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A Note on Assumptions about Skolem Functions
Hans Jürgen Ohlbach and Christoph Weidenbach
Max-Planck-Institut für Informatik
Im Stadtwald
66123 Saarbrücken, Germany
email: (ohlbach,weidenb)@mpi-sb.mpg.de
Journal of Automated Reasoning 15(2):267–275
November 10, 1997
Abstract. Skolemization is not an equivalence preserving transformation. For the purposes of
refutational theorem proving it is sufficient that Skolemization preserves satisfiability and unsatisfiability. Therefore there is sometimes some freedom in interpreting Skolem functions in a particular way. We show that in certain cases it is possible to exploit this freedom for simplifying formulae
considerably. Examples for cases where this occurs systematically are the relational translation
from modal logics to predicate logic and the relativization of first-order logics with sorts.
Key words: Skolemization, Refutational Theorem Proving
1. Introduction
Refutational theorem proving has a certain degree of freedom which so far is
not very often exploited. All kinds of transformations preserving satisfiability and
unsatisfiability of the formulae to be refuted are allowed. Skolemization is a typical
transformation which is not equivalence preserving but satisfiability and unsatisfiability preserving. But this is usually the only routinely applied transformation
with this property.
For an existential quantification, Skolemization introduces a new function symbol whose interpretation is in general not completely determined. Sometimes it is
possible to make additional assumptions about the new Skolem function without
changing satisfiability and unsatisfiability of the formulae. As an example consider
the formula
∀x(Φ(x) ⊃ ∃y(B(x, y) ∧ C(x, y)))
Skolemization and clausification yields as an intermediate result
Φ(x) ⊃ B(x, f (x))
Φ(x) ⊃ C(x, f (x))
(1)
(2)
If Φ is a big formula this duplication may be disastrous. If B is serial, i.e. ∀x∃y B(x, y)
holds, we claim that the following optimized transformation is possible:
B(x, f (x))
Φ(x) ⊃ C(x, f (x))
(3)
(4)
2
Hans Jürgen Ohlbach and Christoph Weidenbach
where one occurrence of Φ is dropped or, equivalently, the Skolem form B(x, f (x))
of B(x, y) is moved to the top-level of the formula. That means the axiomatization
of f is made stronger and the clauses become shorter. The nontrivial part of the
correctness proof, which also shows the idea behind the transformation, amounts
to transforming a model for (1) and (2) into a model for (3) and (4). Since f is
a Skolem function, we use the freedom to define a suitable interpretation for f .
Suppose Φ(x) is true for some assignment x/a. Then by the clause (1), B(x, f (x))
is true as well, i.e. (3) is true, where f (a) is the same value in (1) and (3). This is
the unproblematic case. Now suppose Φ(x) is false for the assignment x/a. Then
(4) is true independently of the meaning of f . But what about (3)? Here we use
the seriality assumption for B. We assume ∀x∃y B(x, y). This tells us that there
is some c such that B(x, y) is true for the assignment [x/a, y/c]. That means we
can define f (a) to be just this c and then (3) becomes true as well.
This informal description is made precise in the next section. In section 3 we
show that certain well known transformations in sorted logics and modal logics are
in fact instances of this optimized Skolemization. We give several examples which
show that optimized Skolemization can improve the proof search considerably.
2. Optimized Skolemization
For a precise definition of the optimized Skolemization we need to manipulate
subformulae of a formula at a certain position inside the subformula and with
a certain polarity. To this end we introduce the standard definitions of formula
occurrences and polarities.
