COMPLEXITY AND EXPRESSIVE POWER OF SECOND-ORDER
EXTENDED HORN LOGIC
FENG SHIGUANG, ZHAO XISHUN
Institute of Logic and Cognition, Sun Yat-Sen University, 510275 Guangzhou, P.R. China
E-mail: [email protected]; [email protected]
1. Introduction
Descriptive complexity is one of the central issues in finite model theory. While
computational complexity is concerned with the computational resources such as
time, space or the amount of hardware that are necessary to decide a property,
descriptive complexity asks for the logic resources that are necessary to define it.
One goal of descriptive complexity is to characterize complexity from the point of
view of logic by providing for each important complexity class, logical systems which
captures the complexity class, that is, whose expressive power (on finite structures)
coincides precisely with that class.
The first capturing result is Fagin’s theorem which says that existential secondorder logic captures NP[13]. Now there have been many capturing results for logic
systems and complexity classes[1, 15, 16, 19, 21]. For example, on ordered finite structures, deterministic transitive closure logic(FO (DTC)), transitive closure
logic(FO (TC)), inflationary fixed-point logic(FO (IFP)), and partial fixed-point
logic(FO (PFP)) captures LOGSPACE, NLOGSPACE, P, and PSPACE, respectively.
It’s well known that second-order logic (SO) captures PH, the polynomial hierarchy (see e.g. L. J. Stockmeyer [18]). More precisely, for each k, Σ1K − SO captures
1
the complexity class ΣP
k , where Σk − SO is the set of second-order sentences with
second-order quantifier prefix
∃R1,1 · · · ∃R1,m1 ∀R2,1 · · · ∀R2,m2 · · · Qk Rk,1 · · · Qk Rk,mk .
Grädel and others[5, 9, 10] had given elegant characterizations of P and NL by
using of fragments of second-order logic. For example, on ordered structures,
second-order Horn logic (SO − HORN) captures P, whereas second-order Krom
logic (SO − KROM) captures NLOGSPACE.
In quantified Boolean formulas (QBF), SO − HORN and SO − KROM have their
counterparts, quantified Horn formulas (QHORN) and quantified 2-CNF formulas
(Q2 − CNF). The evaluation problem for QHORN and Q2 − CNF have been proved
to be poly-time solvable [7, 23, 24]. In QBF, QHORN (resp. Q2 − CNF) is generalized to QEHORN (resp. QE2 − CNF), the class of quantified extended Horn
formulas, in which each clause contains at most one positive existential literal (resp.
each clause contains at most two existential literals). It has been proved that the
evaluation problem for QEHORN (resp. QE2 − CNF) remains PSPACE-complete
[8]. However, when we fix the quantifier prefix type (i.e. the number of alternation
of existential and universal quantifiers), it becomes co − NP-complete [27].
1
2
FENG SHIGUANG, ZHAO XISHUN
In this paper we will introduce second-order Extended revised Horn logic
(SO − EHORNr ) and show that SO − EHORNr captures co − NP on ordered finite
structures. The paper is organized as follows: In section 2, we recall some notations
and previous results. In section 3, SO − EHORNr is defined and we prove that for
each formula in SO − EHORNr , the set of models of the formula is in co − NP. In
section 4, we prove the capturing result of SO − EHORNr . We conclude this paper
in section 5.
2. Preliminaries
In this paper we restrict ourself to ordered finite structures. For a vocabulary τ ,
a finite structure A of τ is said to be ordered , if A is a (τ ∪ {<, succ, min, max})structure, and the {<, succ, min, max}-structure obtained from A by forgetting the
interpretations of the symbols in τ is an ordering.
Second-order logic(SO), is an extension of first order logic which allows to quantify over relation variables. Any second-order formula can be transformed into an
equivalent formula with all second-order quantifiers in front[1]. If all these front
second-order quantifiers are existential then we say the formula is second-order
existential. We use Σ11 to denote the set of all second-order existential formulas.
