Automated Reasoning Systems

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Automated Reasoning Systems

• For first order Predicate Logic

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AR general context.

• Given is a knowledge base in predicate logic T

• is a set of formulae in first order logic
• formally also called a Theory

• Given is also an additional first order formula F

• Notation T F (T implies F)
• Find reasoning techniques that allow to decide on
  this for EACH F and T.
• Requirements
• correctness -- completeness --
  efficiency

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AR decidability.

• Theorem Church 36

• BUT semi-decidable !
• Completeness Theorem of Goedel 31

• SO if F follows from T, then we find a proof,
  else it is possible that the procedure doesnt
  terminate.

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Wait a second ...

• Wrong !
• Let T smart(Kelly) en F strong(Kelly)
• Although strong(Kelly) ? strong(Kelly) is always
  true, we have
• neither smart(Kelly) strong(Kelly)
• nor smart(Kelly) strong(Kelly)

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AR general outline (1).

• First we sketch the most generally used approach
  for automated reasoning in first-order logic

• The different technical components will only be
  explained in full detail in a second pass
  (outline (2)).

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AR general outline (2) .

• We study different subsets of predicate logic

• Horn clause logic
• Clausal logic
• full predicate logic

• In each case we study semi-deciding procedures.
• Each extension requires the introduction of new
  techniques.

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Backward Reasoning Resolution

• in a nutshell

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Backward resolution
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0) The TASK (example)

• Are axioms describe knowledge about some world.
  

• How to prove such theorems in general?

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1) Proof by inconsistency

• But add the negation of F to the axioms and prove
  that this extension is inconsistent.

• the 4 axioms are never true in 1 same
  interpretation.

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2) Clausal form

• Normalize the formulae to a (more simple)
  standard form.

• ? only left ? only right
• no no ?

which is inconsistent if and only if the
original set was inconsistent.
Notice ?x ?y ?z can be dropped.
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Example

• ?u r(f(u))

• ?z q(z)

• ?y p(f(y))

is already in clausal form ( P ? P ? true)

• ?x p(x) ? q(x) ? r(x)

Ps usually requires much more work!
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3) Unification

• Given 2 atomic formulae

• find their most general common instance.

• Most general unifier (mgu) x -gt f(A)
• 
  y -gt g(f(A))

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4) The resolution step

• Proposition logic

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De resolutie step (2)

• Predicate logic
• Example

mgu applied to r(z) ? q(g(x))
Clauses on which resolution is performed must
not have any variables in common.
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5) Resolution proofs

• In order to prove a set of clauses inconsistent

• apply resolution and add the result to the set
• if you obtain the clause false ? STOP !

This means inconsistency of the last set
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Example
So inconsistent !
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A deeper study
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Horn clause logic

• where A, B1, B2,,Bn are atoms.
• An atom is a formula of the form p(t1,,tm), with
  p a predicate symbol and t1,,tm terms.
• Horn clause formulae are universally quantified
  over all variables that occur in them.
• B1,,Bn are called body-atoms of the Horn clause
  A is the head of the Horn clause.
• n may be 0 in this case we say that the Horn
  clause is a fact.

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Alternative notation

• This is a special case of the conjunctive normal
  form (only disjunctions and negations), with
  only 1 positive disjunct.

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Wich kind of formulaecan we prove?

• All variables are existentially quantified !

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A very simple example

• Bosmans is a showmaster
  (1)
• Showmasters are rich
  (2)
• Rich people have big houses
  (3)
• Big houses need a lot of maintenance (4)

• Goal automatically deduce that Bosmans house
  needs a lot of maintenance.

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Representatie in Horn logica

• Bosmans is a showmaster
  (1)
• Showmasters are rich
  (2)
• Rich people have big houses
  (3)
• Big houses need a lot of maintenance
  (4)
• To prove

showmaster(Bosmans)
?p rich(p) ? showmaster(p)
?p big(house(p)) ? rich(p)
?p lot_maint(house(p)) ? big(house(p))
Lot_maint(house(Bosmans))
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AR for ground Horn clause logic

• Backward (and forward) reasoning
• proof procedures based on generalized Modus Ponens

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Restricting to groundHorn clauses
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Easy with modus ponens !
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Modus ponens in AR
A is also true in this interpretation (see truth
tables)

• Problem how to organize this into a procedure
  which is also complete (for ground Horn clauses)?

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Generalized Modus ponens

• Is obviously still correct (truth tables).

