Predicate Calculus

Models and Formalisms
S. Garlatti
MR2A Informatique
2012

Progress
1 Introduction
2 Formal Systems
3 Propositional Logic
4 The Predicate Calculus or First Order Logic
2 / 109 SG Models and Formalisms

Introduction
Concatenation Modeling?
Answer from a logical point of view:
Conc([X|L], L1, [X|L2]) :− Conc(L, L1, L2).
Conc([], L, L).
Conc([a,b], [c,d], L)?

Introduction
History of Logic
Plato’s logic, Aristotle’s logic, Stoic logic
Medieval logic, etc.
Rise of Modern Logic, the mid-nineteenth century:
Rigorous and formalistic discipline whose exemplar was the exact
method of proof used in mathematics
The modern so-called "symbolic" or "mathematical" logic
Modern logic: a calculus whose rules of operation are determined
only by the shape and not by the meaning of the symbols it
employs, as in mathematics.

Progress
1 Introduction
2 Formal Systems

Formal Systems
Formal Systems (or Formalism) consist of :
An Axiomatic Theory

Formal Systems
An Axiomatic Theory
A ﬁnite set of symbols for constructing formulas.

Formal Systems
An Axiomatic Theory
A grammar to deﬁne well-formed formulas (abbreviated wﬀ ).

Formal Systems
An Axiomatic Theory
A set of Axioms or axiom schemata: each axiom must be a wﬀ .

Formal Systems
An Axiomatic Theory
A set of Inference rules

Formal Systems
An Axiomatic Theory
From Axioms and Inferences Rules, Theorems are deduced.

Formal Systems
An Axiomatic Theory
From Axioms and Inferences Rules, Theorems are deduced.
Properties: Consistency and Decidability

Formal Systems
The Formal Systems MIU, Axiomatic Theory
Alphabet A = {M, I, U}.
A grammar : wff consist of all series of the three alphabet
elements.
A set of axioms= {MI}.
A set of inference rules: Let f , g variables representing wff , g
could be empty
R1: f fU
R2: Mf Mff
R3: fIIIg fUg
R4: fUUg fg

Formal Systems
The Formal Systems MIU, Axiomatic Theory
Some Theorems:
R2: MI MII
R1: MII MIIU
R1: MIIU MIIUU
R4: MIIUU MII
R2: MII MIIII
R3: MIIII MIU
MU is a theorem?

Formal Systems
Formal Systems consist of : A Model Theory or Semantics
A standard model M:
It is a structure that gives a concrete interpretation of the theory
(axiomatic theory).
For instance, M = (D, I)
D is the interpretation domain.
An Interpretation function I that assigns:
- Let A a set of alphabet elements, (A)I
: A → D
- Let F a set of wﬀ , (F)I
: F → {True, False}
Properties: consistency, soundness, completeness

Formal Systems
The Axiomatic Theory of Formal Systems pg
Alphabet A = {p, g, −}
wﬀ all series of p, g, −
x ∈ {−, −−, ..., −n
}, Axiom Schema:
xp − gx−
x, y, z ∈ {−, −−, ..., −n
},Inference Rules :
xpygz xpy − gz−

Formal Systems
The Model Theory of the formal system pg
An Interpretation function I that assigns:
(p)I : plus, (g)I : equal, (−n)I : n
(− − −p − −g − − − −−) ⇒ (3 plus 2 equal 5) ⇒
(− − −p − −g − − − −)I = True
(− − −p − g − − − −−) ⇒ (3 plus 1 equal 5) ⇒
(− − −p − g − − − −−)I = False

Formal Systems
Axiomatic and model theory and their properties:
Model Theory
Satisﬁability
Truth
Validity
Axiomatic Theory
Axioms
Inference rules
Deduction
Theorems
Completeness
Soundness

Progress
1 Introduction
2 Formal Systems

The Propositional Logic
The Language and the well-formed formulae or wff
A set of propositional symbols S
A set of connectors: ¬, →, ↔, ∨, ∧
Let p, q, r ∈ S, p, q, r are atoms
An atom is a wff , then p, q, r are wff
Let P be a wff , then ¬P is a wff
Let P, Q be wff , then (P → Q), (P ↔ Q), (P ∨ Q), (P ∧ Q)
All wff are generated by applying the above rules

The Model Theory of Propositional Logic
In Propositional logic, I is an Interpretation function that assigns:
A truth value in {T, F}, to an atom or a well-formed formula,
Let P, Q be wﬀ , assignment of Truth Values are made as follows:
P Q ¬P (P ∧ Q) (P ∨ Q) (P → Q)
T T F T T T
T F F F T F
F T T F T T
F F T F F T
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or falsity
of its components.

