CS 4700: Foundations of Artificial Intelligence

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CS 4700Foundations of Artificial Intelligence

• Carla P. Gomes
• gomes_at_cs.cornell.edu
• Module
• Propositional Logic
• Syntax and Semantics
• (Reading RN Chapter 7)

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

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

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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)

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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) ? ,?, ?, ?, ?

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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).

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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.

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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!

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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.
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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
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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
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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.
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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.
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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.
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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).
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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
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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.
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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.
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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

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• 
  
  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

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Inference
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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)

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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)

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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 interpretations check if P22 is
true in all the KB models
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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
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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
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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.
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Is P12 Entailed from KB?Is P22 Entailed from
KB?Given R1, R2, R3, R4, R5
What does the KB entail wrt P12?
What does the KB entail wrt P22?
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.
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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
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Inference by enumeration
TT-Entails Truth Table enumeration algorithm
for deciding propositional entailment
Processed all symbols
Depth-first enumeration of all models is sound
and complete TT Truth Table PL-True returns t
rue 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
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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.
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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 iff (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 iff (KB ??a) is unsatisfiable (Reductio ad
  absurdum
• Proof by refutation or Proof by contradiction)