Download Logic Logical Concepts Deduction Concepts Resolution

Logical Concepts
Logic
If a formula ϕ is true under a given interpretation M ,
one says that that M satisfies ϕ, M |= ϕ
Representation and Reasoning
A formula is satisfiable if there is some interpretation
under which it is true
Otherwise, it is unsatisfiable (inconsistent)
A formula is valid (a tautology), denoted by |= ϕ, if it is
true in every interpretation
for all M : M |= ϕ
A formula ϕ is entailed (or is a logical consequence) by
a set of formulae Γ, denoted by Γ |= ϕ, if
for all M : M |= Γ then M |= ϕ
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Deduction Concepts
Resolution
A deductive system S is a set of axioms and rules of
inference for deriving theorems
A formula ϕ can be deduced by a set of formulae Γ if ϕ
can be proven using a deduction system S , written as
Γ ⊢S ϕ
A deductive system S is sound if
Automated reasoning: proving logic theorems by means
of a computer program
Γ ⊢? ϕ
Proving unsatisfiability is enough, because:
Γ |= ϕ ⇔ Γ ∪ {¬ϕ} |= ⊥
Γ ⊢S ϕ ⇒ Γ |= ϕ
Resolution (J.A. Robinson, 1965):
A deductive system S is complete if
C ∨ {L}, C ′ ∨ {¬L}
D
Γ |= ϕ ⇒ Γ ⊢S ϕ
A deductive system S is refutation-complete if
where D is equal to C ∨ C ′ without the double literals
Resolution is a refutation-complete inference rule
Γ |= ⊥ ⇒ Γ ⊢S ⊥
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Logic Programming
Motivation for Resolution
In first-order logic:
Logic programming: based on logic for programming
If S is inconsistent, then resolution will terminate within
finite time:
S ⊢R 2
Provides a computational interpretation of resolution
Fundamentally, we cannot do much better than that
Suppose P = {r ← p. q ← p. s ← q, r. p.}
Basically: it searches for a resolution proof (recall
tracing)
Call: (7) s ? creep
Call: (8) q ? creep
Call: (9) p ? creep
Exit: (9) p ? creep
Exit: (8) q ? creep
Call: (8) r ? creep
Call: (9) p ? creep
Exercise: Proof s from P using resolution. Compare your
answer.
Many applications:
Mathematics: Robbins’ conjecture
Proving that medical procedures are correct
Programming!
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Logic Programming Concepts
SLD Resolution
Horn clause: clause with maximally one positive literal
¬A1 ∨ · · · ∨ ¬Am ∨ B , also denoted by B ← A1 , . . . , Am
Fact: expressing relations between objects (literals)
p ← (= p)
SLD resolution (for Horn clauses):
Rule: defines relations in terms of other relations
(clauses)
p ← q, ¬r, s
← B1 , . . . , Bn ,
Bi ← A1 , . . . , Am
← B1 , . . . , Bi−1 , A1 , . . . , Am , Bi+1 , . . . , Bn
Query/Goal: negated fact
SLD derivation: a sequence G0 , G1 , . . . and C1 , C2 , . . .
← p (= ¬p)
Exercise:
Negations have a special meaning: logic not part of the
course
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S = {R ← T, T, P ← R}
Derive P from S using SLD resolution.
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First-order logic
First-order logic - syntax
Allow the representation of entities and their properties,
and relations among such entities
Constant symbol - names exaclty one entity: a, b, . . .
More expressive than propositional logic
Functions - any given object is related to exactly one
other object by the relation: Cos, Father-of, Plus, . . .
Distinguished from propositional logic by its use of
quantifiers
Each interpretation of first-order logic includes a
domain of discourse over which the quantifiers range
Variable - refers to generic entities: x, y, z . . .
Predicates - resemble a function that returns either true
or false: Brother-of, Bigger-than, Has-color, . . .