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A glimpse at a Forward Reasoning Modus Ponens
strategy
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A forward proof procedure

• Given theory T and formula F

• If all atoms from F are in Derived at
  termination, then T implies F, otherwise it
  doesnt.

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Extended example

• One possible derivation

Step 0 Derived Step 1 Derived
showm(Bos) Step 2 Derived Derived ?
belg(Bos) Step 3 Derived Derived ?
european(Bos) Step 4 Derived Derived ?
rich(Bos) Step 5 Derived Derived ?
big(house(Bos)) End Derived Derived ?
lot_maint(house(Bos))
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Remarks

• Correctness generalized modus ponens is correct

• Completeness intuition
• for a finite ground Horn clause theory, only a
  finite number of ground atoms are implied
• these are all derived after finite time

• Efficiency
• can be extremely slow !
• If T contains many Horn clauses unrelated to F,
  then the procedure derives many irrelevant (for F
  ) atoms.

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Backward reasoning

• For ground Horn Clause logic

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Inconsistency

• A theory T is inconsistent if it has NO model.

• Proof

T implies F iff Each model of T makes F
true iff Each model of T makes
F false iff T ? F has no
model iff T ? F is
inconsistent
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The example again
showm(Bos) belg(Bos) european(Bos) ?
belg(Bos) rich(Bos) ? showm(Bos) ?
european(Bos) big(house(Bos)) ?
rich(Bos) lot_maint(house(Bos)) ?
big(house(Bos)) lot_maint(house(Bos))

• is inconsistent.

• Problem this is NOT a Horn clause theory !?

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Refutation proofsthe false predicate

• We agree that false has the truth value false
  under every interpretation.
• Imagine that we defined false as false ?
  p ? p for some predicate p

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definite goals

• So what is the form of F?

(?x1 ?xm B1 ? B2 ? ? Bn) ? ?x1 ?xm
(B1 ? B2 ? ? Bn) ? ?x1 ?xm false ?
(B1 ? B2 ? ? Bn) ? ?x1 ?xm false ?
B1 ? B2 ? ? Bn
A ? B ? A ? B

• Observe F is again a Horn clause !!

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In conjunctive normal form
0 positive disjuncts !

• Implicative en disjunctive form remain consistent
• (an empty disjunction is always false)

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Back to the example

• a ground Horn clause theory !

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Modus ponensgeneralized some more

• Ordinary Modus ponens is the special case with
• n i 1 and m 0
• Correctness via truth tables

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Some backward reasoning steps in the example
false ? big(house(Bos))
false ? rich(Bos)

• and so on ...

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The backward procedurethe idea

• Apply generalized modus ponens to the body-atoms
  Bi of the goal, using the Horn clauses of T

• Then a false formula ia a consequence of T ? F

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Backwards procedure

• On top of this you need to apply backtracking
  over the selected clauses and the selected body
  atoms.
• If the algorithm stops because it has tried all
  these alternatives F was not implied!

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Back to the example
Step 0 Goal false ? lot_maint(house(Bos))
select lot_maint(house(Bos)) ?
big(house(Bos)) Step 1 Goal false ?
big(house(Bos)) select
big(house(Bos)) ? rich(Bos) Step 2 Goal false
? rich(Bos) select rich(Bos) ?
showm(Bos) ? european(Bos) Step 3 Goal false
? showm(Bos) ? european(Bos)
select showm(Bos) Step 4 Goal false ?
european(Bos) select
european(Bos) ? belg(Bos) Step 5 Goal false ?
belg(Bos) select
belg(Bos) Step 6 Goal false ?
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Another example (propositional)

• Prove p
• Observe non-determinism on both atom selection
  and on clause selection !
• we only illustrate the clause selection here

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Search tree traversed bythe backward procedure
false ? p
p ? q ? r q ? t q ? s r ? n r ? o s o n
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Backward procedure is more efficient

• The proof is now goal directed towards the
  theorem.
• no more exploration of irrelevant rules
• Different search methodes can be used to traverse
  this search tree.
• Atom-selection may influence efficiency too
• ex. by detecting a failing branch sooner
• but has no impact on whether or not we find a
  solution
• (in case there are only finitely many ground
  Horn clauses)

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Completeness

• Example

false ? p p ? p
(1) p (2)

• Possible derivations

false ? p

• Is only complete if the search tree is traversed
  using a complete search method.

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Representation-power of ground Horn clauses

• Is ? a subset of propositional logic.

• In general, more expressive logics are needed.
• Essence with variables, one formula may be
  equivalent to a very large number of
  propositional formulae.