Proof systems
Download
1 / 15

Proof Systems - PowerPoint PPT Presentation


  • 380 Views
  • Uploaded on

Proof Systems KB |- Q iff there is a sequence of wffs D1, ..., Dn such that Dn is Q and for each Di in the sequence: a) either Di is in KB or b) Di can be inferred from a wff (or wffs) earlier in the sequence by using one of the rules of inference in R , or

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Proof Systems' - omer


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
Proof systems l.jpg
Proof Systems

KB |- Q iff there is a sequence of wffs D1, ..., Dn

such that Dn is Q and for each Di in the sequence:

a) either Di is in KB or

b) Di can be inferred from a wff (or wffs) earlier in

the sequence by using one of the rules of inference

in R, or

c) Di is an instance of a logical axiomin AX

The sequence (if exists) D1, ..., Dn is called a proof or a deduction of Q from KB.

Q is said to be a theorem of KB.

KB |- Q :

a) by the definition of entailment:


What is soundness l.jpg
What is soundness?

For every KB and Q, if KB |- Q then KB |= Q

Informally, a proof system is sound if it only generates entailed wffs

(every positive answer is correct)

(remember that the semantical system is the reference)

A sound proof system is truth-preserving:

any model for the original set of wffs (KB) is also a model for the derived set of wffs (Q).


Completeness l.jpg
Completeness

  • One other question we can ask is whether using our proof system we can generate all of the entailed wffs

    (the system can give all the correct answers)

  • If we are able to do so, we say that our inference procedure is complete:

    For every KB and Q, if KB |= Q then KB |- Q

    Equivalent form: if KB |/- Q then KB ||/- Q


Complexity l.jpg
Complexity

  • Truth-tables are exponential in the number of atoms: 2n interpretations

  • [Cook 71] showed that Satisfiability is a

    NP-complete problem.

  • But in many cases answers can be found very quickly (Horn-Sat is solvable linear time)

  • in fact really hard problems are quite rare (see hw).


Proof systems5 l.jpg
Proof Systems

Several proof systems in the literature:

  • Resolution (the only one we will study)

    SLD resolution - basis of PROLOG

  • Tableaux

  • Natural Deduction

  • Sequent Calculus (Gentzen)

  • Axiomatic (Hilbert)


Clauses as wffs l.jpg
Clauses as wffs

  • More adequate for computation - canonical form

  • A literal is either an atom (positive literal) or the negation of an atom (negative literal).

  • A clause is a disjunction of literals; the empty clause is equivalent to False.

  • A wwf is in Conjunctive Normal Form (CNF) iff it is a set of clauses (the set is abreviating the conjunction of all the clauses).


Converting arbitrary wffs to cnf l.jpg
Converting arbitrary wffs to CNF

  • Eliminate implications:

    A  B becomesA  B

  • Move  inwards:

    • Apply De Morgan’s :

      (A v B) becomes (A B)

      (A  B) becomes (A v B)

    • Apply double negation rule:

        A becomes A


Converting arbitrary wffs to cnf8 l.jpg
Converting arbitrary wffs to CNF

Distribute  over v :

(A  B) v C becomes (A v C)  (B v C)

Flatten nested conjunctions and disjunctions:

(A v B) v C becomes (A v B v C)

(A  B)  C becomes (A  B  C)

At this point we have a conjunction of clauses;

We must have a set of clauses!

separate the conjuncts


Important theorem l.jpg
Important Theorem

  • Let S be a set of wffs and S’ the set of clauses obtained by converting S to CNF.

  • In Propositional Logic S and S’ are equivalent; but in FOL they are not equivalent in general

  • But in both logics we have:

    S is unsatisfiable iff S’ is unsatisfiable.

  • Therefore, KB |= Q iff S = KB U { Q}is unsat

    iff S’ is unsat


Resolution system l.jpg
Resolution System

  • Language: Clauses

  • Logical Axioms: AX = { }

  • Inference Rules:

    R = {Resolution}

    Notice that since the language is clausal, resolution is applied only to clauses:

    P1 v ... v Pi v ... v Pn , Q1 v ... v  Pi v ... v Qm

    ---------------------------------------------------------

    P1v...vPi-1vPi+1v...vPn vQ1v...vQj1vQj+1v...vQm

    The conclusion is called the resolvent


Resolution system11 l.jpg
Resolution System

  • Soundness

    Since its only rule is resolution and there are no logical axioms, it is easy to show that the resolution system is sound:

    show the soundness of the resolution inference rule

    (show by truth-table that the premisses entail the

    conclusion)

    and then show by induction on the length of a proof

    that if S’ |- False then S’ ||= False.


Resolution system12 l.jpg
Resolution System

  • Completeness:

    Resolution is not complete

    P , R |= P V R but P , R |/- P V R

    But Resolution is Refutation Complete:

    Let S’ = CNF(KB U { Q})

    If KB |= Q then S’ |- False

    P , R,  P,  R |- False


Resolution system13 l.jpg
Resolution System

To answer if KB |= Q:

  • Convert S = KB U { Q} into S’ = CNF(S)

    convert each formula of S into clauses

  • Iteratively apply resolution to the clauses in S’ and add the results to S’ either until there are no more resolvents that can be added or until the empty clause is produced.


Refinement strategies l.jpg
Refinement Strategies

  • The procedure described above is inefficient because some resolutions need not be performed at all (are irrelevant).

  • Refinement strategies disallows certain kinds of resolutions to take place.

  • Linear resolution with initial set of support


Proof as a search task l.jpg
Proof as a search task

  • State representation:

    a set of wffs (considered to to be true)

  • Operators: inference rules

  • Start state: an initial set of wffs

    (what is initially considered to to be true)

  • Goal state: the wff to prove is in our state’s set of known wffs


ad