F22H1 Logic and Proof Week 7

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F22H1 Logic and Proof Week 7

Clausal Form and Resolution
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Isnt English Great!

• We have many ways to say the same thing
• If you miss tutorials, you will do badly in the
  exam
• Anyone who doesnt attend tutorials will do
  poorly in the module
• Doing well in the exam only happens if you attend
  tutorials
• Either you attend tutorials, or you do badly in
  the course.
• Etc
• When we formalise this in predicate logic, things
  are just the same

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Isnt this a pain!

• But this is not much fun when you have to work
  out if one logical statement is the same as
  another, or even if one English statement means
  the same as another.
• Luckily there is a way to convert any predicate
  logic statement into a unique form. This means
  that if we suspect two logical statements may be
  equivalent, we can convert each into their unique
  form if they end up the same, then they are
  equivalent.
• This process has various applications it can be
  helpful in automated reasoning (trying to prove
  things). Also, the unique form is quite simple,
  and without quantifiers, so is easier for
  computer manipulation.

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Clausal Form

• The unique form is called Clausal Form (since it
  involves a combination of clauses). There are
  various versions, and we will learn a common
  version.
• A conjunction of disjunctions looks like this
• I.e. a list of things joined by ANDs, where each
  thing is either an atomic predicate (maybe
  negated) or a list of atomic predicates joined by
  Ors.
• ANY statement in predicate logic cam be
  converted to a such a conjunction of
  disjunctions. The result is in what we call
• Conjunctive Normal Form

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A Procedure for Clausal Form

• There are six stages to the conversion
• Remove
• De Morgans to move negation to atomic
  propositions
• Skolemizing (gets rid of )
• Eliminating universal quantifiers
• Distributing AND over OR
• Arrange into clauses and maybe reorder

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Lets go through the stages with
1. Remove implications (i.e. use the implication
equivalence)
2. Use De Morgans to move negation to atomic
propositions
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3. Skolemizing.
We were here In this step, we simply replace
existentially quantified variables
by constants. Generally, if we have we can
replace this with
I.e. we invent a specific individual for whom we
can say that something is true, and give it a
unique name.
In this case, choosing any name that comes into
my head, we get
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• 4. Eliminate universal quantifiers
• Since there are no longer any existential
  quantifiers, the order of the universal
  quantifiers doesnt matter. We may as well move
  them all the way to the left, like this
• Meanwhile, it is guaranteed that every variable
  here is universally quantified (we only apply
  this process to propositions). So, why not just
  get rid of them, and we will take them as
  implicit

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• Now we distribute AND over OR i.e. we mainly
  use this equivalence

We had this In this case were fine already,
since we dont have an AND within brackets. It
is already in Conjunctive normal form (CNF), we
just have to tidy up Now we have removed
useless brackets, predicates within clauses are
in alphabetical order, and the clauses
themselves are in alphabetical order
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Skolemisation II

• Actually Skolemisation is slightly more
  complicated. Consider this
• If just replace the existential quantification
  with a constant we get
• Which says M302 is everyones mother.
• What we should do when there are universally
  quantified variables in the surrounding scope is
  make the existential variable a function (called
  a Skolem function) of those universally
  quantified variables. So
• Here, f1(k) is the Skolem function, outputting
  the individual who is the mother of the input
  variable k.

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Better do another one
Step 1 Remove implication
Step 2 De Morgans
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We were here

• Next Step 3, Skolemization
• In this case there are no universally quantified
  variables in the surrounding scope I.e.
  everything here is true for a single individual
  z, not depending on x or g. So we dont need a
  function
• Next Step 4, eliminate universal quantifiers

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Step 5 use the distributivity equivalences to
get closer to CNF
We were here
Recall the distributivity equivalence on the
previous slide. If we think of
as P, as Q, and
as R, then we get
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We were here
If we now think of
as P, and think of
as the Q and R, we get

We are now in CNF !! Lets tidy up and finish by
placing it into some form of reproducible order
just a slight change to get it in a kind of
alphabetical order
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Resolution how we do proofs in CNF

Converting predicate logic statements to CNF is
very convenient, because it simplifies the
process of trying to prove things automatically.
With fewer connectives and no quantifiers to
worry about, there are fewer applicable
equivalences or inference rules that can be used.
In fact, we can do a great deal of automated
proof work by using just one inference rule. This
is called Resolution, and it is usually used in
conjunction with Reasoning by Contradiction (see
slides for week 6) The Resolution inference rule
is basically If, in CNF form, one clause
contains X and another clause contains ,
then we can infer the union of those
clauses, without the X and
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Resolution
To help understand, note that we will implicitly
often use AND-introduction and AND-elimination.
That is, if A AND B is true, then we know that A
on its own is true, and that B on its own is
true, and vice versa.
Suppose we have these two clauses 1) 2) The
resolution rule allows us to infer the following
by noting that one of them contains something
that is negated in the other. We join the
clauses, eliminating that pair. 3)
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Is it valid?
Of course it is. But it is worth thinking about
it to convince yourself

• Think of the simple case where you have
• Clearly, resolution now gives us
• 3)
• We could prove it in a longer way by
• Implication rule, from (1)
• 4)
• follows from 2 and 4
• 5)

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Proof by Contradiction using resolution
Suppose we have a complex predicate logic
statement, and we want to prove that a certain
clause is true. (e.g. the statement may set out a
state of a computer program, such as the
constraints on variable values for a heart
monitor application, and the clause may refer to
the state of the warning indicator. Or the
statement may set out various aspects of
immigration law, and the clause express be
whether a certain client is legal or not) In
automated proof, we would typically (have a
program that did) 1. convert the complex
statement(s) to CNF 2. assume the NEGATION of the
clause 3. Apply resolution several times, 4. If
and when we find a contradiction, we have proved
the clause.
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E.g. Suppose our statement is And we want to
see if is true.
We start with assuming the negation of what we
want to prove.

• Lets also write down what we already know to be
  true
• 2)
• Resolution twice on the third clause of 2, using
  each part of 1, gets
• Resolution using 3 and the second clause of 2
  gives
• Resolution using 4 and the first clause of 2
  gives
• -- this is the negation of
  1), so we have a contradiction!
• (we could infer F from 1 and 6, and so have
  proved the negation
• of our assumption)

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NOTE
In reality, Resolution is sound, but not
complete we wont necessarily always be
able to prove what we want to prove. At any
particular point, there will typically be many,
many ways that the rule could be used with the
clauses that are known or derived so far.
There is no obvious way to decide which is the
best move to make -- thats a research issue,
beyond our scope for the moment.