Propositional Logic
or how to reason correctly
Chapter 8 (new edition)
Chapter 7 (old edition)
Goals
• Feigenbaum: In the knowledge lies the
power. Success with expert systems. 70’s.
• What can we represent?
– Logic(s): Prolog
– Mathematical knowledge: mathematica
– Common Sense Knowledge: Lenat’s Cyc has a
million statement in various knowledge
– Probabilistic Knowledge: Bayesian networks
• Reasoning: via search
History
• 300 BC Aristotle: Syllogisms
• Late 1600’s Leibnitz’s goal: mechanization
of inference
• 1847 Boole: Mathematical Analysis of
Logic
• 1879: Complete Propositional Logic: Frege
• 1965: Resolution Complete (Robinson)
• 1971: Cook: satisfiability NP-complete
• 1992: GSAT Selman min-conflicts
Syllogisms
• Proposition = Statement that may be either
true or false.
• John is in the classroom.
• Mary is enrolled in 270A.
• If A is true, and A implies B, then B is true.
• If some A are B, and some B are C, then
some A are C.
• If some women are students, and some
students are men, then ….
Concerns
• What does it mean to say a statement is
true?
• What are sound rules for reasoning
• What can we represent in propositional
logic?
• What is the efficiency?
• Can we guarantee to infer all true
statements?
Semantics
•
•
•
•
Model = possible world
x+y = 4 is true in the world x=3, y=1.
x+y = 4 is false in the world x=3, y = 1.
Entailment S1,S2,..Sn |= S means in every
world where S1…Sn are true, S is true.
• Careful: No mention of proof – just
checking all the worlds.
• Some cognitive scientists argue that this is
the way people reason.
Reasoning or Inference Systems
• Proof is a syntactic property.
• Rules for deriving new sentences from old
ones.
• Sound: any derived sentence is true.
• Complete: any true sentence is derivable.
• NOTE: Logical Inference is monotonic.
Can’t change your mind.
Proposition Logic: Syntax
• See text for complete rules
• Atomic Sentence: true, false, variable
• Complex Sentence: connective applied to
atomic or complex sentence.
• Connectives: not, and, or, implies,
equivalence, etc.
• Defined by tables.
Propositional Logic: Semantics
• Truth tables: p =>q |=
~p or q
p
q
p =>q
~p or q
t
t
t
t
t
f
t
t
t
f
t
t
t
f
t
t
Implies =>
• If 2+2 = 5 then monkeys are cows. TRUE
• If 2+2 = 5 then cows are animals. TRUE
• Indicates a difference with natural
reasoning. Single incorrect or false belief
will destroy reasoning. No weight of
evidence.
Inference
•
•
•
•
Does s1,..sk entail s?
Say variables (symbols) v1…vn.
Check all 2^n possible worlds.
In each world, check if s1..sk is true, that s
is true.
• Approximately O(2^n).
• Complete: possible worlds finite for
propositional logic, unlike for arithmetic.
Translation into Propositional Logic
• If it rains, then the game will be cancelled.
• If the game is cancelled, then we clean house.
• Can we conclude?
– If it rains, then we clean house.
• p = it rains, q = game cancelled r = we clean
house.
• If p then q.
not p or q
• If q then r.
not q or r
• if p then r.
not p or r (resolution)
Concepts
• Equivalence: two sentences are equivalent
if they are true in same models.
• Validity: a sentence is valid if it true in all
models. (tautology) e.g. P or not P.
– Sign: Members or not Members only.
– Berra: It’s not over till its over.
• Satisfiability: a sentence is satisfied if it true
in some model.
Validity != Provability
• Goldbach’s conjecture: Every even number
(>2) is the sum of 2 primes.
• This is either valid or not.
• It may not be provable.
• Godel: No axiomization of arithmetic will
be complete, i.e. always valid statements
that are not provable.
Natural Inference Rules
• Modus Ponens: p, p=>q |-- q.
– Sound
• Resolution example (sound)
– p or q, not p or r |-- q or r
• Abduction (unsound, but common)
– q, p=>q |-- p
– ground wet, rained => ground wet |-- rained
– medical diagnosis
Natural Inference Systems
• Typically have dozen of rules.