Fagin’s theorem states that Σ11 captures NP. However, on ordered finite structures,
NP is captured by Σ11 formulas with the following form:
∃P1 · · · ∃Pm ∀x1 · · · ∀xt (C1 ∧ · · · ∧ Cn ),
where∀x1 · · · ∀xt (C1 ∧ · · · ∧ Cn ) is a first-order formula, each Ci is a disjunction of
of atomic or negated atomic formulas[9, 20].
Next we recall some notions and results in quantified Boolean formulas (QBF).
In this paper we are interested in QBF in the following form:
Φ = Q1 x1 · · · Qn xn ϕ,
where Qi ∈ {∀, ∃}, ϕ is a CNF formula over Boolean variables x1 , · · · , xn . A literal
xi or ¬xi is called universal (resp. existential) if Qi is ∀ (resp. ∃). If every clause
in ϕ contains at most one positive literal (resp. positive existential literal) then
Φ is called a quantified Horn formula(QHORN)(resp. quantified extended Horn
formula (QEHORN)). If every clause in ϕ contains at most two literals (resp. two
existential literals) then Φ is called a quantified 2 − CNF formula(Q2 − CNF)(resp.
quantified extended 2 − CNF formula (QE2 − CNF)).
It is well-known that the evaluation problem for QBF (QSAT) is PSPACEcomplete [7, 28]. QSAT for QHORN and Q2 − CNF becomes poly-time solvable,
whereas it remains PSPACE-complete for QEHORN and QE2 − CNF. However,
for each fixed m ≥ 1, QSAT is co − NP-complete for QEHORN and QE2 − CNF
formulas with prefix type ∀x1 ∃y 1 · · · ∀xm ∃y m (here xi , y i are sequence of Boolean
variables)[3, 4].
3. Extended HORN Logic
First we recall the SO − HORN logic (see e.g. [9]).
Definition 1. Second-order Horn logic, denoted by SO − HORN, is the set of
second-order sentences of the form
Q1 R1 · · · Qm Rm ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Cn ),
COMPLEXITY AND EXPRESSIVE POWER OF SECOND-ORDER EXTENDED HORN LOGIC3
where Qi ∈ {∀, ∃}, Ri are relation symbols and Ci are Horn clauses with respect to
R1 , · · · , Rm , more precisely, each Ci is an implication of the form
β1 ∧ · · · ∧ βq → H,
where
(1) For each j ∈ {1, . . . , q}, either βj is Rx in which R is from{R1 , . . . , Rm }, or
βj is P x or ¬P x in which P is from the underling vocabulary.
(2) H is either an atomic formula Rk z in which Rk is from {R1 , . . . , Rm }, or the
Boolean constant ⊥ (for false).
It is well-known that SO − HORN captures Ptime over ordered finite structures[10].
The first-order part of SO − HORN sentence is an universal Horn sentences. However, we can enlarge SO − HORN a little bit without increase its expressive power.
In Definition 1, if we replace condition (1) by
(1’) For each j ∈ {1, . . . , q}, either βj is Rx or ∀yRyz in which R is from{R1 , . . . , Rm },
or βj is P x or ¬P x in which P is from the underling vocabulary.
then we call the logic second-order revised Horn Logic, denoted as SO − HORNr .
Example 2. Given a structure < A, S, f, a >. S is a unary relation over the
universe A. f is a binary function. a is a constant. We call a set X ⊆ A is
generated by S and closed under f means that S ⊆ X and for any x, y ∈ A, if
x, y ∈ X then f (x, y) ∈ X. X is the smallest if X ⊆ Y for any set Y that is
generated by S and closed under f . GEN[10, 29] is the decision problem for the set
{< A, S, f, a > |a is in the smallest set X generated by S and closed under f }
GEN is P-complete. The SO − HORN formula
∃R∀x∀y ((S (x) → R (x)) ∧ (R (x) ∧ R (y) → R (f (x, y))) ∧ (R (a) → ⊥))
defines the complement of GEN which is also P-complete.
Proposition 3. SO − HORNr still captures Ptime on ordered structures.