An occurrence is a word over IN. Let denote the empty word. Then we define
the set of occurrences occ(Φ) of a formula Φ as follows: (i) the empty word is in
occ(Φ) (ii) i.π is in occ(Φ) iff Φ = Ψ1 ∧ . . . ∧ Ψn or Φ = Ψ1 ∨ . . . ∨ Ψn , 1 ≤ i ≤ n
and π ∈ occ(Ψi ) (iii) 1.π (2.π) is in occ(Φ) iff Φ = ∀x Ψ or Φ = ∃x Ψ or Φ = ¬Ψ
or Φ = Ψ ⊃ Θ or Φ = Ψ ≡ Θ and π ∈ occ(Ψ) (π ∈ occ(Θ)). Now if π ∈ occ(Ψ)
then Ψ| = Ψ and Ψ|i.π = Ψi|π where Ψi is the ith subformula of Ψ (see above).
Intuitively, the polarity of some formula Φ|π = Ψ in Φ is 1 if Ψ occurs below an
even number of (explicit or implicit) negation symbols, it is -1 if Ψ occurs below
an odd number of negation symbols and it is 0 if Ψ occurs below an equivalence
symbol. We define the polarity pol(Φ, π) of a formula Φ |π in a formula Φ by:
(i) pol(Φ, ) = 1 (ii) pol(Φ, i.π) = pol(Φ|i , π) if Φ is a conjunction, disjunction,
quantifier formula or Φ is an implication and i = 2 (iii) pol(Φ, i.π) = −pol(Φ|i , π)
if Φ is a negation or Φ is an implication and i = 1 (iv) pol(Φ, i.π) = 0 if Φ is an
equivalence.
We also use occurrences to define the replacement of a formula inside another
formula. Φ[π/Ψ] with π ∈ occ(Φ) is the formula obtained from Φ by replacing Φ|π
in Φ with Ψ.
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A Note on Assumptions about Skolem Functions
3
Usually, Skolemization is defined with respect to a formula in negation normal
form. We generalize this definition to the case of arbitrary formulae. The polarity
function tells us whether a formula remains unchanged by producing the negation
normal form. A formula Φ|π , π ∈ occ(Φ), is in the scope of a universal quantifier, if
there exists a prefix µ of π such that Φ|µ = ∀xΘ and pol(Φ, µ) = 1 or Φ|µ = ∃xΘ and
pol(Φ, µ) = −1 or pol(Φ, µ) = 0 and Φ|µ is an arbitrary quantificational formula.
It is in the scope of an existential quantifier, if there exists a prefix ν of π such that
Φ|ν = ∀xΘ and pol(Φ, ν) = −1 or Φ|ν = ∃xΘ and pol(Φ, ν) = 1 or pol(Φ, µ) = 0
and Φ|µ is an arbitrary quantificational formula.
LEMMA 1. Let Φ|π = Ψ with π ∈ occ(Φ) and pol(Φ, π) = 1 and M be an interpretation. Then
if M |= Φ ∧ (Ψ ⊃ Θ) then M |= Φ[π/Θ]
Proof. The proof is due to Loveland [2, Lemma 1.5.1, p. 40].
THEOREM 2 (Optimized Skolemization). Let Φ|π = ∃y(Ψ∧Θ), π ∈ occ(Φ), pol(Φ, π) =
1, ∃y(Ψ ∧ Θ) is not in the scope of an existential quantifier and let x1 , . . . , xn be
the universally quantified variables which occur freely in ∃y(Ψ ∧ Θ). In addition,
we assume the seriality condition |= Φ ⊃ ∀x1 , . . . , xn ∃y Θ. Then
Φ is satisfiable
iff
Φ[π/Ψ{y/f (x1 , . . . , xn )}] ∧ ∀x1 , . . . , xn Θ{y/f (x1 , . . . , xn )} is satisfiable
where f is a new n-place Skolem function.