• Difficult for people to use.
• Expensive for computation.
– e.g. a |-- a or b
– a and b |-- a
• All known systems take exponential time in
worse case. (co-np complete)
Full Propositional Resolution
•
•
•
•
•
•
clause 1: x1 +x2+..xn+y (+ = or)
clause 2: -y + z1 + z2 +… zm
clauses contain complementary literals.
x1 +.. xn +z1 +… zm
y and not y are complementary literals.
Theorem: If s1,…sn |= s then
s1,…sn |-- s by resolution.
Refutation Completeness.
Factoring: (simplifying: x or x goes to x)
Conjunctive Normal Form
• To apply resolution we need to write what
we know as a conjunct of disjuncts.
• Pg 215 contains the rules for doing this
transformation.
• Basically you remove all  and => and
move “not’s” inwards. Then you may need
to apply distributive laws.
Proposition -> CNF
Goal: Proving R
•
•
•
•
•
•
P
(P&Q) =>R
(S or T) => Q
T
Distributive laws:
(-s&-t) or q
(-s or q)&(-t or q).
•
•
•
•
•
•
P
-P or –Q or R
-S or Q
-T or Q
T
Remember:implicit
adding.
Resolution Proof
•
•
•
•
•
•
P (1)
-P or –Q or R (2)
-S or Q (3)
-T or Q (4)
T (5)
~R (6)
•
•
•
•
•
-P or –Q : 7 by 2 & 6
-Q : 8 by 7 & 1.
-T : 9 by 8 & 4
empty: by 9 and 5.
Done: order only
effects efficiency.
Resolution Algorithm
To prove s1, s2..sn |-- s
1. Put s1,s2,..sn & not s into cnf.
2. Resolve any 2 clauses that have
complementary literals
3. If you get empty, done
4. Continue until set of clauses doesn’t grow.
Search can be expensive (exponential).
Forward and Backward Reasoning
• Horn clause has at most 1 positive literal.
– Prolog only allows Horn clauses.
– if a, b, c then d => not a or not b or not c or d
– Prolog writes this:
• d :- a, b, c.
– Prolog thinks: to prove d, set up subgoals a, b,c
and prove/verify each subgoal.
Forward Reasoning
•
•
•
•
•
•
From facts to conclusions
Given s1: p, s2: q, s3: p&q=>r
Rewrite in clausal form: s3 = (-p+-q+r)
s1 resolve with s3 = -q+r (s4)
s2 resolve with s4 = r
Generally used for processing sensory
information.
Backwards Reasoning:
what prolog does
•
•
•
•
•
•
•
From Negative of Goal to data
Given s1: p, s2: q, s3: p&q=>r
Goal: s4 = r
Rewrite in clausal form: s3 = (-p+-q+r)
Resolve s4 with s3 = -p +-q (s5)
Resolve s5 with s2 = -p (s6)
Resolve s6 with s1 = empty. Eureka r is true.
Davis-Putnam Algorithm
• Effective, complete propositional algorithm
• Basically: recursive backtracking with tricks.
– early termination: short circuit evaluation
– pure symbol: variable is always + or – (eliminate the
containing clauses)
– one literal clauses: one undefined variable, really
special cases of MRV
• Propositional satisfication is a special case of
Constraint satisfication.
WalkSat
• Heuristic algorithm, like min-conflicts
• Randomly assign values (t/f)
• For a while do
– randomly select a clause
– with probability p, flip a random variable in
clause
– else flip a variable which maximizes number of
satisfied clauses.
• Of course, variations exists.
Hard Satisfiability Problems
• Critical point: ratio of clauses/variables =
4.24 (empirical).
• If above, problems usually unsatsifiable.
• If below, problems usually satisfiable.
• Theorem: Critical range is bounded by
[3.0003, 4.598].
What can’t we say?
• Quantification: every student has a father.
• Relations: If X is married to Y, then Y is
married to X.
• Probability: There is an 80% chance of rain.
• Combine Evidence: This car is better than
that one because…
• Uncertainty: Maybe John is playing golf.