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Propositional Logic Reasoning correctly computationally. Chapter 7 or 8. Natural Reasoning. John plays tennis if sunny and weekend day. If John plays tennis, Mary goes shopping. It is Saturday. It is sunny. Specific: Does John play tennis? All: what may one conclude?.

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natural reasoning
Natural Reasoning

John plays tennis if sunny and weekend day.

If John plays tennis, Mary goes shopping.

It is Saturday.

It is sunny.

  • Specific: Does John play tennis?
  • All: what may one conclude?
state space model
State-Space Model?
  • What are the States?
  • What are the legal operators?
  • What is an appropriate search?
  • What do we want?
states
States
  • Collection of boolean formula in boolean variables.
  • Proposition variables stand for a statement that may be either true or false.
  • Ex. It is the weekend. Q
  • Ex. It is Saturday. P
  • Ex. It is Saturday implies is weekend:

P =>Q

Initial State: what you know

{ P, P=>Q} meaning clauses are true.

operators
Operators
  • Operators take a previous state (collection of formula) and add new formula.
  • Modus Ponens: If A is true, and A implies B, then B is true.
  • Model:

A = it is Saturday, B = it is weekend

and A is true, and A=>B is true, then B is true.

what are the right operators
What are the right operators?
  • If some A are B, and some B are C, then some A are C.
  • If A implies B, and B is false, then A is false.
a model
A model
  • Models are particular instantiations of the variables.
  • If some A are B, and some B are C, then some A are C.
  • A = women, B= students, C = men
  • If some women are students, and some students are men, then ….
  • Bad Rule.
concerns
Concerns
  • What does it mean to say a statement is true?
  • What are a good set of operators?
  • What can we say in propositional logic?
  • What is the efficiency?
  • Can we guarantee to infer all true conclusions?
semantic definition of truth
Semantic definition of Truth
  • 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 = 2.
  • 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
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
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
Propositional Logic: Semantics
  • Truth tables: p =>q |= ~p or q
beware implies
Beware: 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
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.
  • Complexity: approximately O(2^n).
  • Complete: possible worlds finite for propositional logic, unlike for arithmetic.
translation into propositional logic
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
Concepts
  • Equivalence: two sentences are equivalent if they are true in same models or worlds.
  • Validity: a sentence is valid if it is 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
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
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
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
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)

horn clauses prolog program
Horn Clauses = Prolog program
  • Horn clauses have 1 positive literal.
  • They have the form a,b,c,…=> d
  • Modus Ponens is “Horn Clause” complete.
  • Means: If KB is a set of horn clauses, and KB => horn clause c, then KB -> c by modus ponens.
  • Resolution is also “horn clause” complete since it yields modus ponens.
conjunctive normal form
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.

Proposition -> CNFGoal: Proving R
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 Proof
resolution algorithm
Resolution Algorithm

To prove s1, s2..sn |-- s

  • Put s1,s2,..sn & not s into cnf.
  • Resolve any 2 clauses that have complementary literals
  • If you get empty, done
  • Continue until set of clauses doesn’t grow.

Search can be expensive (exponential).

forward and backward reasoning
Forward and Backward Reasoning

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
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
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.
what can t we say
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.
  • Changing world: actions
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