Truth, Validity
A wff P is said to be True under an interpretation I iff P is
evaluated to T in the interpretation;
Otherwise, P is said to be False under the interpretation.
A wff is said to be Valid iff it is true under all its interpretations
A wff is said to be Inconsistent or Unsatisfiable iff it is false
under all its interpretations
A wff is said to be Consistent or Satisfiable iff it is not
Inconsistent

Truth, Validity
Examples of Valid or inconsistent formulae
(P ∧ ¬P)
(P → P)
((P ∧ ¬P) → Q)
(P ∨ ¬P)

Outline
1 Introduction
2 Formal Systems
The Axiomatic Theory
Propositional Logic Properties
Resolution Principles
The Model Theory or Semantics
Predicate Calculus Properties
Resolution Strategy
Prolog Modeling

Axiomatic Theory
An axiomatic theory is composed of:
A language and wﬀ
A set of axioms
Inference rules

Axioms
Let P, Q, R be wﬀ
A1 : (P → (Q → P))
A2 : (P → (Q → R)) → ((P → Q) → (P → R)))
A3 : (¬P → ¬Q) → ((¬P → Q) → P))
One inference rule
Modus Ponens: P, (P → Q) Q
From hypothesis P and (P → Q), Q is deduced

Deduction
Theorem demonstration: (A → A)
A1 : (A → ((A → A) → A))
A2 : ((A → ((A → A) → A)) →
((A → (A → A) → (A → A))))
A1, A2 ((A → (A → A)) → (A → A))
A1” : (A → (A → A))
A1”, ((A → (A → A)) → (A → A)) (A → A)
(A → A)

Deduction
Theorem demonstration: (A → A)
A1 : (A → ((A → A) → A))
A2 : ((A → ((A → A) → A)) → ((A → (A → A) → (A → A))))
((A → (A → A)) → (A → A))
A1” : (A → (A → A)) ((A → (A → A)) → (A → A))
(A → A)

Theorems
Some theorems
The law of noncontradiction: ¬(P ∧ ¬P)
Or ((P ∧ ¬P) → Q)
The law excluded middle: (P ∨ ¬P)

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Completeness Theorem
Every Valid formula of the predicate calculus is a theorem.
- if |= F then F
Soundness Theorem
Every theorem of the predicate calculus is a Valid formula
- if F then |= F
The propositional Logic is decidable

Model Theory
Satisﬁability
Truth
Validity
Axiomatic Theory
Axioms
Inference rules
Deduction
Theorems
≡ |=
Completeness
Soundness

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Normal Forms
Deﬁnition: A Literal is an atomic formula or Atom or the
negation of an Atom
A Formula F is called a Clause if and only if F is composed of a
disjunction of literals, (F1 ∨ F2, ... ∨ Fn), n 1
An empty clause, denoted , is unsatisﬁable or always
interpreted as False, (deduced from the conjunction of two
complementary literals P and ¬P)

Normal Forms
Deﬁnition: A formula F is in a Conjunctive Normal Form if
and only if F has the following form (F1 ∧ F2, ... ∧ Fn), n 1,
where each of F1, F2, ..., Fn is a disjunction of literals or a clause.
Any formula can be transformed into a normal forms

One possible resolution rule: the Cut Elimination Rule of
Gentzen
Let D and E be clauses without any occurrence of literals P or
¬P
(¬P ∨ E), (P ∨ D) (E ∨ D)
The clause (E ∨ D) is called the resolvent
This rule is a generalization of the Modus Ponens Rule (obtained
by elimination of the clause D)

Let F be a set of clauses and A be a clause
To demonstrate F A
By means of the Cut Elimination Rule, the goal is to show that
{F, ¬A} is unsatisfiable.
It is a refutation
Demonstration of the unsatisfiability of {F, ¬A}
Obtain a resolvent equal to the empty clause , by elimination
of complementary literals in two different clauses.

