2. SYLLABUS
The resolution principles
Binary resolution
Unit resolution
Linear resolution
Set of support strategy
Non-deductive inference methods : induction , deduction and analogy.
3. The resolution principles
Resolution yields a complete inference algorithm when coupled with any
complete search algorithm.
Resolution makes use of the inference rules. Resolution performs deductive
inference.
Resolution uses proof by contradiction. One can perform Resolution from a
Knowledge Base.
A Knowledge Base is a collection of facts or one can even call it a database
with all facts.
Resolution basically works by using the principle of proof by contradiction. To
find the conclusion we should negate the conclusion. Then the resolution rule
is applied to the resulting clauses.
4. CONT…..
Each clause that contains complementary literals is resolved to produce a two
new clause , which can be added to the set of facts (if it is not already
present).This process continues until one of the two things happen:
There are no new clauses that can be added
An application of the resolution rule derives the empty clause
An empty clause shows that the negation of the conclusion is a complete
contradiction ,hence the negation of the conclusion is invalid or false or the
assertion is completely valid or true.
5. Binary resolution
Example:
Consider the following axioms:
All hounds howl at night.
Anyone who has any cats will not have any mice.
Light sleepers do not have anything which howls at night.
John has either a cat or a hound.
(Conclusion) If John is a light sleeper, then John does not have any mice
6. CONT….
The conclusion can be proved using Resolution as shown below. The first step
is to write each axiom as a well-formed formula in first-order predicate
calculus. The clauses written for the above axioms are shown below, using
LS(x) for `light sleeper'.
∀ x (HOUND(x) → HOWL(x))
∀ x ∀ y (HAVE (x,y) ∧ CAT (y) → ¬ ∃ z (HAVE(x,z) ∧ MOUSE (z)))
∀ x (LS(x) → ¬ ∃ y (HAVE (x,y) ∧ HOWL(y)))
∃ x (HAVE (John,x) ∧ (CAT(x) ∨ HOUND(x)))
LS(John) → ¬ ∃ z (HAVE(John,z) ∧ MOUSE(z))
7. Unit resolution
Unit Resolution Strategy.
• A unit resolvent – resolvent in which at least one of. the parent clauses is
a unit clause i.e. is a clause containing a single literal.
Resolution is an inference rule (with many variants) that takes two or
more parent clauses and soundly infers new clauses.
A special case of resolution is when the parent causes are contradictory, and
an empty clause is inferred.
Resolution is a general form of modus ponens.
8. Linear resolution
Linear resolution with selection function (SL-resolution) is a restricted form
of linear resolution.
The main restriction is effected by a selection function which chooses from
each clause a single literal to be resolved upon in that clause.
9. Set of support strategy
Resolution Strategies: Set of support: The set of support partitions all
clauses into two sets, the set of support and auxiliary set.
We initialize the auxiliary set to be the set S (our starting set of clauses).
Used for large clause spaces.
One of the resolvents in each resolution have an ancestor in the set of
support.
10. Non-deductive inference methods : induction
, deduction and analogy.
Induction produces generalizations from special cases. Example: from "Robins are
birds" and "Robins have feather" to derive "Birds have feather".
Abduction produces explanations for given cases. ...
Analogy produces similarity-based judgments.
An analogy is a comparison between two objects, or systems of objects, that
highlights respects in which they are thought to be similar.
Analogical reasoning is any type of thinking that relies upon an analogy.
An analogical argument is an explicit representation of a form of analogical
reasoning that cites accepted similarities between two systems to support the
conclusion that some further similarity exists.
In general (but not always), such arguments belong in the category of ampliative
reasoning, since their conclusions do not follow with certainty but are only
supported with varying degrees of strength.
11. CONT…..
Analogical reasoning is fundamental to human thought and, arguably, to some
nonhuman animals as well.
Historically, analogical reasoning has played an important, but sometimes
mysterious, role in a wide range of problem-solving contexts.
The explicit use of analogical arguments, since antiquity, has been a
distinctive feature of scientific, philosophical and legal reasoning.
