Complete Problems of Propositional Logic for the Exponential Hierarchy

Martin Lück

Institut für Theoretische Informatik Leibniz Universität Hannover, DE [email protected]

Abstract. Large complexity classes, like the exponential time hierarchy, received little attention in terms of finding complete problems. In this work a generalization of propositional logic is investigated which fills this gap with the introduction of Boolean higher-order quantifiers or equivalently Boolean Skolem functions. This builds on the important results of Wrathall and Stockmeyer regarding complete problems, namely QBF and QBF푘, for the polynomial hierarchy. Furthermore it generalizes the Dependency QBF problem introduced by Peterson, Reif and Azhar which is complete for NEXP, the first level of the exponential hierarchy. Also it turns out that the hardness results do not collapse at the consideration of conjunctive and disjunctive normal forms, in contrast to plain QBF.

1 Introduction

The class of problems decidable in polynomial space, PSPACE, can equivalently be defined as the class AP, i.e., via alternating machines with polynomial P P runtime and no bound on the alternation number. The classes 훴푘 and 훱푘 of the polynomial hierarchy are then exactly the restrictions of AP to levels of bounded alternation [5]. The problem of quantified Boolean formulas, often called QBF resp. QBF푘, is complete for these classes. The subscript 푘 denotes the number of allowed quantifier alternations of a qbf in prenex normal form, whereas QBF imposes no bound on quantifier alternations. For this correspondence Stockmeyer called QBF the 휔-jump of the bounded QBF푘 variants, and similar the class PSPACE the 휔-jump of the polynomial hierarchy [15], in reference to the arithmetical hierarchy. On the scale of exponential time the alternation approach leads to discrep- ancies regarding natural complete problems. Unbounded alternations in exponential time (AEXP) leads to the same class as exponential space, in symbols AEXP = EXPSPACE, and therefore EXPSPACE is analogously the 휔-jump of exponential time classes with bounded alternations [5]. Complete problems arXiv:1602.03050v2 [cs.CC] 27 May 2016 for EXPSPACE are rare, and often artificially constructed, frequently just succinctly encoded variants of PSPACE-complete problems [1,7, 10]. If the number of machine alternations is bounded by a polynomial then this leads to the class AEXP(poly) which in fact lies between the exponential time hierarchy and its 휔-jump. 2 M. Lück

In this paper a natural complete problem is presented, similar to QBF푘, which allows quantification over Boolean functions and is complete for the levels ofthe exponential time hierarchy. The first appearance of such Boolean formulas with quantified Boolean functions was in the work of Peterson, Reif and Azhar who modeled games of imperfect information as a problem they called DQBF or Dependency QBF [13]. The basic idea is that in a game the player ∃ may or may not see the whole state that is visible to player ∀, and hence her next move must depend only on the disclosed information. The existence of a winning strategy in a game can often be modeled as a formula with a prefix of alternating quantifiers corresponding tothe moves. This naturally fits games where all information about the state ofthe game is visible to both players, as quantified variables always may depend on each previously quantified value. Any existentially quantified proposition 푥 can equivalently be replaced by its Skolem function which is a function depending on the ∀-quantified propositions to the left of 푥. To model imperfect information in the game, all one has to do is now to restrict the arguments of the Skolem function. For first-order predicate logic several formal notions have been introduced to accommodate this semantics, e.g., Henkin’s branching quantifiers (see [4]) or Hintikka’s and Sandu’s Independence Friendly Logic [9].

Contribution. The presented problem is a generalization of the QBF problem where Skolem functions of variables are explicit syntactical objects. This logic will be called QBSF as in Quantified Boolean Second-order Formulas. It is shown that this introduction of function quantifiers to QBF (reminding of the step from first-order predicate logic to second-order predicate logic) yields enough expressive power the encode alternating quantification of exponentially large words. The problem of deciding the truth of a given QBF with higher-order quantifiers is complete for the class AEXP(poly), but has natural complete fragments for every level of the exponential hierarchy. The complexity of the problem is classified for several fragments, namely bounded numbers of function quantifiers and proposition quantifiers, as wellas the restriction to formulas where function variables only occur as the Skolem functions of quantified propositions, i.e., always with the same arguments. The latter fragment is used as an alternative hardness proof for the original DQBF problem by Peterson, Reif and Azhar [13].