Example : let F be a conjunctive normal form as follows:
F = ((P ∨ Q) ∧ (¬P ∨ Q) ∧ (P ∨ ¬Q) ∧ (¬P ∨ ¬Q))
F unsatisﬁable?
(P ∨ Q) (¬P ∨ Q)
Q
(P ∨ ¬Q) (¬P ∨ ¬Q)
¬Q

Progress
1 Introduction
2 Formal Systems

The Predicate Calculus
Predicate calculus (ﬁrst-order logic): an extension of the
propositional logic
The Language
A set of Constants C
A set of Variables V
A set of function symbols F
A set of Terms T
A set of predicate symbols P
A set of connectors: ¬, →, ↔, ∨, ∧
A set of Quantiﬁers: ∀, ∃

The Language
The Terms
A Constant a is a term
A Variable x is a term
if t1, t2, ..., tn ∈ T, (n > 0), then f (t1, t2, .., tn) ∈ T
where f is a function symbol of arity n.
Well-Formed Formula or wff
If P is a predicate symbol and if t1, t2, . . . , tn ∈ T(n > 0) then
P(t1, t2, ..., tn) is an atomic formula
If G and H are wff then
(G → H), (G ↔ H), (G ∨ H), (G ∧ H), ¬G, ∀xG(x) and ∃xG(x)
are wff .

The Language
Quantification theory
Scope of quantifiers
Free and bound variables
Examples of wff :
∀x∃y(P(x, a, f (y), z) → ∀yQ(y, g(x, a)))
(P(x, f (z)) ∨ ∀xQ(x, z))
∀z(∃yP(y, b) ∧ Q(y, f (a, z))

The Language
Quantification theory
∀x∃y(P(x, a, f (y), z) → ∀yQ(y, g(x, a)))
x, y are bound variables, z is a free variable, the second variable y
is bound to the quantification forally, a is a constant, f and g are
functional symbols.
(P(x, f (z)) ∨ ∀xQ(x, z))
the first occurrence of x is free, the second is bound, the two
occurrences of z are free, but they do not represent the same
variable.
∀z(∃yP(y, b) ∧ Q(y, f (a, z))
The second occurrence of y is free, the first one bound, z is bound.

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Standard Model
A standard model is a structure M = (D, I) Where
D is a nonempty set called the Interpretation Domain
I is an Interpretation function that assigns:
A truth value in {T, F}, to a well-formed formula,
A n-place relation to a nary predicate symbol,
An n-place operation to a nary function symbol
An element of D to a constant
An element of D to variable
An element of D to a nary functional term
The interpretation function I is also Truth-Functional

Satisfiability, Truth, Models, Validity
Let s be a denumerable sequence of D elements, s = (s1, s2, ..., sn)
The idea is: from a sequence s, the corresponding n-tuple
< s1, s2, ..., sn > will satisfy a wff F with free variables
(x1, x2, ..., xn)
For instance, the sequence s = (s1, s2, ..., sn) of elements of the
Domain D will satisfy the wff F(x2, x5)
if the ordered pair < s2, s5 > is in the relation (F)I
assigned to
the predicate symbol F by the interpretation I.

An Interpretation I:
a ∈ C, (a)I
: C → D
x ∈ V , (x)I
: V → D
t1, t2, . . . , tn and f (t1, t2, .., tn) are terms, f ∈ F,
(f (t1, t2, .., tn))I : Dn → D
with (f (t1, t2, .., tn))I ≡ ((f )I((t1)I, (t2)I, .., (tn)I))
if ti is the variable xi , (t)I is equal to the sequence element si

Satisfiability
If F(t1, t2, .., tn) is an atomic formula and (F)I
is the
corresponding n-place relation then the sequence
s = (s1, s2, ..., sn) satisfies F(t1, t2, .., tn) if and only if
(F)I
((t1)I
, (t2)I
, .., (tn)I
)), that is to say the n-tuple
< s1, s2, ..., sn > is in (or verify) the relation (F)I
.
s satisfies ¬F if and only if s does not satisfies F
s satisfies (F → G) if and only if s does not satisfy F or s
satisfies G
s satisfies ∀xi F if and only if every sequence s that differs from s
in at most the ith
component satisfies F

Let F be the set of wff , S be the set of denumerable sequences
A wff F is true for the interpretation I (written |=I F) if and
only if every sequence in S satisfies F.
F is said to be false for the interpretation I if and only if no
sequence in S satisfies F.
An interpretation I is said to be a model for a set F of wff if
and only if every wff is true for I.
A wff F is valid (written |= F) if and only if F is true for all
interpretation I, over all domains.

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Axiomatic Theory
An axiomatic theory is composed of:
A language and wﬀ
A set of axioms
Inference rules

Axioms
Let P, Q, R be wﬀ and x be a variable
A1 : (P → (Q → P))
A2 : (P → (Q → R)) → ((P → Q) → (P → R)))
A3 : (¬P → ¬Q) → ((¬P → Q) → P))
A4 : ∀xP(x) → P(x)
A5 : ∀x(P → Q) → (P → ∀xQ)
where P contains no free occurrence of x.