Proof. “⇒” Assume M |= Φ. Since f is new to Φ it is sufficient to construct an
interpretation M0 which is like M, but in addition specifies an interpretation for f
such that M0 |= Φ[π/Ψ{y/f (x1 , . . . , xn )}] ∧ ∀x1 , . . . , xn Θ{y/f (x1 , . . . , xn )}. Consider domain elements a1 , . . . , an as assignments for the universally quantified variables x1 , . . . , xn . If M[x1 /a1 , . . . , xn /an ] |= ∃y(Ψ ∧ Θ), then there exists some b as
assignment for y such that M[x1 /a1 , . . . , xn /an , y/b] |= Ψ ∧ Θ. We choose b as val0
def
ue for f , i.e. f M (a1 , . . . , an ) = b. If M[x1 /a1 , . . . , xn /an ] 6|= ∃y(Ψ ∧ Θ) we choose
0
def
f M (a1 , . . . , an ) = c, where M[x1 /a1 , . . . , xn /an , y/c] |= Θ. Such a c always exists
by the seriality assumption M |= ∀x1 , . . . , xn ∃y Θ which implies, as f is new to Φ,
0
M0 |= ∀x1 , . . . , xn ∃y Θ. Now by construction of f M we have M0 |= Φ ∧ (∃y(Ψ ∧
Θ) ⊃ Ψ{y/f (x1 , . . . , xn )}) and thus by Lemma 1, M0 |= Φ[π/Ψ{y/f (x1 , . . . , xn )}].
In addition, M0 |= ∀x1 , . . . , xn ∃yΘ ∧ (∃y Θ ⊃ Θ{y/f (x1 , . . . , xn )}) and thus again
by Lemma 1, M0 |= ∀x1 , . . . , xn Θ{y/f (x1 , . . . , xn )}.
“⇐”Assume M |= Φ[π/Ψ{y/f (x1 , . . . , xn )}] ∧ ∀x1 , . . . , xn Θ{y/f (x1 , . . . , xn )}.
Then we have: M |= Ψ{y/f (x1 , . . . , xn )} ⊃ ∃y(Ψ∧Θ) by choosing y/f M (a1 , . . . , an )
for y in ∃y(Ψ ∧ Θ) and any assignment a1 , . . . , an of the x1 , . . . , xn . Now by Lemma 1 we conclude M |= Φ.
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Hans Jürgen Ohlbach and Christoph Weidenbach
EXAMPLE 3. We apply the optimized Skolemization to Pelletier’s [6] problem
no 29:
(1)
(2)
(3)
∃x F (x)
∃x G(x)
¬[(∀x(F (x) ⊃ H(x)) ∧ ∀x(G(x) ⊃ J(x))) ≡ ∀x, y((F (x) ∧ G(y)) ⊃ (H(x) ∧
J(y)))]
Elimination of the equivalence symbol in (3) by ¬[Φ ≡ Ψ] iff (Φ ∨ Ψ) ∧ (¬Φ ∨ ¬Ψ)
gives:
(4)
(5)
∀x(F (x) ⊃ H(x)) ∧ ∀x(G(x) ⊃ J(x))) ∨ ∀x, y((F (x) ∧ G(y)) ⊃ (H(x) ∧ J(y)))
¬(∀x(F (x) ⊃ H(x)) ∧ ∀x(G(x) ⊃ J(x))) ∨ ¬∀x, y((F (x) ∧ G(y)) ⊃ (H(x) ∧
J(y)))
For the purpose of readability we move the negation symbols and quantifiers occurring in (5) inside:
(6)
∃x(F (x)∧¬H(x))∨∃x(G(x)∧¬J(x))∨∃x(F (x)∧∃y(G(y)∧(¬H(x)∨¬J(y))))
There are the following occurrences of existential formulae:
(6)|1
(6)|2
(6)|3
(6)|312
=
=
=
=
∃x(F (x) ∧ ¬H(x))
∃x(G(x) ∧ ¬J(x))
∃x(F (x) ∧ ∃y(G(y) ∧ (¬H(x) ∨ ¬J(y))))
∃y(G(y) ∧ (¬H(x) ∨ ¬J(y)))
Theorem 2 is applicable to all these occurrences, because all occurrences have
polarity 1 and the formulae (1), (2) guarantee the seriality condition for the atoms
of the form F (x), G(x), which we want to move outside. Thus we get after optimized Skolemization:
(7)
(8)
¬H(a) ∨ ¬J(b) ∨ ¬H(c) ∨ ¬J(d)
F (a) ∧ G(b) ∧ F (c) ∧ G(d)
It is obvious that a refutation of the formulae (1), (2), (4), (7), (8) is much simpler
than refuting (1), (2), (4), (5). In fact, OTTER [3] (version 3.0, auto mode) needed
half of the time and clauses to refute the formulae with optimized Skolemization
compared to the formulae translated with OTTER’s standard Skolemization procedure. In addition, the optimized Skolemization proof is shorter and has a lower
proof complexity.