2 Preliminaries

The reader is assumed to be familiar with usual notions of Turing machines (TMs) and complexity classes, especially in the setting of alternation introduced by Chandra, Kozen and Stockmeyer [5]. In accordance to the original definition of alternating machines (ATMs) we distinguish them by the type of their initial state. We abbreviate alternating Turing machines that start in an existential Complete Problems for the Exponential Hierarchy 3

state as 훴 type machines (훴-ATMs), and those which start in a universal state as 훱 type machines (훱-ATMs). We define EXP and NEXP as the classes of problems which are decidable by a (non-)deterministic machine in time 2푝(푛) for a polynomial 푝.

Definition 1. For 푄 ∈ {훴, 훱} 푔(푛) ≥ 1, define ATIME푄 (푡(푛), 푔(푛)) as the class of all problems 퐴 for which there is a 푄-ATM deciding 퐴 in time 풪(푡(푛)) with at most 푔(푛) alternations. The number of alternations is the maximal number of transitions between universal and existential states or vice versa that 푀 does on inputs of length 푛, counting the initial configuration as first alternation. A polynomial time 훴-ATM P P (훱-ATM) with 푔(푛) alternations is also called 훴푔 -machine (훱푔 -machine). For 푝(푛) E exponential time, i.e., 2 for al polynomial 푝, we analogously write 훴푔 resp. E 훱푔 . Definition 2 ([5]). ⋃︁ AEXP := ATIME훴 (푡, 푡) , 푡∈2푛풪(1) ⋃︁ AEXP(poly) := ATIME훴 (푡, 푝) . 풪(1) 푡∈2푛 푝∈푛풪(1)

In this work we further require the notion of oracle Turing machines. An oracle Turing machine is an ordinary Turing machine which additionally has access to an oracle language 퐵. The machine queries 퐵 by writing an instance 푥 on a special oracle tape and moving to a query state 푞?. But then instead of 푞? itself, one of two states, say, 푞+ and 푞−, is assumed instantaneously to identify the answer if 푥 ∈ 퐵 or not. There is no bound on the number of queries during a computation of an oracle machine, i.e., the machine can erase the oracle tape and ask more questions. If 퐵 is a language, then the usual complexity classes P, NP, NEXP etc. are generalized to P퐵, NP퐵, NEXP퐵 etc. where the definition is just changed from ordinary Turing machines to corresponding oracle machines with oracle 퐵. If 풞 풞 ⋃︀ 퐵 is a class of languages, then P := 퐵∈풞 P and so on. To classify the complexity of the presented decision problems we require some standard definitions. Definition 3. A logspace-reduction from a language 퐴 to a language 퐵 is a function 푓 that is computable in logarithmic space such that 푥 ∈ 퐴 ⇔ 푓(푥) ∈ 퐵. log log If such 푓 exists then write 퐴 ≤m 퐵. Say that 퐵 is ≤m -hard for a complexity log log log class 풞 if 퐴 ∈ 풞 implies 퐴 ≤m 퐵, and 퐵 is ≤m -complete for 풞 if it is ≤m -hard for 풞 and 퐵 ∈ 풞. Definition 4 (The Polynomial Hierarchy [15]). The levels of the polynomial hierarchy are defined inductively as follows, where 푘 ≥ 1: 4 M. Lück