Inference Rules
Two rules
Modus Ponens
P, (P → Q) Q
- from hypothesis P and (P → Q), Q is deduced
Generalization Rule
P(x) ∀P(x)
- from hypothesis P(x), ∀P(x) is deduced

Theorems
Some theorems
The law of noncontradiction: ¬(P ∧ ¬P)
Or ((P ∧ ¬P) → Q)
The law excluded middle: (P ∨ ¬P)
(¬∀xP(x) → ∃x¬P(x))
(¬∃xP(x) → ∀x¬P(x))

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Completeness Theorem
Every Valid formula of the predicate calculus is a theorem.
- if |= F then F
Soundness Theorem
Every theorem of the predicate calculus is a Valid formula
- if F then |= F

Decidability : the Predicate Calculus is undecidable
There is no decision procedure to determine whether arbitrary
formulas are theorems (Church, Turing, Post, Markov).
Nevertheless, if a formula F is valid, there is a constructive proof
that F is valid.
- From the soundness theorem, we can conclude that F is a theorem.
- The predicate calculus is semidecidable.

Model Theory
Satisﬁability
Truth
Validity
Axiomatic Theory
Axioms
Inference rules
Deduction
Theorems
≡ |=
Completeness
Soundness

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

For predicate calculus, the resolution is as follows:
Conjunctive normal forms are used
It is a refutation demonstrating that a set of clauses is
unsatisfiable
Conjunctive normal forms : 3 stages are necessary to deal with
quantifiers and variables
1. Prenex normal forms
A formula F is in Prenex Normal Form if and only if the formula
is in the form of (Q1x1)(Q2x2), ..., (Qnxn)(M) where
- (Qi xi ), i = 1, ..., n, is either (∀xi ) or (∃xi )
- M is a formula containing no quantifiers

2. Skolem Standard Forms
A formula F is in Skolem Normal Form iff the formula is in the
form of (∀x1)(∀x2), ..., (∀xn)(M) where M is a formula containing
no quantifier
In a Prenex Standard Form, for an existential quantifier (∃x)
having n universal quantifiers on variables (x1, x2, ..., xn) appearing
before, all occurrences of x are replaced by a n-place function
f (x1, x2, ..., xn) and (∃x) is eliminated.
If n = 0, all occurrences of x are replaced by a constant.
3. Conjonctive Normal Form: a Skolem Standard Form is rewritten
to obtain a Conjunctive Normal Form

The Herbrand Universe of a set of clauses
A set F of clauses is unsatisﬁable if and only if it is false under
all interpretations I over all domains
Impossible to consider all interpretations I over all domains
There is a special domain H, called the Herbrand Universe of F,
"replacing all domains"

H, the Herbrand Universe of a set of clauses
1. H0 is the set of constants appearing in F. If no constant is
appearing in F, H0 is to consist to a single constant, H0 = {a}
2. Hi+1 be the union of Hi and the set of all terms of the form
f n
(t1, t2, ..., tn) for all n-place functions f n
occurring in F, where
t1, t2, ..., tn ∈ Hi
3. H∞ is the Herbrand Universe of F
4.

H, the Herbrand Universe of a set of clauses
F = {P(a), ¬P(x) ∨ P(f (x))}
1. H0 = {a}
2. H1 = {a, f (a)}
3. H2 = {a, f (a), f (f (a))}
4. H3 = {a, f (a), f (f (a)), f (f (f (a)))}
5. ..
6. ...
7. H∞ = {a, f (a), f (f (a)), f (f (f (a))), .....}

Herbrand’s Theorem
Definition: A ground instance of a clause C of a set of clauses F
is a clause obtained by replacing variables in C by members of the
Herbrand Universe of F.
Herbrand’s Theorem: A set F of clauses is unsatisfiable if and only
if there is a finite unsatisfiable set F of ground instances of clauses
of F.

Deﬁnition: A substitution is a ﬁnite set of the form
{t1/x1, ..., tn/xn} where every ti is a term and every xi is a variable

Deﬁnition: Let θ = {t1/x1, ..., tn/xn} be a substitution et E be an
expression, Eθ is an expression obtained from E by replacing
simultaneously each occurrence of the variable xi (1 ≤ i ≤ n) by
the term ti in E. Eθ is called an instance of E.