Proof. Obviously, SO − HORN ⊆ SO − HORNr . The proof for the data complexity
of SO − HORNr is similar to Theorem 6 below, we will give a short explanation
after the proof of Theorem 6.
In the definition of SO − HORN, if we only demand that the first-order clauses
are Horn with respect to only existentially quantified second-order relations, then
we denote the logic as SO − EHORN. More precisely,
Definition 4. Second-order Extended Horn logic, denoted by SO − EHORN, is
the set of second-order sentences of the form
Q1 R1 · · · Qm Rm ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Cn ),
where Qi ∈ {∀, ∃}, Ri are relation symbols and Ci are Horn clauses with respect
to the relation symbols bounded by ∃ quantifier, more precisely, each Ci is an
implication of the form
α1 ∧ · · · αt ∧ β1 · · · ∧ βq → H,
where
(1) For each j ∈ {1, . . . , q}, βj is Rx in which R is from{R1 , . . . , Rm }and bounded
by ∃ quantifier.
4
FENG SHIGUANG, ZHAO XISHUN
(2) For each s ∈ {1, . . . , t}, αs is P x or ¬P x in which P is from the underling
vocabulary or from{R1 , . . . , Rm } and bounded by ∀ quantifier.
(3) H is either an atomic formula Rk z in which Rk is from {R1 , . . . , Rm } and
bounded by ∃ quantifier, or the Boolean constant ⊥ (for false).
Similar to the definition SO − HORNr , if we replace the condition (1) in Definition 4 by
(1’) For each j ∈ {1, . . . , q}, βj is Rx or ∀yRyz in which R is from {R1 , . . . , Rm }
and bounded by ∃ quantifier.
then we call the logic Second-order Extended revised Horn logic, denoted by
SO − EHORNr
We give an example below to show the expressive power of SO − EHORNr .
Example 5. The decision problem for the set {φ|φ is an unsatisfiable CNF formula}
is co − NP complete[7]. We encode φ via the structure Aφ =< A, P, N >[20]. The
universe A is a finite set of natural numbers which represent clauses and variables.
The relation P (i, j) means that the jth variable occurs positively in the ith clause
and N (i, j) means that the jth variable occurs negatively in the ith clause. For
example, the CNF formula
φ = ((p1 ∨ p3 ∨ ¬p4 ) ∧ (p2 ∨ ¬p3 ) ∧ (p1 ∨ p5 ))
can be encoded as
Aφ =< {1, 2, 3, 4, 5} , P, N >
P = {(1, 1) , (1, 3) , (2, 2) , (3, 1) , (3, 5)}
N = {(1, 4) , (2, 3)}
The formula
∀X∃Y ∀x∀y (∃z¬Y (z) ∧ (¬Y (x) ∧ P (x, y) ∧ → ¬X (y)) ∧ (¬Y (x) ∧ N (x, y) ∧ → X (y)))
means that for any valuation X, where X (i) holds iff the ith variable is True under
this valuation, there exists a set of clauses Y , such that any clause that doesn’t
occur in Y is False under this valuation. The above formula is equivalent to
∀X∃Y ∀x∀y ((∀zY (z) → ⊥) ∧ (X (y) ∧ P (x, y) ∧ → Y (x)) ∧ (¬X (y) ∧ N (x, y) ∧ → Y (x)))
which is a SO − EHORNr formula and for any CNF formula φ, Aφ satisfies this
formula iff φ is unsatisfiable.
Theorem 6. If Φ ∈ SO − EHORNr , then the set of finite models of Φ is in co-NP.
Proof. Given any finite structure A of appropriate vocabulary τ , we shall reduce
the problem deciding whether A |= Φ to the problem for QEHORN formulas.
Suppose Φ = ∀P1 ∃R1 · · · ∀Pm ∃Rm ∀x1 · V
· · ∀xs (CV1 ∧ · · · ∧ Cn ).
Replace ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Cn ) by a∈As ( 1≤i≤n Ci )[a], then substitute in
each clause the formulas that do not involves P1 , R1 , · · · , Pm , Rm by their truth
values in A.