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A Note on Assumptions about Skolem Functions
5
As pointed out by a reviewer, our Skolemization technique is not new in the sense
that it can be simulated by standard Skolemization and equivalence preserving
transformations. The formula Φ of Theorem 2 is equivalent to the formula
∀x1 , . . . , xn ∃y Φ[π/(Ψ ∧ Θ)] ∧ ∀x1 , . . . , xn ∃y Θ
(5)
because |= Φ ⊃ ∀x1 , . . . , xn ∃y Θ. Now (5) is equivalent to the formula
∀x1 , . . . , xn ∃y [Φ[π/Ψ] ∧ Θ]
(6)
This can be proved by techniques similar to those used to prove Theorem 2. Eventually standard Skolemization yields
∀x1 , . . . , xn [Φ[π/Ψ{y/f (x1 , . . . , xn )}] ∧ Θ{y/f (x1 , . . . , xn )}]
(7)
which is exactly the result of Theorem 2 if the universal quantifiers are moved
inside. However, we prefer the formulation of Theorem 2, because we interpret
∀x1 , . . . , xn Θ{y/f (x1 , . . . , xn )} as a (stronger) definition of the Skolem function
f . In addition, the formulation of Theorem 2 is compatible with the usual techniques for clause normal form, e.g. anti-prenexing, whereas the above argumentation requires to move quantifiers outwards.
3. Déja vu
The optimized Skolemization has been used implicitly in some other systems.
3.1. Sorted Logic
The fact that information about Skolem functions can be moved from a local
context to the top-level has been implicitly exploited in sorted logic. Consider the
formula Φ ⊃ ∃xB C(xB ), which is the sorted formalization of Φ ⊃ ∃x(B(x) ∧
C(x)). In sorted logics, where all sorts are a priori assumed non-empty, it gets
Skolemized to Φ ⊃ C(a) and the sort declaration B(a) is added to the top-level
sort declarations [9, 8]. Thus, the sort declaration for a does not depend on the
condition Φ anymore. That means global sort declarations about Skolem functions
implicitly apply the optimized Skolemization.
Weidenbach [10, 11] shows how sorted Skolemization is applied if the sorts
are not a priori assumed non-empty. Then it is only possible to move the sort
declarations of Skolem functions outside, if the sort of the existential variable can
be proved non-empty. Otherwise the sort declaration remains inside the formula
and makes a more general approach to sorted reasoning necessary.
The example of Section 2 is an instance of Weidenbach’s approach to sorted logic. The unary predicates can be translated into sorts. This enables further
simplifications of formula (4):
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Hans Jürgen Ohlbach and Christoph Weidenbach
(40 )
(∀xF H(xF ) ∧ ∀yG J(yG )) ∨ ∀xF , yG (H(xF ) ∧ J(yG ))
Since the sorts F and G are non-empty (see (1),(2)), (40 ) is further simplified to
(400 )
∀xF H(xF ) ∧ ∀yG J(yG )
Now a refutation of (1), (2), (400 ), (7), (8) by resolution extended with sorts
[11] yields no search anymore. Every possible resolution step contributes to the
proof.
3.2. Modal Logic
Modal Logic is an extension of predicate logic with the two operators 2 and 3 [1].