Deﬁnition: Let θ = {t1/x1, ..., tn/xn} be a substitution et E be an
expression, Eθ is an expression obtained from E by replacing
simultaneously each occurrence of the variable xi (1 ≤ i ≤ n) by
the term ti in E. Eθ is called an instance of E.
The instance deﬁnition is compatible with that of ground instance
of a clause.

A Substitution
{f (y)/y, a/z, f (g(b))/x}
An instance
θ = {a/x, f (b)/y, b/z},
E = Q(x, y) ∨ P(y, z)
Then, Eθ = Q(a, f (b)) ∨ P(f (b), b)

Deﬁnition: Composition of substitutions
Let θ = {t1/x1, ..., tn/xn} and λ = {u1/y1, ..., um/ym} be two
substitutions;
then, the substitution composed of θ et λ, denoted by θ ◦ λ, that is
obtained from the set {t1λ/x1, ..., tnλ/xn, u1/y1, ..., um/ym}
by deleting any element ti λ/xi for which ti λ = xi and any element
uj/yj such that yj is among{x1, x2, ..., xn}

Composition of Substitutions
θ = {t1/x1, t2/x2} = {f (y)/x, z/y}
λ = {u1/y1, u2/y2, u3/y3} = {a/x, b/y, y/z}
{t1λ/x1, ..., tnλ/xn, u1/y1, ..., um/ym} =
{f (b)/x, y/y, a/x, b/y, y/z}
After deletions: θ ◦ λ = {f (b)/x, y/z}
Comments: θ ◦ (λ ◦ β) = (θ ◦ λ) ◦ β; ◦ β = β ◦ with the
empty substitution.

Definition: Unifier
A substitution θ is called a unifier for a set {E1, E2, ..., Ek} if
and only if E1θ = E2θ = ... = Ekθ. The set {E1, E2, ..., Ek} is
said to be unifiable if there is a unifier for it.
Several unifiers may be exist for a set F of formulae
Definition: Most General Unifier
A unifier σ for a set {E1, E2, ..., Ek} of formulae is a Most
General Unifier if and only if each unifier θ there is a
substitution λ such that θ = σ ◦ λ

Let P(g(y)) et P(x) be formulae, σ, λ, θ can be defined as follows:
θ = {g(a)/x}, λ = {a/y}, σ = {g(y)/x}
σ is the most general substitution
Most General Unifier
Let U be the set of unifiers, F be a set of formulae and S be the
set of substitutions on F.
∀α ∈ U, ∃β ∈ S such that α = β ◦ σ
The most general unifier (when it exists), denoted MGU (Most
General Unifier) is the most general substitution that match
formulae and it is unique.

The resolution principle is composed of two inference rules: the
resolution rule and the reduction rule
Resolution Rule: the formula C is the resolvent of the two clauses
A and B if and only if
A = P ∨ D
B = P1 ∨ E
C = (Dθ ∨ E)σ
θ is a renaming substitution so that Dθ and E are formulae with
disjoint variables. σ is the most general uniﬁer of P et P1 such
that P et P1 are complementary literals (or complementary pairs)

The resolution rule can be rewrite as follows:
P ∨ D, P1 ∨ E (Dθ ∨ E)σ
Example
Let A = P(x, c) ∨ R(x) and B = ¬P(c, c) ∨ Q(x)
Let D = R(x) and E = Q(x)
C = R(c) ∨ Q(x) with θ = {y/x} in A and σ = {c/y}

Reduction rule
Let A = P ∨ P1 ∨ B be a clause, C is the resolvent of A if and only
if:
Pσ = P1σ, σ Most General Uniﬁer of P et P1
C = Pσ ∨ Bσ
The rule may be rewrite as follows:
P ∨ P1 ∨ B Pσ ∨ Bσ with Pσ = P1σ
Example
Let A = P(x, g(y)) ∨ P(f (c), z) ∨ R(x, y, z)
C = R(f (c), y, g(y)) ∨ P(f (c), g(y)) with
σ = {f (c)/x, g(y)/z}

Theorem: a set F of clauses is unsatisfiable if and only if there is a
deduction of the empty clause from F.
Properties: the resolution principle is sound and complete, that is
to say:
If F is satisfiable, the empty clause cannot be deduced
vide.
If F is unsatisfiable, the empty clause is deduced

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Resolution Strategy
The two main methods are: the management of a set of clauses
and the exploration of deduction tree.
We focus on the later. Two possible choices are available for this
strategy:
Selection of a sub-tree of the deduction tree, limitation of the
possible deductions
Selection of an exploration strategy of a sub-tree (Depth- ﬁrst
search, Breadth-ﬁrst search, ...)

