentence)
• x2 y is true iff the number x2 is no less
  than the number y
• x2 y is true in a world where x 7, y 1
• x2 y is false in a world where x 0, y 6

8

Language to Specify Systems as Constrained
Featured Sets

• Several systems biological, mechanical,
  electric, etc --- can
• be represented by appropriate sets of features
  with
• constraints among the features encoding physical
  or other
• laws relevant to the organism or device
• Reasoning can then be used among other purposes,
  to
• diagnose malfunctions in these systems for
  example, features
• associated with causes can be inferred from
  features
• associated with symptoms. This general approach
  is key to
• an important class of AI applications.

9
Simple Robot Domain

• Consider a robot that is able to lift a block, if
  that block is liftable (i.e., not
• too heavy), and if the robots battery power is
  adequate. If both of these
• conditions are satisfied, then when the robot
  tries to lift a block it is
• holding, its arm moves.

Feature 1 BatIsOk (True or False) Feature 2
BlockLiftable (True or False) Feature 3
RobotMoves (True or False)
10
Simple Robot Domain
We need a language to express the
features/properties/assertions and constraints
among them also inference mechanisms,
i.e, principled ways of performing reasoning.
Example logical statement about the robot
(BatIsOk and BlockLiftable) implies RobotMoves
11
Binary valued featured descriptions

• Consider the following description
• The router can send packets to the edge system
  only if it supports the new address space. For
  the router to support the new address space it is
  necessary that the latest software release be
  installed. The router can send packets to the
  edge system if the latest software release is
  installed. The router does not support the new
  address space.
• Features
• Router
• Feature 1 router can send packets to the edge
  of system
• Feature 2 router supports the new address space
• Latest software release
• Feature 3 latest software release is installed

12
Binary valued featured descriptions

• Constraints
• The router can send packets to the edge system
  only if it supports the new address space.
  (constraint between feature 1 and feature 2)
• It is necessary that the latest software release
  be installed for the router to support the new
  address space . (constraint between feature 2 and
  feature 3)
• The router can send packets to the edge system if
  the latest software release is installed.
  (constraint between feature 1 and feature 3)
• How can we write these specifications in a formal
  language and reason about the system?

13
Truth of Sentence vs. Satisfaction of Constraints

• Truth of a sentence vs. Satisfaction of
  Constraints - we can think of a logic sentence,
  e.g., an arithmetic sentence or a general logic
  sentence, as a constraint the sentence is true
  if and only if the constraint is satisfied.
• We will talk more about constraint languages,
  particular
• kinds of logics, and constraint solving as a form
  of logical
• reasoning.
• Standard logics ? every sentence must be either
  true or false in each possible world there is
  no in between.

14
SudokuA constraint language
15
SudokuSolving Sudoku by logic inference
16
Logical ReasoningEntailment

• Entailment means that one thing follows from
  another
• KB a
• A Knowledge base KB entails sentence a iff (if
  and only if) a is true in all worlds where KB is
  true
• E.g., the KB containing Giants won and Reds
  won entails Either the Giants won or the Reds
  Won
• E.g., xy 4 entails 4 xy
• 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
• Example
• x y 7 , is true in all the models in which
  x 7 - y, assuming that we are dealing with
  real numbers, in particular x 7 and y 0 or x
  8 and y 1, etc
• Basically, each model corresponds to a different
  assignment to the variables satisfying the
  constraints note each assignment determines the
  truth or falsehood of the arithmetic sentence.
• We say m is a model of a sentence a if a is true
  in m
• M(a) is the set of all models of a (i.e., models
  that assign true to a ).

