1. CS 4700: Foundations of Artificial Intelligence
Carla P. Gomes
Module:
Propositional Logic:
Syntax and Semantics
(Reading R&N: Chapter 7)

2. Propositional Logic

3. Syntax: Elements of the language
Primitive propositions --- statements like:
Bob loves Alice
Alice loves Bob P Q
Propositional Symbols
(atomic propositions)
Compound propositions
Bob loves Alice and Alice loves Bob
P  Q ( - stands for and)

4. Connectives ¬ - not  - and  - or  - implies
 - equivalent (if and only if)

5. 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)

6. 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):  ,, , , 

7. Propositional Logic: Syntax 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).

8. Propositional Logic: Semantics
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.

9. Propositional Logic: Semantics
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!

10. Propositional Logic: Semantics
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.

11. Contra-positive: q   p; Inverse  p   q;
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;

12. 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

13. Propositional Logic: Semantics
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”.

14. Exclusive Or P  Q is equivalent to (P ¬Q)  (¬PQ)
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.

15. Propositional Logic: 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.

16. 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).

17. Propositional Logic: Inconsistency (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; (P  P); [P  (Q  P)]; [(P  Q) P) P]

18. 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.

19. Truth Tables Truth table for connectives False
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.

20. 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

21. Feature 2 (Q) (router supports the new address space )
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

22. Inference

23. 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)

24. 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)

25. Inference by enumeration
The goal of logical inference is to decide whether KB╞ α, 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 breezes 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;

26. 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

27. Is P12 Entailed from KB. Is P22 Entailed from KB
Is P12 Entailed from KB? Is P22 Entailed from KB? Given R1, R2, R3, R4, R5
P11
P12
P21
P22
P31
B11
B21
R1:P11
R2:B11
R3:B21
R4:B11(P12  P21)
R5:B21P11  P22P31 KB
False
True
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

28. Is P12 Entailed from KB. Is P22 Entailed from KB
Is P12 Entailed from KB? Is P22 Entailed from KB? Given R1, R2, R3, R4, R5
P11
P12
P21
P22
P31
B11
B21
R1:P11
R2:B11
R3:B21
R4:B11(P12  P21)
R5:B21P11  P22P31 KB
False
True
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.

29. Is P12 Entailed from KB. Is P22 Entailed from KB
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?
P11
P12
P21
P22
P31
B11
B21
R1:P11
R2:B11
R3:B21
R4:B11(P12  P21)
R5:B21P11  P22P31 KB
False
True
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.

30. 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…

31. Inference by enumeration
TT-Entails – Truth Table enumeration algorithm for
deciding propositional entailment;
Processed all symbols
We only care about models
For which KB is True
Depth-first enumeration of all models is sound and complete
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;

32. Models KB ╞ α iff M(KB)  M(α) Note: The empty set or null set ( Ø )
is a subset of every set.
An inconsistent KB entails every possible sentence.

33. 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 ╞ α iff (KB  α) 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 ╞ α iff (KB α) is unsatisfiable (Reductio ad absurdum;
Proof by refutation or Proof by contradiction)