Resolution Strategy
Linear Strategy
Deﬁnition: Let S be a set of clauses and C0 a clause in S, a linear
deduction of Cn from S with C0 as initial clause has the following
properties: Ci+1 is the resolvent of Ci and Bi (resolution rule) with
Bi ∈ S or Bi = Cj such that 1 ≤ i ≤ n − 1, j < i.

Resolution Strategy
Linear deduction properties (cf. Chang and Lee or Loveland):
If there is a deduction of the empty clause for S, then there is a
linear deduction of the empty clause for S.
Moreover, if S = S ∪ {C0} with S satisfiable and S unsatisfiable
then there is a linear deduction from S having C0 as initial clause.
Breadth-first exploration strategy: the linear deduction is sound
and complete.
Depth-first exploration strategy: the linear deduction is sound,
but not complete.

Resolution Strategy
Input Strategy
Deﬁnition: Let S be a set of clauses and C0 be a clause in S, an
Input deduction of Cn from S with C0 as initial clause has the
following properties: Ci+1 is a resolvent obtained by the application
of the rule resolution and Bi with Bi ∈ S such that 1 ≤ i ≤ n − 1.
Even, with a Breadth-ﬁrst exploration strategy, an Input
deduction is not complete.
The Input deduction tree does not contain the empty clause. but
the tree of all deductions contain the empty clause.

Resolution Strategy
Input Strategy
Nevertheless, there is an interesting result in a particular case::
If S = S ∪ {C0} is unsatisﬁable, C0 is a negative clause that
only contains negative literals and S only contains clauses with
exactly one positive literal (Horn clauses) then there is an input
deduction with C0 as initial clause which gives the empty clause
clause and does not use the reduction rule.

Resolution Strategy
Ordered Strategy
Deﬁnition: An ordered resolution between two ordered clauses
(without common variables) C1 and C2 deduces the ordered clause
C with the following conditions:
C1 = L1 ∨ L2 ∨ ... ∨ Ln
C2 = ¬L1 ∨ L2 ∨ ... ∨ Lm
C = (L2 ∨ ... ∨ Ln ∨ L2 ∨ ... ∨ Lm)θ avec θ Most General Uniﬁer
of L1 and ¬L1
The clause C is ordered: the clause containing the positive literal is
located at the beginning of the resolvent and the respective orders
of the two clauses are kept

Resolution Strategy
Deﬁnition: Let S be a set of clauses and C0 be a clause in S, a
linear ordered deduction of Cn from S with C0 as initial clause is a
deduction obtained by applying n ordered resolutions.
Linear ordered Strategy
Deﬁnition: Let S be a set of clauses and C0 be a clause in S, an
ordered deduction of Cn from S with C0 as initial clause has the
following properties: Ci+1 is an ordered resolvent of ordered clauses
Ci and Bi (ordered resolution) with Bi ∈ S or Bi = Cj such that
1 ≤ i ≤ n1, j < i.

Resolution Strategy
A linear ordered refutation is a linear ordered deduction of the
empty clause.
With a Breadth-ﬁrst exploration strategy, the linear ordered
deduction is sound and complete.
The most important result is as follows:
If S = S ∪ {C0} is unsatisﬁable, C0 is a negative literal and S
does not have clauses with exactly one positive literal (Horn
clauses) then there is an input and linear ordered with the initial
clause C0 leading to the empty clause.

Resolution Strategy
SLD Resolution
A SLD resolution is a linear resolution with a selection function on
a set of Horn clauses.
Deﬁnition: a SLD refutation of P ∪ {¬G} is a ﬁnite SLD deduction
of P ∪ {¬G} of the empty clause via a selection function.

Resolution Strategy
SLD Resolution
Deﬁnition of Derivation: Let Gi be a goal such that
A1, ..., Am, ...Ak → Gi , the clause Ci+1 such that B1, ..., Bq → A
and a selection function R. The derivation of the goal Gi+1 is
obtained from Gi with Ci+1 by using the MGU θi+1 via R if the
following conditions are ﬁlled:
Am is an atom selected by a selection function
Amθi+1 = Aθi+1 tel que θi+1 is the MGU of A et Am
Gi+1 = (A1, ..., Am−1, B1, ..., Bq, Am+1, ..., Ak)θi+1
Gi+1 is the resolvent of Gi et Ci+1. Ci+1 is a variant of A that is
to say a clause such that all variables of A has been renamed.