18
Models

• KB a iff M(KB) ? M(a)
• E.g. KB Giants won and Redswon a Giants won
• Other ways of talking about entailment
• KB a
• If a is true, then KB must be true
• (Informally the truth of a is contained in the
  truth of KB)

We can think of a knowledge base as a statement
and we talk about a knowledge base entailing a
sentence.
19
2 - Wumpus World
20
Wumpus World

• Performance measure
• gold 1000,
• death -1000
• (falling into a pit or being eaten by the wumpus)
• -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, Shootnnnnnnnn

21
Exploring a wumpus world
The knowledge base of the agent consists of the
rules of the Wumpus world plus the percept
nothing in 1,1
None, none, none, none, none
Stench, Breeze, Glitter, Bump, Scream
22
Exploring a wumpus world
The knowledge base of the agent consists of the
rules of the Wumpus world plus the percept
nothing in 1,1 by inference, the agents
knowledge base also has the information that
2,1 and 1,2 are okay.
None, none, none, none, none
Stench, Breeze, Glitter, Bump, Scream
23
Exploring a wumpus world
P?
P?
A/B
V
None, breeze, none, none, none
None, none, none, none, none
A Agent V visited B - Breeze
Stench, Breeze, Glitter, Bump, Scream
Pit in (2,2) or (3,1)
24
Exploring a wumpus world
4
3
W
S
P?
2
P
1
P
P?
1 2 3 4
S (Stench, none, none, none, none)
S ? Wumpus nearby
Wumpus cannot be in (1,1) or in (2,2) (Why?)?
Wumpus in (1,3) Not breeze in (1,2) ? no pit in
(2,2) but we know there is a pit in (2,2) or
(3,1) ? pit in (3,1)
25
Exploring a wumpus world
Difficult inference, because it
combines knowledge gained at different times
in difference places the inference is beyond the
abilities of most animals.
A
none, none, none, none, none
Assumption the agent turns and go to square
(2,3) ?!
In each case where the agent draws a conclusion
from the available information, that conclusion
is guaranteed to be correct if the initial
information is correct - fundamental property of
logical reasoning!
How to build logical agents that can represent
the necessary information and draw conclusions?
26
Entailment in the wumpus world
Knowledge Base in the Wumpus World ? Rules of the
wumpus world new percepts

• Situation after detecting nothing in 1,1,
  moving right, breeze in 2,1
• Consider possible models for KB with respect to
  the cells (1,2), (2,2) and (3,1), with respect
  to the existence or non existence of pits
• 3 Boolean choices ?
• 8 possible models (enumerate all the models)

27
Wumpus models
Why is KB false in these models?

• KB wumpus-world rules observations

28
Wumpus models
Models of the KB and a1

• KB wumpus-world rules observations
• a1 "1,2 has no pit", KB a1,
• In every model in which KB is true, a1 is True
  (proved by model checking)

29
Wumpus models
Models of the KB and a2

• KB wumpus-world rules observations
• a2 "2,2 has no pit", this is only True in
  some
• of the models for which KB is True, therefore KB
  a2
  
• Inference algorithm used to reason about a1 and
  a2 ?
• Model Checking

30
InferenceModel Checking

• Inference by Model checking
• we enumerate all the KB models and check if
• a1 and a2 are True in all the models (which
• implies that we can only use it when we have
• a finite number of models).

31
Inference

• KB i a we say sentence a can be derived from KB
  by procedure i
• Soundness (or Truth preservation) i is sound if
  whenever KB i a, it is also true that KB a an
  unsound procedure can conclude statements that
  are not true.
• Completeness i is complete if whenever KB a, it
  is also true that KB i a a complete procedure
  is able to derive any sentence that is entailed.
  That is, the procedure will answer any question
  whose answer follows from what is known by the
  KB.
• Note first-order logic which is expressive
  enough to say almost anything of interest, and
  for which there exists a sound and complete
  inference procedure.