Additionally, it covers predicates
Used to represent either a property or a relation
between entities
Quantifiers - allow the representation of properties that
hold for a collection fo objects. Consider a variable x,
Existential: ∃x, "there is an x"
Universal: ∀x, "for all x"
Sufficient for building knowledge-based systems
Logical connectives and auxiliary symbols
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First-order logic - semantics
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First-order logic - semantics
A domain of discourse D is defined
An interpretation assigns a denotation to elements in
the language
Each term is assigned an object that it represents,
and each sentence is assigned a truth value
The domain of discourse D is a nonempty set of entities
(of some kind)
For instance, one can take D to be the set of integer
numbers
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The interpretation of a function symbol is a function
If D consists of integers, a function symbol f /2 can
be interpreted as "+"
The interpretation of a constant symbol can be simply
identified with an object in D
An interpretation may assign the value I(c) = 5
Given an interpretation, a predicate symbol, and n
elements of D, one can tell whether the predicate is
true according to the given interpretation
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Semantics - example
Resolution in first-order logic
Let D be the domain of natural numbers. Consider the
formula ∀x∃yP (x, y)
In order to evaluate if this formula is true or false, we
need to give the predicate symbol P an interpretation
Suppose we interpret P as the < relation, i.e., P (x, y)
means "x is less than y" Under this interpretation, the
formula above is clearly true
Note, however, that this would not be the case if we
had, for instance, interpreted P as "greater than"
Intuitively, this is how resolution works in first-order
logic,
All Humans are Mortals
Socrates is Human
Therefore, Socrates is Mortal
More generally,
∀xH(x) → M (x)
H(Socrates)
Therefore, M (Socrates)
To recast the reasoning using the resolution technique,
first the clauses must be converted to conjunctive
normal form
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Clausal form
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Skolemisation
A clause is a formula of the form
When transforming a set of formulae into a set of
clauses, existentially quantified variables are replaced
by Skolem functions
∀x1 . . . ∀xs (L1 ∨ . . . ∨ Lm )
where each Li is a literal, Li 6= Lj , and x’s are variables
occurring in those literals.
In this form, all quantification becomes implicit
Existentially quantified variables are replaced by the
so-called Skolem functions
The simplest form of Skolemisation is for existentially
quantified variables which are not inside the scope of a
universal quantifier. These can simply be replaced by
creating new constants
E.g., ∃xP (x) can be changed to P (c), where c is a new
constant
After replacement of all existentially quantified
variables, any remaining variables are (implicitly)
universally quantified
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Skolemisation (cont.)
Skolemisation (cont.)
More generally, Skolemisation is performed by
replacing every existentially quantified variable y with a
term f (x1 , . . . , xn ) whose function symbol f is new
What you have is that,
∀x g(x) ∨ ∃yR(x, y) ⇐⇒ ∀x g(x) ∨ R(x, f (x))
x1 , . . . , xn are the variables that are universally
quantified and whose quantifiers precede that of y
where f (x) is a function that maps x to y .
The function f introduced in this process is called a
Skolem function
"For every x there is a y s.t. R(x, y)" is converted into
"There is a function f mapping every x into a y s.t. for
every x R(x, f (x)) holds"
∀x1 . . . ∀xn ∃yR(x1 , . . . , xn , y) is satisfied by a model M iff
For each possible value for x1 , . . . , xn there is a value
for y that makes R(x1 , . . . , xn , y) true
By the axiom of choice, there exists a function f s.t.
y = f (x1 , . . . , xn )
The formula obtained by this transformation is
satisfiable iff the original formula is
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Resolution method in first-order logic
Unification
Unification algorithm computes, if possible, the
substitution which is needed in order to make the given
expressions syntatically identical
A substitution is a finite set
{t1 /x1 , . . . , tn /xn }
where each xi is a variable, being xi 6= xj , and each ti is
a term not equal to xi
Remember: a term is a constant, a variable, or
f (t1 , . . . , tn ) (where f is a n-arity function symbol and
t1 , . . . , tn are terms)
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In summary, this is what occurs,
Find two clauses containing the same predicate, where
such predicate is negated in one clause but not in the
other
Perform a unification on the two complementary
predicates
If the unification fails, you might have made a bad
choice of predicates
Go back to the previous step and try again
Discard the unified predicates, and combine the
remaining ones from the two clauses into a new clause,
also joined by the or-operator
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