Resolution Strategy
SLD Resolution
Definition: a SLD derivation of P ∪ {¬G} is composed of a finitite
or infinite serie G0, G1, G2, ... of goals, a serie C0, C1, C2, ... of
clauses’ variants of the program P and a serie θ1, θ2, ... of MGU
such that Gi+1 is derived of Gi and Ci+1 and uses θi+1 via R.
A selection function is an application of a set of goals to a set of
atoms, such that the function value for a goal is alsways an atom.
The later is called the selected atom.
Independence of the function selection: if P ∪ {¬G} is
unsatisfiable then there is always a SLD refutation that will use
this function selection. The space search traversed by a SLD
refutation is called a SLD tree.

Resolution Strategy
SLD Resolution
Let P be a program, G be a goal and R a selection fucntion, the
SLD tree for P ∪ {¬G} via R is deﬁned as follows:
Each tree node is a goal (possibly empty)
The root is G
Let A1, A2, ..., Ak → Gi , (k ≥ 1) be a tree node and Am be the
selected atom via R, this node is a descendant for each clause
B1, ..., Bq → A such that Am et A is uniﬁable. the descendant is
(A1, ..., Am−1, B1, ..., Bq, Am+1, ..., Ak)θi+1 where θi+1 is the
MGU of Am and A
The empty nodes do not have descendant

Resolution Strategy
SLD Resolution
The soundness of the SLD refutation rely upon the unification.
Thus, if it unifies non unifiable terms, the result will be wrong.
Occur check:
test ← P(x, x)
P(x, f (x)) ← P(x, x))
The unification algorithm has to made the occur check. Let x be a
variable that must be unified with the term t1, the ccur check aim
at detecting the presence of x into the term t1. If one consider the
above mentioned example, for demonstrating test it is necessary to
unify P(x, x) with P(y, f (y)) - after renaming the variables. These
two terms are not unifiale because x = f (x). Without the occur
check, the algorithm will unify the two terms.

Outline
1 Introduction
2 Formal Systems
Resolution Strategy
Prolog Modeling

Prolog Modeling
Concatenation Program
Conc([], L, L).

Prolog Modeling
Ancestor Program, Version 1
Ancestor(X,Y) :− Parents(X,Z), Ancestor(Z,Y).
Ancestor(X,Y) :− Parents(X,Y).
Parents(X,Y) :− Father(X,Y).
Parents(X,Y) :− Mother(X,Y).
Version 2
Version 3
Ancestor(X,Y) :− Ancestor(Z,Y), Parents(X,Z).

Prolog Modeling
Eight Queens’ Problem
1 2 3 4 5 6 7 8
1 X
2 X
3 X
4 X
5 X
6 ? ? ?
7 ? ? ?
8 ? ? ?

Prolog Modeling
Eight Queens’ Program
Put(Listdd, Listdg ,Col,8, Result ).
Put(Listdd, Listdg ,Col,Row,Result) :−
Row_1 is Row + 1,
column(I), not(member(I,Col)),
Dd is Row_1 + I, Dg is I − Row_1,
not(member(Dd,Listdd)),
not(member(Dg,Listdg)),
Put([Dd|Listdd ],[ Dg|Listdg ],[ I |Col ], Row_1,[[Row_1,I]|Result]).
Column(1). Column(2). Column(3). Column(4).
Column(5). Column(6). Column(7). Column(8).

Prolog Theorem Proving
Concatenation Program
Conc([], L, L).
∀X, L, L1, L2 (Conc(f (X, L), L1, f (X, L2)) ∨ ¬Conc(L, L1, L2))∧
∀L (Conc(f (empty), L, L)∧
∀L ¬Conc(f (a, f (b, f (empty))), f (c, f (d, f (empty))), L)

Concatenation Demonstration
¬Conc(f (a, f (b, f (empty))), f (c, f (d, f (empty))), L )
Conc(f (X, L), L1, f (X, L2)) ∨ ¬Conc(L, L1, L2))
¬Conc(f (b, f (empty)), f (c, f (d, f (empty))), L2)
σ =
{a/X, f (b, f (empty))/L, f (c, f (d, f (empty)))/L1, f (a, L2)/L }
Resolvent : ¬Conc(f (b, f (empty)), f (c, f (d, f (empty))), L2)

Conc(f (X, L), L1, f (X, L2 )) ∨ ¬Conc(L, L1, L2 )
¬Conc(f (empty), f (c, f (d, f (empty))), L2 )
σ = {b/X, f (empty)/L, f (c, f (d, f (empty)))/L1, f (b, L2 )/L2}
Resolvent : ¬Conc(f (empty), f (c, f (d, f (empty))), L2 )