32
Propositional Logic
33
Syntax Elements of the language
Primitive propositions --- statements like Bob
loves Alice Alice loves Bob
Compound propositions Bob loves Alice and Alice
loves Bob
34
Connectives

• - not
• ? - and
• ? - or
• ? - implies
• ? - equivalent (if and only if)

35

Syntax

• Syntax of Well Formed Formulas (wffs) or
  sentences
• Atomic sentences are wffs
• Propositional symbol (atom)
• Example P, Q, R, BlockIsRed SeasonIsWinter
• Complex or compound wffs.
• Given w1 and w2 wffs
• ? w1 (negation)
• (w1 ? w2) (conjunction)
• (w1 ? w2) (disjunction)
• (w1 ? w2) (implication w1 is the antecedent w2
  is the consequent)
• (w1 ? w2) (biconditional)

36
Propositional logic Examples
Examples of wffs

• P ? Q
• (P ? Q) ? R
• P ? Q ? P
• (P ? Q) ? (?Q ? ?P)
• ? ?P
• P ? ? this is not a wff.

• Note1 atoms or negated atoms are called
  literals examples p and ?p are literals. P ? Q
  is a compound statement or proposition.
• Note2 parentheses are important to ensure that
  the syntax is unambiguous. Quite often
  parentheses are omitted The order of precedence
  in propositional logic is (from highest to
  lowest) ? ,?, ?, ?, ?

37
Propositional LogicSyntax vs. Semantics

• Semantics has to do with meaning
• ? it associates the elements of a logical
  language with the elements of a domain of
  discourse.
• Propositional Logic we associate atoms with
  propositions / assertions about the world
  (therefore propositional logic).

38
Propositional LogicSemantics

• Interpretation or Truth Assignment
• Assignment of truth values (True or False) to
  every proposition.
• So if for n atomic propositions, there are
  2n truth assignments or interpretations. This
  makes the representation powerful the
• propositions implicitly capture 2n possible
  states of the world.

39
Propositional LogicSemantics

• Example
• We might associate the atom (just a symbol!)
  BlockIsRed with the proposition The block is
  Red, but we could also associate it with the
  proposition The block is Black even though this
  would be quite confusing BlockIsRed has value
  True just in the case the block is red otherwise
  BlockIsRed is False. (Aside computers manipulate
  symbols. The string BlockIsRed does not mean
  anything to the computer. Meaning has to come
  from how to come from relations to other symbols
  and the external world. Hmm.
• How can a computer / robot obtain the
  meaning The block is Red? The fact that
  computers only push around symbols led to quite
  a bit of confusion in the early days or
  Artificial Intelligence, Robotics, and natural
  language understanding.
• Which ones are propositions?
• Cornell University is in Ithaca NY
• 1 1 2
• what time is it?
• 2 3 10
• watch your step!

40
Propositional LogicSemantics
Truth table for connectives Given the values of
atoms under some interpretation, we can use a
truth table to compute the value for any wff
under that same interpretation the truth table
establishes the semantics (meaning) of the
propositional connectives.
?
?
We can use the truth table to compute the value
of any wff given the values of the constituent
atom in the wff. Note In table, P and Q can be
compound propositions themselves. Note
implication not necessarily aligned with English
usage.
41

Implication (p ? q)

• This is only False (violated) when q is False and
  p is True.
• Related implications
• Converse q ? p
• Contra-positive ?q ? ? p
• Inverse ? p ? ? q

Important only the contra-positive of p ? q is
equivalent to p ? q (i.e., has the same truth
values in all models) the converse and the
inverse are equivalent
42
Implication (p ? q)

• Implication plays an important role in reasoning
  a variety of terminology is used to refer to
  implication

• conditional statement
• if p then q
• if p, q
• p is sufficient for q
• q if p
• q when p
• a necessary condition for p is q ()

• p implies q
• p only if q ()
• a sufficient condition for q is p
• q whenever p
• q is necessary for p ()
• q follows from p

Note the mathematical concept of implication is
independent of a cause and effect relationship
between the hypothesis (p) and the conclusion
(q), that is normally present when we use
implication in English. Note Focus on the case,
when is the statement False. I.e., p is True and
q is False, should be the only case that makes
the statement false.
() assuming the statement true, for p to be
true, q has to be true
43
Propositional LogicSemantics
Notes Bi-conditionals (p ? q)