The standard Kripke semantics of normal modal systems interprets the 2-operator
as a universal quantification over accessible worlds and the 3-operator as an existential quantification over accessible worlds. This semantics can be exploited to
define a “relational” translation from modal to predicate logic. For example 23P
is translated into ∀w(R(o, w) ⊃ ∃v(R(w, v) ∧ P (v))). R denotes the accessibility
relation and o some initial world.
Notice that the translation of the 3-operator has the typical pattern where
our optimized Skolemization is applicable — provided the accessibility relation is
serial, i.e. we have modal systems above D. The overall effect of the optimized
Skolemization is that the conditions on R coming from 3-operators become positive unit clauses. From the 2-operator we obtain only negative literals in the
clauses. Then the negative R literals can be viewed as constraints over the theory
consisting of the positive R-unit clauses and the formulae of the specific modal
logic. This approach has been studied by Scherl [7].
EXAMPLE 4. We show the power of the optimized Skolemization by an example taken from modal logic KD45 [1]. In modal logic KD45 the formula 32P ≡
3232P is a theorem. The theorem can be translated into first-order logic by
introducing an accessibility relation R. Then the theorem is:
∃x(R(o, x) ∧ ∀y(R(x, y) ⊃ P (y)))
≡
∃x(R(o, x) ∧ ∀y(R(x, y) ⊃ ∃z(R(y, z) ∧ ∀u(R(z, u) ⊃ P (u)))))
Here o names the initial world and R(x, y) means that world y is accessible from
world x. The properties of R in modal logic KD45 are expressed by the following
formulae:
(1)
(2)
(3)
∀x∃y R(x, y)
∀x, y, z (R(x, y) ∧ R(y, z) ⊃ R(x, z))
∀x, y, z (R(x, y) ∧ R(x, z) ⊃ R(y, z))
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A Note on Assumptions about Skolem Functions
7
Now we apply Theorem 2 to the negated theorem. In order to get positive polarities
for the existential subformulae, we eliminate the equivalence symbol by ¬[Φ ≡ Ψ]
iff (Φ ∨ Ψ) ∧ (¬Φ ∨ ¬Ψ). For better readability we move the negation sign inside.
The result is:
(4)
(5)
∃x(R(o, x) ∧ ∀y(R(x, y) ⊃ P (y))) ∨
∃x(R(o, x) ∧ ∀y(R(x, y) ⊃ ∃z(R(y, z) ∧ ∀u(R(z, u) ⊃ P (u)))))
∀x(R(o, x) ⊃ ∃y(R(x, y) ∧ ¬P (y))) ∨
∀x(R(o, x) ⊃ ∃y(R(x, y) ∧ ∀z(R(y, z) ⊃ ∃u(R(z, u) ∧ ¬P (u))))
There are the following occurrences in (4) and (5) which name existentially quantified subformulae:
(4)|1 = ∃x(R(o, x) ∧ ∀y(R(x, y) ⊃ P (y)))
(4)|2 = ∃x(R(o, x) ∧ ∀y(R(x, y) ⊃ ∃z(R(y, z) ∧ ∀u(R(z, u) ⊃ P (u)))))
(4)|21212 = ∃z(R(y, z) ∧ ∀u(R(z, u) ⊃ P (u)))
(5)|112 = ∃y(R(x, y) ∧ ¬P (y))
(5)|212 = ∃y(R(x, y) ∧ ∀z(R(y, z) ⊃ ∃u(R(z, u) ∧ ¬P (u))))
(5)|2121212 = ∃u(R(z, u) ∧ ¬P (u))
All occurrences have polarity 1. They are either of the form ∃w(R(o, w) ∧ Ψ) or
∃w(R(v, w) ∧ Ψ), where v, w are variables. In order to apply Theorem 2 and to
move the formula R(o, w) (R(v, w)) outside, we must prove the seriality condition
|= ((1)∧(2)∧(3)∧(4)∧(5)) ⊃ ∃wR(o, w) (|= ((1)∧(2)∧(3)∧(4)∧(5)) ⊃ ∃wR(v, w)).
