A note on assumptions about skolem functions

A Note on Assumptions about Skolem Functions
Hans J¨u rgen Ohlbach and Christoph Weidenbach
Max-Planck-Institut f¨u r Informatik
Im Stadtwald
66123Saarbr¨u cken,Germany
email:(ohlbach,weidenb)@mpi-sb.mpg.de
Journal of Automated Reasoning15(2):267–275
November10,1997
Abstract.Skolemization is not an equivalence preserving transformation.For the purposes of refutational theorem proving it is sufficient that Skolemization preserves satisfiability and unsatis-fiability.Therefore there is sometimes some freedom in interpreting Skolem functions in a particu-lar 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 offirst-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 unsatis-fiability preserving.But this is usually the only routinely applied transformation with this property.
For an existential quantification,Skolemization introduces a new function sym-bol 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))(1)
Φ(x)⊃C(x,f(x))(2) IfΦis 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))(3)
Φ(x)⊃C(x,f(x))(4)
2Hans J¨u rgen Ohlbach and Christoph Weidenbach
where one occurrence ofΦis 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.
SupposeΦ(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 supposeΦ(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 section3we
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 I N.Let denote the empty word.Then we define
the set of occurrences occ(Φ)of a formulaΦas follows:(i)the empty word is in occ(Φ)(ii)i.πis in occ(Φ)iffΦ=Ψ1∧...∧Ψn orΦ=Ψ1∨...∨Ψn,1≤i≤n andπ∈occ(Ψi)(iii)1.π(2.π)is in occ(Φ)iffΦ=∀xΨorΦ=∃xΨorΦ=¬Ψ
orΦ=Ψ⊃ΘorΦ=Ψ≡Θandπ∈occ(Ψ)(π∈occ(Θ)).Now ifπ∈occ(Ψ)
thenΨ| =ΨandΨ|i.π=Ψi|πwhereΨi is the i th subformula ofΨ(see above).
Intuitively,the polarity of some formulaΦ|π=ΨinΦis1ifΨoccurs below an
even number of(explicit or implicit)negation symbols,it is-1ifΨoccurs below
an odd number of negation symbols and it is0ifΨoccurs below an equivalence
symbol.We define the polarity pol(Φ,π)of a formulaΦ|πin a formulaΦby:
(i)pol(Φ, )=1(ii)pol(Φ,i.π)=pol(Φ|i,π)ifΦis a conjunction,disjunction,
quantifier formula orΦis an implication and i=2(iii)pol(Φ,i.π)=−pol(Φ|i,π)
ifΦis a negation orΦis an implication and i=1(iv)pol(Φ,i.π)=0ifΦis an
equivalence.
We also use occurrences to define the replacement of a formula inside another
formula.Φ[π/Ψ]withπ∈occ(Φ)is the formula obtained fromΦby replacingΦ|πinΦwithΨ.
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A Note on Assumptions about Skolem Functions3
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 formulaΦ|π,π∈occ(Φ),is in the scope of a universal quantifier,if there exists a prefixµofπsuch thatΦ|µ=∀xΘand pol(Φ,µ)=1orΦ|µ=∃xΘand
pol(Φ,µ)=−1or pol(Φ,µ)=0andΦ|µis an arbitrary quantificational formula.
It is in the scope of an existential quantifier,if there exists a prefixνofπsuch that
Φ|ν=∀xΘand pol(Φ,ν)=−1orΦ|ν=∃xΘand pol(Φ,ν)=1or pol(Φ,µ)=0
andΦ|µis an arbitrary quantificational formula.
LEMMA1.LetΦ|π=Ψwithπ∈occ(Φ)and pol(Φ,π)=1and M be an inter-pretation.Then
if M|=Φ∧(Ψ⊃Θ)then M|=Φ[π/Θ]
Proof.The proof is due to Loveland[2,Lemma1.5.1,p.40].
THEOREM2(Optimized Skolemization).LetΦ|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,...,x n be
the universally quantified variables which occur freely in∃y(Ψ∧Θ).In addition,
we assume the seriality condition|=Φ⊃∀x1,...,x n∃yΘ.Then
Φis satisfiable
iff
Φ[π/Ψ{y/f(x1,...,x n)}]∧∀x1,...,x nΘ{y/f(x1,...,x n)}is satisfiable
where f is a new n-place Skolem function.
Proof.“⇒”Assume M|=Φ.Since f is new toΦit is sufficient to construct an interpretation M which is like M,but in addition specifies an interpretation for f such that M |=Φ[π/Ψ{y/f(x1,...,x n)}]∧∀x1,...,x nΘ{y/f(x1,...,x n)}.Con-sider domain elements a1,...,a n as assignments for the universally quantified vari-ables x1,...,x n.If M[x1/a1,...,x n/a n]|=∃y(Ψ∧Θ),then there exists some b as assignment for y such that M[x1/a1,...,x n/a n,y/b]|=Ψ∧Θ.We choose b as val-
ue for f,i.e.f M (a1,...,a n)def=b.If M[x1/a1,...,x n/a n]|=∃y(Ψ∧Θ)we choose
f M (a1,...,a n)def=c,where M[x1/a1,...,x n/a n,y/c]|=Θ.Such a c always exists
by the seriality assumption M|=∀x1,...,x n∃yΘwhich implies,as f is new toΦ,
M |=∀x1,...,x n∃yΘ.Now by construction of f M we have M |=Φ∧(∃y(Ψ∧Θ)⊃Ψ{y/f(x1,...,x n)})and thus by Lemma1,M |=Φ[π/Ψ{y/f(x1,...,x n)}].
In addition,M |=∀x1,...,x n∃yΘ∧(∃yΘ⊃Θ{y/f(x1,...,x n)})and thus again
by Lemma1,M |=∀x1,...,x nΘ{y/f(x1,...,x n)}.