• Variety of terminology

• p is necessary and sufficient for q
• if p then q, and conversely
• p if and only if q
• p iff q

p ? q is equivalent to (p?q) ? (q ?p)
Note the if and only if construction used in
biconditionals is rarely used in common
language Example if you finish your meal, then
you can play what is really meant is If you
finish your meal, then you can play and You
can play, only if you finish your meal.
44
Exclusive Or

• Truth Table
• P Q P ? Q
• _____________
• T T F
• T F T
• F T T
• F F F

P ? Q is equivalent to (P ?Q) ? (P?Q) and also
equivalent to (P ? Q) Use a truth table to
check these equivalences.
45
Propositional LogicSatisfiability and Models
Satisfiability and Models
An interpretation or truth assignment satisfies
a wff, if the wff is assigned the value True,
under that interpretation. An interpretation that
satisfies a wff is called a model of that wff.
Given an interpretation (i.e., the truth values
for the n atoms) the one can use the truth
table to find the value of any wff.
46
The truth table method
(Propositional) logic has a truth compositional
semantics Meaning is built up from the meaning
of its primitive parts (just like English text).
47
Propositional LogicInconsistency
(Unsatisfiability) and Validity

• Inconsistent or Unsatisfiable set of Wffs

• It is possible that no interpretation satisifies
  a set of wffs ?
• In that case we say that the set of wffs is
  inconsistent or unsatisfiable or a contradiction
• Examples
• 1 P ? ?P
• 2 P ? Q, P ??Q, ?P ? Q, ?P ??Q
• (use the truth table to confirm that
  this set of wffs is inconsistent)

• Validity (Tautology) of a set of Wffs

If a wff is True under all the interpretations of
its constituents atoms, we say that the wff is
valid or it is a tautology. Examples 1- P ?
P 2 - ?(P ? ?P) 3 - P ? (Q ? P) 4-
(P ? Q) ?P) ?P
48
Logical equivalence

• Two sentences p an q are logically equivalent (?
  or ?) iff p ? q is a tautology
• (and therefore p and q have the same truth
  value for all truth assignments)

?
Note logical equivalence (or iff) allows us to
make statements about PL, pretty much like we
use in in ordinary mathematics.
49
Truth Tables
Truth table for connectives
We can use the truth table to compute the value
of any wff given the values of the constituent
atom in the wff. Example Suppose P and Q are
False and R has value True. Given this
interpretation, what is the truth value of ( P ?
Q) ? R ? P?
False
If a system is described using n features
(corresponding to propositions), and these
features are represented by a corresponding set
of n atoms, then there are 2n different ways
the system can be. Why? Each of the ways the
system can be corresponds to an interpretation.
Therefore there are , i.e., 2n interpretations.
50
Example Binary valued featured descriptions

• Consider the following description
• The router can send packets to the edge system
  only if it supports the new address space. For
  the router to support the new address space it is
  necessary that the latest software release be
  installed. The router can send packets to the
  edge system if the latest software release is
  installed. The router does not support the new
  address space.
• Features
• Router
• P - router can send packets to the edge of
  system
• Q - router supports the new address space
• Latest software release
• R latest software release is installed

51

• 
  
  Formal
• The router can send packets to the edge system
  only if it supports
• the new address space. (constraint between
  feature 1 and feature 2)
• If Feature 1 (P) (router can send packets to the
  edge of system) then P ? Q
• Feature 2 (Q) (router supports the new address
  space )
• For the router to support the new address space
  it is necessary that the
• latest software release be installed.
  (constraint between feature 2 and feature 3)
• If Feature 2 (Q) (router supports the new address
  space ) then
• Feature 3 (R) (latest software release is
  installed) Q ? R
• The router can send packets to the edge system if
  the latest software release
• is installed. (constraint between feature 1
  and feature 3)
• If Feature 3 (R) (latest software release is
  installed) then
• Feature 1 (P) (router can send packets to the
  edge of system) R ? P
• The router does not support the new address
  space. Q