The proof is trivial, since (1) already implies the seriality of R. Thus Theorem 2
is applicable to all occurrences of the existential quantifiers. For example we start
with the occurrence 1 of (4). We introduce a new constant a, replace ∃x(R(o, x) ∧
∀y(R(x, y) ⊃ P (y))) with ∀y(R(a, y) ⊃ P (y))) and add R(o, a) as a conjunct to
(4). The procedure can be repeated for the other occurrences. Eventually, we get
(6)
(7)
(8)
(9)
∀y(R(a, y) ⊃ P (y))) ∨ ∀y(R(b, y) ⊃ ∀u(R(h(y), u) ⊃ P (u)))
R(o, a) ∧ R(o, b) ∧ ∀y R(y, h(y))
∀x(R(o, x) ⊃ ¬P (i(x))) ∨ ∀x(R(o, x) ⊃ ∀z(R(f (x), z) ⊃ ¬P (g(x, z))))
∀x R(x, i(x)) ∧ ∀x R(x, f (x)) ∧ ∀x, z R(z, g(x, z))
where (6) is the optimized Skolemization of (4), (8) is the optimized Skolemization
of (5), (7) are the R atoms moved outside (4) and (9) are the R atoms moved
outside (5).
The obvious advantage of this Skolemization technique are the stronger definitions for the Skolem functions (constants). Using standard Skolemization these
definitions usually occur in disjunctions with other literals from the theorem. This
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Hans Jürgen Ohlbach and Christoph Weidenbach
makes a proof of the theorem more complicated. The theorem prover OTTER
(version 3.0, auto mode) proves the theorem with optimized Skolemization in less
than one minute, i.e. it refutes the formulae (1),(2),(3),(6),(7),(8),(9). Although we
tried various parameter settings, OTTER did not find a proof of the theorem in
the version with standard Skolemization, i.e. OTTER fails to refute the formulae
(1),(2),(3),(4),(5) using its standard Skolemization.
However, it should be noted that the special translation techniques developed
for modal logic, e.g. see the work of Nonnengart [4] or the work of Ohlbach [5], are
still more powerful than our optimized Skolemization, because they also eliminate
the formulae coming from the specific modal logics (in our case the formulae (1),
(2), (3)). Translating the above example using Nonnengart’s approach we get
(10 )
(40 )
(50 )
∀x, y, z R(x, y: z)
∀x(R(o: a, x) ⊃ P (x)) ∨ ∀y(R(o: b, y) ⊃ ∀z(R(y: h(y), z) ⊃ P (z)))
∀x(R(o, x) ⊃ P (x: i(x))) ∨ ∀y(R(o, y) ⊃ ∀z(R(y: f (y), z) ⊃ P (z: g(y, z))))
where “:” is a new two-place function symbol written in infix notation. The formula
(10 ) is the translation of (1),(2),(3), (40 ) is the translation of (4) and (50 ) is the
translation of (5). The formulae (10 ), (40 ), (50 ) are refuted by OTTER in less than
one second.
4. Summary
We have presented an optimized Skolemization of existential quantifiers which
moves information about the Skolem function from the local context of the occurrence of the existential quantifier to the top-level of the formula. Instances of this
optimized Skolemization have been used implicitly or explicitly in special applications. We have defined it now in such a way that it can be used as a general
method for arbitrary formulae. However, the proof of the seriality condition may
be as complex as the proof of the input formula, in general. Therefore optimized
Skolemization requires a more sophisticated implementation concept than standard
Skolemization. Nevertheless there are many examples where the seriality condition
can be easily proved (e.g. see the examples above, other problems of the Pelletier
collection) and then optimized Skolemization avoids duplication of literals, yields
shorter clauses, shorter and less complex proofs and a smaller search space. In some
cases optimized Skolemization makes a proof possible where proof procedures using
standard Skolemization fail.
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