Loading in 5 sec....

Propositional Logic or how to reason correctlyPowerPoint Presentation

Propositional Logic or how to reason correctly

- By
**Lucy** - Follow User

- 182 Views
- Updated On :

Download Presentation
## PowerPoint Slideshow about '' - Lucy

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

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.

(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-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 ProofResolution 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

- 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.

Download Presentation

Connecting to Server..