“⇐”Assume M|=Φ[π/Ψ{y/f(x1,...,x n)}]∧∀x1,...,x nΘ{y/f(x1,...,x n)}. Then we have:M|=Ψ{y/f(x1,...,x n)}⊃∃y(Ψ∧Θ)by choosing y/f M(a1,...,a n)
for y in∃y(Ψ∧Θ)and any assignment a1,...,a n of the x1,...,x n.Now by Lem-
ma1we conclude M|=Φ.
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4Hans J¨u rgen Ohlbach and Christoph Weidenbach
EXAMPLE3.We apply the optimized Skolemization to Pelletier’s[6]problem no29:
(1)∃x F(x)
(2)∃x G(x)
(3)¬[(∀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)∀x(F(x)⊃H(x))∧∀x(G(x)⊃J(x)))∨∀x,y((F(x)∧G(y))⊃(H(x)∧J(y)))
(5)¬(∀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 occur-ring 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=∃x(F(x)∧¬H(x))
(6)|2=∃x(G(x)∧¬J(x))
(6)|3=∃x(F(x)∧∃y(G(y)∧(¬H(x)∨¬J(y))))
(6)|312=∃y(G(y)∧(¬H(x)∨¬J(y)))
Theorem2is applicable to all these occurrences,because all occurrences have polarity1and 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 opti-mized Skolemization:
(7)¬H(a)∨¬J(b)∨¬H(c)∨¬J(d)
(8)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](version3.0,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 pro-cedure.In addition,the optimized Skolemization proof is shorter and has a lower proof complexity.
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A Note on Assumptions about Skolem Functions5 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 formulaΦof Theorem2is equivalent to the formula
∀x1,...,x n∃yΦ[π/(Ψ∧Θ)]∧∀x1,...,x n∃yΘ(5)
because|=Φ⊃∀x1,...,x n∃yΘ.Now(5)is equivalent to the formula
∀x1,...,x n∃y[Φ[π/Ψ]∧Θ](6)
This can be proved by techniques similar to those used to prove Theorem2.Even-tually standard Skolemization yields
∀x1,...,x n[Φ[π/Ψ{y/f(x1,...,x n)}]∧Θ{y/f(x1,...,x n)}](7)
which is exactly the result of Theorem2if the universal quantifiers are moved inside.However,we prefer the formulation of Theorem2,because we interpret ∀x1,...,x nΘ{y/f(x1,...,x n)}as a(stronger)definition of the Skolem function f.In addition,the formulation of Theorem2is compatible with the usual tech-
niques for clause normal form,e.g.anti-prenexing,whereas the above argumenta-tion requires to move quantifiers outwards.
3.D´e ja 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 formulaΦ⊃∃x B C(x B),which is the sorted formalization ofΦ⊃∃x(B(x)∧C(x)).In sorted logics,where all sorts are a priori assumed non-empty,it gets Skolemized toΦ⊃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 conditionΦanymore.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 Section2is an instance of Weidenbach’s approach to sort-ed logic.The unary predicates can be translated into sorts.This enables further simplifications of formula(4):
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6Hans J¨u rgen Ohlbach and Christoph Weidenbach
(4 )(∀x F H(x F)∧∀y G J(y G))∨∀x F,y G(H(x F)∧J(y G))
Since the sorts F and G are non-empty(see(1),(2)),(4 )is further simplified to
(4 )∀x F H(x F)∧∀y G J(y G)
Now a refutation of(1),(2),(4 ),(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 operators2and3[1]. The standard Kripke semantics of normal modal systems interprets the2-operator as a universal quantification over accessible worlds and the3-operator as an exis-tential quantification over accessible worlds.This semantics can be exploited to define a“relational”translation from modal to predicate logic.For example23P 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 the3-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 from3-operators become pos-itive unit clauses.From the2-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].
EXAMPLE4.We show the power of the optimized Skolemization by an exam-ple taken from modal logic KD45[1].In modal logic KD45the formula32P≡3232P is a theorem.The theorem can be translated intofirst-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 KD45are expressed by the following formulae:
(1)∀x∃y R(x,y)
(2)∀x,y,z(R(x,y)∧R(y,z)⊃R(x,z))
(3)∀x,y,z(R(x,y)∧R(x,z)⊃R(y,z))
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A Note on Assumptions about Skolem Functions7 Now we apply Theorem2to 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)∃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)))))
(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)))) There are the following occurrences in(4)and(5)which name existentially quan-tified 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 polarity1.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 Theorem2and 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 Theorem2 is applicable to all occurrences of the existential quantifiers.For example we start with the occurrence1of(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)∀y(R(a,y)⊃P(y)))∨∀y(R(b,y)⊃∀u(R(h(y),u)⊃P(u)))
(7)R(o,a)∧R(o,b)∧∀y R(y,h(y))
(8)∀x(R(o,x)⊃¬P(i(x)))∨∀x(R(o,x)⊃∀z(R(f(x),z)⊃¬P(g(x,z))))
(9)∀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 defi-nitions 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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8Hans J¨u rgen Ohlbach and Christoph Weidenbach
makes a proof of the theorem more complicated.The theorem prover OTTER (version3.0,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 notfind 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 (1 )∀x,y,z R(x,y:z)
(4 )∀x(R(o:a,x)⊃P(x))∨∀y(R(o:b,y)⊃∀z(R(y:h(y),z)⊃P(z)))
(5 )∀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 (1 )is the translation of(1),(2),(3),(4 )is the translation of(4)and(5 )is the translation of(5).The formulae(1 ),(4 ),(5 )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 occur-rence of the existential quantifier to the top-level of the formula.Instances of this optimized Skolemization have been used implicitly or explicitly in special appli-cations.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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