52
Inference
53
Entailment in the wumpus world
Knowledge Base in the Wumpus World ? Rules of the
wumpus world new percepts

• Situation after detecting nothing in 1,1,
  moving right, breeze in 2,1
• Consider possible models for KB with respect to
  the cells (1,2), (2,2) and (3,1), with respect
  to the existence or non existence of pits
• 3 Boolean choices ?
• 8 possible models (enumerate all the models)

54
Wumpus world sentences

• Let Pi,j be true if there is a pit in i, j.
• Let Bi,j be true if there is a breeze in i, j.
• Sentence 1 (R1) ? P1,1
• Sentence 2 (R2) ?B1,1
• Sentence 3 (R3) B2,1
• "Pits cause breezes in adjacent squares"
• Sentence 4 (R4) B1,1 ? (P1,2 ? P2,1)
• Sentence 5 (R5) B2,1 ? (P1,1 ? P2,2 ? P3,1)

55
Inference by enumeration

• The goal of logical inference is to decide
  whether KB a, for some sentence ?.
• For example, given the rules of the Wumpus World
  is P22
• entailed?
• Relevant propositional symbols

R1 ? P1,1 R2 ?B1,1 R3 B2,1 "Pits cause breeze
s in adjacent squares" R4 B1,1 ? (P1,2 ?
P2,1) R5 B2,1 ? (P1,1 ? P2,2 ? P3,1)

Inference by enumeration ? we have 7 symbols
therefore 27 models
56
Propositional logic Wumpus World

• Each model specifies true/false for each
  proposition symbol
  
• E.g. P1,2 P2,2 P3,1
• false true false
• With these symbols, 8 interpretations, can be
  enumerated
• automatically.

P12 ? P22 ? P31 P12 ? P22 ? ?P31 P12 ? ?P22 ?
P31 etc
57
Is P12 Entailed from KB?Is P22 Entailed from
KB?Given R1, R2, R3, R4, R5
Consider all possible truth assignments to P12,
P22, P31, and check which assignments lead to
models for the KB then check if P12 and P22 is
true in all the models
58
Is P12 Entailed from KB?Is P22 Entailed from
KB?Given R1, R2, R3, R4, R5
There are only 3 models for the KB i.e., for
which R1, R2, R3, R4, R5 are True In all of
them P12 is false, so there is not pit in 1,2
the KB entails ?P12 on the other hand P22 is
true in two of the three models and false in the
other one so at this point we cannot tell
whether P22 is true or not.
59
Inference by enumeration
TT-Entails Truth Table enumeration algorithm
for deciding propositional entailment
Processed all symbols
TT Truth Table PL-True returns true if a
sentence holds within a model Model represents
a partial model an assignment to some of the
variables EXTEND(P,true,model) returns a
partial model in which P has the value True
60
Models

• KB a iff M(KB) ? M(a)

Note The empty set or null set ( Ø ) is a
subset of every set. An inconsistent KB entails
every possible sentence.
61
Inference by enumeration
TT-Entails Truth Table enumeration algorithm
for deciding propositional entailment
This is a recursive enumeration of a finite
space of assignments to variables depth-first
algorithm it enumerates all models and checks if
the sentence is true in all the models ? sound
? complete For n symbols, time complexity is
O(2n), space complexity is O(n). Worst-case
complexity is exponential for any algorithm. But
in practice we can do better. More later
62
Validity and Satisfiability

• A sentence is valid (or is a tautology) if it is
  true in all interpretations,
• e.g., True, A ??A, A ? A, (A ? (A ? B)) ? B
• Validity is connected to inference via the
  Deduction Theorem
• KB a if and only if (KB ? a) is valid
• A sentence is satisfiable if it is true in some
  model
• e.g., A? B, 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 a if and only if (KB ??a) is unsatisfiable
  (Reductio ad absurdum
• Proof by refutation or Proof by contradiction)