In each clauseV
Ci [a], for every (∀yRi yz)[a], if it occurs in the body of Ci [a], then
we replace it by b∈Ak (Rj yz[ba]), here k is the length of y.
For each relation symbol R ∈ {P1 , R1 , · · · , Pm , Rm }, and each sequence a =
a1 , · · · , ak (here k is the arity of R) of elements in A, we consider the atom Ra as a
proposition variable. Let Pi (resp. Ri ) be a k-ary relation symbol, and a1 , · · · , a|A|k
COMPLEXITY AND EXPRESSIVE POWER OF SECOND-ORDER EXTENDED HORN LOGIC5
be an enumeration of Ak . Then we replace ∀Pi (resp. ∃Ri ) by ∀Pi a1 · · · ∀Pi a|A|k
(resp. ∃Ri a1 · · · ∃Ri a|A|k ).
The resulting formula is denoted by f (Φ, A), which is clearly in QEHORN.
The following two facts are not hard to see:
∗
∗
∗
∗
∗
∗
(1) For any
(A, P1∗V
, R1∗ , · · ·V, Pm
, Rm
) |=
V relations P1 , R1 , · · · , Pm , R∗m on∗ A, we have
∗
∗
∀x1 · · · ∀xs ( 1≤i≤n Ci ) if and only if (A, P1 , R1 , · · · , Pm , Rm ) |= a∈At ( 1≤i≤n Ci )[a].
(2) Each relation R∗ ⊆ Ak determines uniquely a truth assignment t on Boolean
variables Ra, a ∈ Ak as follows: t(Ra) = 1 if and only if a ∈ R∗ . Conversely, for
each truth assignment t on Ra, a ∈ Ak . we can define uniquely a relation R∗ by
a ∈ R∗ if and only if t(Ra) = 1.
Therefore, A |= Φ if and only if f (Φ, A) is true. Since Φ is fixed, the size
of f (Φ, A) is polynomial in the cardinality of A, and the number of alternations
of quantifiers in f (Φ, A) is always m. From the results in [6], we know that the
satisfiability problem for f (Φ, A) is in co-NP. The proof completes.
In the above proof, suppose Φ is in SO − HORNr , then f (Φ, A) is clearly in
QHORN. Because the satisfiability problem for QHORN is tractable, the set of
finite models of Φ is in Ptime[7]. It follows that SO − HORNr captures Ptime since
SO − HORN ⊆ SO − HORNr .
4. Descriptive Complexity
In this chapter we want to show that on ordered structures SO − EHORNr captures co-NP. By Theorem 6, it is sufficient to show the following: Any co-NP time
decidable isomorphism-closed class of ordered structures (over an appropriate vocabulary) is definable in SO − EHORNr . By Fagin’s theorem, it is enough to prove
that for any Σ11 formulas Φ, ¬Φ is equivalent to a SO − EHORNr formula on ordered
structures.
Theorem 7. Any Π11 formula is equivalent to a Π12 − EHORNr formula on ordered
finite structures.
Proof. Consider an arbitrary formula
Φ := ∃P1 · · · ∃Pm ∀x1 · · · ∀xt (C1 ∧ · · · ∧ Cn ),
where Ci is a clause αi,1 ∨ · · · ∨ αi,si in which each αi,j is an atomic formula or its
negation.
Let R0 , R1 , · · · , Rn be new t-ary relation symbols. We associate to each clause
Ci = αi,1 ∨ · · · ∨ αi,si the following si clauses
Di,1 := (αi,1 ∧ Ri−1 x → Ri x), · · · , Di,si := (αi,si ∧ Ri−1 x → Ri x),
Define

Ψ := ∀P1 · · · ∀Pm ∃R0 ∃R1 · · · ∃Rn ∀x1 · · · ∀xt R0 x ∧ (∃y¬Rn y) ∧

^
^
Di,j  .
1≤i≤n 1≤j≤si
Next we show that ¬Φ and Ψ are equivalent on finite structures.