¬Conc(f (empty), f (c, f (d, f (empty))), L2 ))
Conc(f (empty), L, L)
σ = {f (c, f (d, f (empty)))/L, f (c, f (d, f (empty)))/L2 }
Resolvent :

Ancestor Program
Parents(X,Y) :− Father(X,Y).
Parents(X,Y) :− Mother(X,Y).
Father(a,b). Father(b,c). Father(d,f ). Father(e,g).
Mother(a,d). Mother(d,e). Mother(a,h). Mother(b,i).
Ancesotr(X,Y) ?

Ancestor Program
∀X, Y , Z
(Ancestor(X, Y ) ∨ ¬Parents(X, Z) ∨ ¬Ancestor(Z, Y )∧
∀X, Y (Ancestor(X, Y ) ∨ ¬Parents(X, Y )∧
∀X, Y (Parents(X, Y ) ∨ ¬Father(X, Y )∧
∀X, Y (Parents(X, Y ) ∨ ¬Mother(X, Y )∧
Father(a, b) ∧ Father(b, c) ∧ Father(d, f ) ∧ Father(e, g)∧
Mother(a, d) ∧ Mother(d, e) ∧ Mother(a, h) ∧ Mother(b, i)∧
¬Ancestor(X, Y )

Ancestor Demonstration
¬Ancestor(T, U)
(Ancestor(X, Y ) ∨ ¬Parents(X, Z) ∨ ¬Ancestor(Z, Y )
¬Parents(T, Z) ∨ ¬Ancestor(Z, U)
σ = {T/X, U/Y }
Resolvent : ¬Parents(T, Z) ∨ ¬Ancestor(Z, U)

Ancestor Program
∀X, Y , Z
¬Ancestor(X, Y )

¬Parents(T, Z) ∨ ¬Ancestor(Z, U)
(Parents(X, Y ) ∨ ¬Father(X, Y )
¬Father(T, Z) ∨ ¬Ancestor(Z, U)
σ = {T/X, Z/Y }
Resolvent : ¬Father(T, Z) ∨ ¬Ancestor(Z, U)

Ancestor Program
∀X, Y , Z
¬Ancestor(X, Y )

¬Father(T, Z) ∨ ¬Ancestor(Z, U)
Father(a, b)
¬Ancestor(b, U)
σ = {a/T, b/Z}
Resolvent : ¬Ancestor(b, U)

Ancestor Program
∀X, Y , Z
¬Ancestor(X, Y )

¬Ancestor(b, U)
(Ancestor(X, Y ) ∨ ¬Parents(X, Z) ∨ ¬Ancestor(Z, Y )
¬Parents(b, Z) ∨ ¬Ancestor(Z, U)
σ = {b/Y , U/Y }
Resolvent : ¬Parents(b, Z) ∨ ¬Ancestor(Z, U)

Ancestor Program
∀X, Y , Z
¬Ancestor(X, Y )

¬Parents(b, Z) ∨ ¬Ancestor(Z, U)
(Parents(X, Y ) ∨ ¬Father(X, Y )
¬Father(b, Z) ∨ ¬Ancestor(Z, U)
σ = {b/X, U/Y }
Resolvent : ¬Father(b, Z) ∨ ¬Ancestor(Z, U)

Ancestor Program
∀X, Y , Z
¬Ancestor(X, Y )

¬Father(b, Z) ∨ ¬Ancestor(Z, U)
Father(b, c)
¬Ancestor(c, U)
σ = {c/Z}
Resolvent : ¬Ancestor(c, U)

Ancestor Program
∀X, Y , Z
¬Ancestor(X, Y )

¬Ancestor(c, U)
Ancestor(X, Y ) ∨ ¬Parents(X, Z) ∨ ¬Ancestor(Z, Y )
¬Parents(c, Z) ∨ ¬Ancestor(Z, U)
σ = {c/X, U/Y }
Resolvent : ¬Parents(c, Z) ∨ ¬Ancestor(Z, U)

¬Parents(c, Z) ∨ ¬Ancestor(Z, U)
Parents(X, Z ) ∨ ¬Father(Z , Y )
¬Father(c, U) ∨ ¬Ancestor(Z , U )
σ = {c/X, Z/Z }
Resolvent : ¬Father(c, U ) ∨ ¬Ancestor(Z , U )

Predicate Calculus

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