Proof Systems

1
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
2
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).

3
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

4
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).

5
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)

6
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).

7
Converting arbitrary wffs to CNF

• Eliminate implications
• A ? B becomes ?A ? B
• Move ? inwards
• Apply De Morgans
• ?(A v B) becomes (?A ? ?B)
• ?(A ? B) becomes (?A v ?B)
• Apply double negation rule
• ? ? A becomes A

8
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

9
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 ? Qis unsat
• iff S is unsat

10
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-1vPi1v...vPn vQ1v...vQj1vQj1v...vQm
• The conclusion is called the resolvent

11
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.

12
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

13
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.

14
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

15
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 states
  set of known wffs