6
FENG SHIGUANG, ZHAO XISHUN
∗
Suppose A is a model such that A 6|= Ψ. Then for some relations P1∗ , · · · , Pm
,
we have


^
^
∗
(A, P1∗ , · · · , Pm
) 6|= ∃R0 ∃R1 · · · ∃Rn ∀x1 · · · ∀xt R0 x ∧ (∃y¬Rn y) ∧
Di,j  .
1≤i≤n 1≤j≤si
R0∗
R1∗ , · · ·
At first we define
:= A . Then we construct relations
, Rn∗ on A as
t
follows. Consider an arbitrary a ∈ A .
∗
Case 1. (A, P1∗ , · · · , Pm
) |= (C1 ∧ · · · ∧ Cn )[a]. Then we put a into every Ri∗ ,
i = 1, · · · , n.
∗
Case 2. (A, P1∗ , · · · , Pm
) |= ¬(C1 ∧ · · · ∧ Cn )[a]. Let i be the smallest number
∗
∗
such that (A, P1 , · · · , Pm ) |= ¬Ci [a]. Then we just put a into Rj∗ for j < i.
By the definition of clauses Di,j , it is not hard to see that


^
^
∗
Di,j  .
(A, P1∗ , · · · , Pm
, R0∗ , · · · , Rn∗ ) |= ∀x1 · · · ∀xt R0 x ∧
t
1≤i≤n 1≤j≤si
∗
We want to show (A, P1∗ , · · · , Pm
) |= ∀x1 · · · ∀xt (C1 ∧ · · · ∧ Cn ). Suppose by
∗
∗
contrary that (A, P1 , · · · , Pm ) |= ¬(C1 ∧ · · · ∧ Cn )[a] for some a of elements in A.
Then from the definition of Ri∗ we get a 6∈ Rn∗ . Thus,


∗
(A, P1∗ , · · · , Pm
, R0∗ , · · · , Rn∗ ) |= ∀x1 · · · ∀xt R0 x ∧ (∃y¬Rn y) ∧
^
^
Di,j  .
1≤i≤n 1≤j≤si
∗
) |= ∀x1 · · · ∀xt (C1 ∧ · · · ∧ Cn ).
This is a contradiction. Therefore, (A, P1∗ , · · · , Pm
Hence A |= Φ.
Conversely, suppose A is a structure such that A |= Ψ. We want to show A 6|= Φ.
∗
) |= ∀x1 · · · ∀xt (C1 ∧· · ·∧Cn ) for some relations
Suppose by contrary (A, P1∗ , · · · , Pm
∗
on A. Since A |= Ψ, there are relations R0∗ , R1∗ , · · · , Rn∗ such that
P1∗ , · · · , Pm


^
^
∗
(A, P1∗ , · · · , Pm
, R0∗ , · · · , Rn∗ ) |= ∀x1 · · · ∀xt R0 x ∧ (∃y¬Rn y) ∧
Di,j 
1≤i≤n 1≤j≤si
R0∗
t
Rn∗
t
Clearly,
must be the full relation A , whereas
6= A . On the other hand, for
∗
each a and each Ci there must be some j such that αi,j [a] is true in (A, P1∗ , · · · , Pm
),
∗
∗
∗
∗
thus a must belong to every Ri , i = 1, · · · , n. Consequently, R0 , R1 , · · · , Rn are all
the full relation At . This contradicts that Rn∗ 6= At . The proof completes.
Corollary 8. SO − EHORN r captures co-NP.
In the following we show that SO − EHORN and SO − EHORNr have the same
expressive power on ordered structures. From Theorem 6 and Theorem 7, it’s suffice
to prove any formula of Π12 − EHORNr is equivalent to a formula of SO − EHORN.
Proposition 9. SO − EHORN and SO − EHORN r have the same expressive
power on ordered structures.
Proof. SO − EHORN ⊆ SO − EHORNr is obviously.
