Automated Reasoning

1
Declarative Progarmming Languages
COMP3006/3906
Lecture 9-10 Tatjana Zrimec
2
Ground vs non-ground term

• An argument to a functor may be left unspecified
  - it may be a variable.
• A term refers to both terms that contain
  variables and terms that do not.
• If a term contains no variables, then it is said
  to be ground
• otherwise we say the term is non-ground.
• Given a relation r of arity n requiring n
  arguments,
• an atom is an expression r(t1,..., tm) where each
  ti is a term.
• an atom is ground if
• each of its argument ti is ground
• otherwise it is non-ground

3
Substitution

• A substitution is a function that maps variables
  into terms which may or may not be ground.
• Suppose q is the substitution that maps the
  variable X to 3, Y to house-of(W) and
• Z to between(3, W)
• Then one way to write q is
• q(X) 3
• q(Y) house-of(W)
• q(Z) between(3, W)

4
Substitution

• Given an expression E and a substitution q, q(E)
  denotes the expression obtained by simultaneously
  applying q to each variable occurring in E.
• The expression q(E) is called an instance of E.
• If q(E) is ground variable free, then q is said
  to be a grounding substitution for E and q(E) is
  said to be a ground instance of E.

5
Substitution

• As an example
• Suppose E is the variable X and q is as defined
  above.
• Then q(E) is simply the term 3.
• On the other hand, suppose
• E is an atom r(X, Y).
• Then q(r(X, Y)) is the non-ground atom
• r(3, house-of(W)).
• The substitution q is an example of a grounding
  substitution for E in the first case, but not in
  the latter case.

6
Unification

• A binding will be a pair V e.
• (variable V is bound to the expression, e.
• e is a term.
• X f(X) is not a legal binding.)
• A set of bindings referred to as a substitution.
• V1 e1 , V2 e2 ,V3 e3 , ... Vn en
• q(e) is equal to eq.
• To expressions, e1 and e2, unify (or are
  unifiable) if exists a set of bindings, such
  that
• e1q e2q q is called a unifier
  of e1 and e2

7
Most General Unifier

• We are interested in the most general unifier of
  two expressions
• p(X, f(Y), b) and p(X, f(b), b)
• the set of bindings X c, Y b is the
  unifier.
• Y b is also a unifier
• Y b is a more general unifier than X
  c, Y b

8
Most General Unifier

• It can be shown that for any two unifiable terms,
  there exists a substitution that makes the fewest
  possible substitutions for variable of two terms.
• The unifying substitution is called the Most
  General Unifier , or MGU.

Another way to denote a substitution
9
Resolution

• Resolution, as defined in Languages and Logic, is
  a deduction method which uses only one rule, the
  resolution rule.
• Resolution is an inference rule that derives one
  clause from two clauses.
• A poof theory for Clausal Form Logic - CFL is
  based on the resolution.
• In CFL,
• a literal is an atom or the negation of an atom
• A clause is therefor a disjunction of literals
• A ground clause/literal/atom/term/wff is one
  without variables

10
Resolution

• Example ( without variables)
• (1) and (2) are parents
• p1 and p1 are the selected literals
• p1 and p1 are a complementary pair
• (3) is a resolvent

From resolution of 1 and 2
Ø
Ø
11
Resolution

• Example
• The derivation of the empty clause o means that
  the original clauses are contradictory.

From resolution of 1 and 2
o
12
Resolution with variables

• Clause can be resolved if there is some pair in
  the parents that is complementary when some MGU
  is applied to them.
• The MGU is also applied to the resolvent.
• Example

From resolution of 1 and 2 with MGU X a (X
/ a)
13
Resolution

• Exercise 1
• Example

Resolution from 1 and 2 with MGU X U, Y V
o
14
Resolution

• Unification in resolution is a sophisticated
  form of pattern matching operation (used in
  Prolog).
• When matching literals, we look for variable
  substitutions that will make the two expressions
  identical. Eg.
• runs_faster_than(X, zeno)
• runs_faster_than(tortoise, Y)
• are identical under the substitution X/tortoise,
  Y/zeno

15
Resolving Clauses

• To resolve two non-ground clauses, you must find
  a unifier for complimentary literals. Eg.
• beats_in_race(X, zeno), younger_than(X,
  zeno)
• and
• beats_in_race(tortoise, Y),
  philosopher(Y)
• have unifier n X/tortoise, Y/zeno and
  generate the resolvent
• philosopher(zeno), younger_than(tortoise
  , zeno)

16
Proofs

• We can prove a formula, p, if we can derive it
  from a theory,T, by a sequence of resolution
  steps.
• Written as T O p.
• If the theory is very large, there may be many
  ways of deriving a proof.
• How can we find a short derivation?
• We try a proof by refutation, ie. add negation of
  goal to theory and show that the new theory is
  inconsistent, ie. implies false.
• The empty clause, , is interpreted as false.
  So if theory derives false, we have an
  inconsistent theory.

o
17
Resolution

• Resolution is very good at demonstrating
  inconsistency (unsatisfiability).
• If a set of clauses is unsatisfiable, then it is
  always possible by resolution alone to derive a
  contradiction from the clauses in the set.
• In clausal from a contradiction takes the from of
  the empty clause, .

o
18
Resolution refutation

• To prove p follows from some theory, T, assume
  p and then try to derive a contradiction from
  its conjunction with T.
• This way of doing proof by contradiction is
  known as refutation theorem proving.
• Resolution refutation theorem proving is
  refutation theorem proving where the inference
  rule is resolution.

19
Resolution refutation tree

• Resolution that lead to the derivation of
  can be shown as a refutation tree.
• Show that the following clauses are inconsistent

o
o
20
Horn Clauses

• A Horn clause is one which only has a single
  positive literal, eg.

p

q
,
K
,
q

1
1
n

• The programming language, Prolog, consists of
  Horn clause definitions, eg.

• on(a, b).
• on(b, c).
• above(X, Y) - on(X, Y).
• above(X, Y) - on(Z, Y), above(X, Z).

Let ask Prolog
?- above(a, c).
- above(a, c).
21
A Prolog Proof Tree
- above(a, c).
above(X, Y) - on(Z, Y), above(X, Z).
- on(Z, c), above(a, Z).
on(b, c).
- above(a, b).
above(X, Y) - on(X, Y).
- on(a, b).
on(a, b).
o
22
Resolution Search

• Resolution uses backward chaining to focus search
  for clauses to resolve.
• There are many refinements to this search.
• We will stick to the Prolog method which resolves
  clauses and their literals in input order, ie,
  top-to-bottom, left-to-right.

23
Soundness and Completeness

• A proof procedure is sound if every formula it
  derives is true. Ie. it cannot prove something it
  shouldn't.
• A proof procedure is complete if it can derive
  every thing that is possible to derive from a
  theory. Ie. There is no true statement that it
  cannot prove.
• Decidability means that we can always show if a
  proposition follows from a theory.
• Prolog's proof procedure is sound and complete
  for Horn clauses.
• Unrestricted first-order logic is undecidable.