Consider a Π12 − EHORNr formula
Φ := ∀P1 · · · ∀Pn ∃R1 · · · ∃Rm ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Ck ),
COMPLEXITY AND EXPRESSIVE POWER OF SECOND-ORDER EXTENDED HORN LOGIC7
We fix P1 , . . . , Pn , then the sentence φ = ∃R1 · · · ∃Rm ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Cn ) is
a sentence of SO − HORNr . We have known that SO − HORN and SO − HORNr
coincide on ordered structures from Proposition 2, so there exists a formula φ0 ∈
SO − HORN, where φ0 = ∃R1 · · · ∃Rm0 ∀x1 · · · ∀xs0 (C1 ∧ · · · ∧ Ck0 ) such that φ and
φ0 are equivalent.
Let Φ0 = ∀P1 · · · ∀Pn φ0 , A is an ordered structure. A |= Φ ⇐⇒ (A, P1∗ , . . . , Pn∗ ) |=
φ for any P1∗ , . . . , Pn∗ over A ⇐⇒ (A, P1∗ , . . . , Pn∗ ) |= φ0 for any P1∗ , . . . , Pn∗ over A
⇐⇒ A |= Φ0 . Φ and Φ0 are equivalent, and Φ0 is a formula of SO − EHORN.
Corollary 10. SO − EHORN captures co-NP.
The above results that we get are over ordered structures. It seems that the
ordering is necessary in these proofs. Without this requirement we show that
SO − EHORNr are strictly more expressive than SO − EHORN for any finite structures.
Definition 11. We say a formula φ is preserved under substructures, if A |= φ,
then for any substructure B of A, B |= φ.
Lemma 12. If a formula φ is preserved under substructures, then ∃P φ and ∀P φ
are preserved under substructures, where P is a relation symbol in φ.
Proof. Suppose φ is a formula preserved under substructures, A is a structure of
an appropriate vocabulary.
If A |= ∃P φ, then there is a P ∗ over A such that (A, P ∗ ) |= φ. For any substructure B of A, because φ is preserved under substructures, then (B, P ∗ |B) |= φ,
where P ∗ |B is the restriction of P ∗ on B, i.e. B |= ∃P φ. So ∃P φ is preserved
under substructures.
If A |= ∀P φ, then (A, P ∗ ) |= φ for any relation P ∗ over A. For any substructure
B of A and any relation P ∗ over B, because φ is preserved under substructures, we
have (B, P ∗ ) |= φ, i.e. B |= ∀P φ. So ∀P φ is preserved under substructures.
Proposition 13. SO − EHORN are preserved under substructures.
Proof. Any formula of SO − EHORN has the form
Q1 R1 · · · Qm Rm ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Cn ),
where Qi ∈ {∀, ∃} and (C1 ∧· · ·∧Cn ) is a first-order formula that doesn’t contain any
quantifier. We see that ∀x1 · · · ∀xs (C1 ∧· · ·∧Cn ) is an universal formula, from[22]we
know that ∀x1 · · · ∀xs (C1 ∧ · · · ∧ Cn ) is preserved under substructures.Using the
lemma above we can get by induction that every formula of SO − EHORN are
preserved under substructures.
However,∃R∀x((P x → Rx) ∧ (∃y¬Ry)) is a formula of SO − HORNr , but it is
not preserved under substructures.
Corollary 14. Over any finite structures, SO − EHORNr (resp. SO − HORNr ) are
strictly more expressive than SO − EHORN(resp. SO − HORN).
5. Conclusion
We introduce SO − EHORN, an extended version of SO − HORN, and the revised version SO − HORNr (resp. SO − EHORNr ) of SO − HORN(resp. SO − EHORN).
We prove that SO − HORNr captures Ptime, SO − EHORNr captures co − NP.
8
FENG SHIGUANG, ZHAO XISHUN
SO − HORN and SO − HORNr , SO − EHORN and SO − EHORNr are coincide
on ordered finite structures. In fact, from Theorem 6 and Theorem 7, every
SO − EHORNr formula is equivalent to a Π12 −EHORN formula. Thus Π12 −EHORN
captures co − NP. The ordering is essential in the proof. We proved that
SO − EHORNr (resp. SO − HORNr ) are strictly more expressive than SO − EHORN
(resp. SO − HORN) over any finite structures.
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