P P P – 훴0 = 훱0 = 훥0 := P. P P P P 훴푘−1 P 훴푘−1 P 훴푘−1 – 훴푘 := NP , 훱푘 := coNP , 훥푘 := P . Definition 5 (The Exponential Hierarchy [11, 14]). The levels of the exponential hierarchy are defined inductively as follows, where 푘 ≥ 1: E E E – 훴0 = 훱0 = 훥0 = EXP. P P P E 훴푘−1 E 훴푘−1 E 훴푘−1 – 훴푘 := NEXP , 훱푘 := coNEXP , 훥푘 := EXP . Theorem 6 ([5]). For all 푘 ≥ 1:

P ⋃︁ 훴푘 = ATIME훴 (푝, 푘) , 푝∈푛풪(1) P ⋃︁ 훱푘 = ATIME훱 (푝, 푘) . 푝∈푛풪(1) The next two lemmas characterize the classes of the exponential hierarchy similar to the characterization of the polynomial hierarchy in [15, 16]. The proofs are rather straightforward adaptions of the characterization of the polynomial hierarchy. First, it is possible to reduce a language recognized in alternating exponential time down to a language with deterministic polynomial time complexity by introducing additional word quantifiers. These words roughly correspond to the “choices” of an encoded alternating machine, hence for the polynomial hierarchy words of polynomial length are quantified. To encode machines deciding problems E E in 훴푘 or 훱푘 we require, informally spoken, large word quantifiers, i.e., we quantify words of exponential length w. r. t. to the input.

E 푛풪(1) Lemma 7. For 푘 ≥ 1, 퐴 ∈ 훴푘 if and only if there is 푡 ∈ 2 and 퐵 ∈ P s. t.

푥 ∈ 퐴 ⇔ ∃푦1∀푦2 ... a푘푦푘 : ⟨푥, 푦1, . . . , 푦푛⟩ ∈ 퐵,

where a푘 = ∀ for even 푘 and a푘 = ∃ for odd 푘, and all 푦푖 have length bounded in 푡(|푥|). E Proof. We have to show that for all 푘 ≥ 1 it is 퐴 ∈ 훴푘 if and only if there is 풪(1) 푡 ∈ 2푛 , 퐵 ∈ P s. t.

푥 ∈ 퐴 ⇔ ∃푦1∀푦2 ... a푘푦푘 : ⟨푥, 푦1, . . . , 푦푛⟩ ∈ 퐵,

where a푘 = ∀ for even 푘 and a푘 = ∃ for odd 푘, and all 푦푖 have length bounded in 푡(|푥|). “⇐”: Define {︁ ⃒ }︁ 푡(|푥|) ⃒ 퐷 := ⟨0 , 푥, 푦1⟩ ⃒ ∀푦2 ... a푘푦푘 : ⟨푥, 푦1, . . . , 푦푛⟩ ∈ 퐵 ,

푡(|푥|) where 0 is the string consisting of 푡(|푥|) zeros and the quantified 푦푖 are P length-bounded by 푡(|푥|). Then 퐷 ∈ 훱푘−1, and the algorithm that guesses a 푦1 P 훱푘−1 E of length ≤ 푡(|푥|) and queries 퐷 as oracle witnesses that 퐴 ∈ NEXP = 훴푘 . Complete Problems for the Exponential Hierarchy 5

“⇒”: Let 퐴 be decided by some non-deterministic Turing machine 푀 with P 푝(푛) oracle 퐶 ∈ 훴푘−1. Assume that 푀 has runtime 푡(푛) = 2 for some polynomial 푝. Consider now words of the form 푧 = ⟨푑, 푞, 푎⟩ of length 풪(푡2(|푥|)) where 푑 encodes 푡(푛) non-deterministic choices in a computation of 푀, 푞 encodes the oracle questions asked, and 푎 encodes the answers used by 푀. Then 푥 ∈ 퐴 if and only if there is such a word 푧 = ⟨푑, 푞, 푎⟩ s. t. 푀 accepts on the computation encoded by the choices 푑, and 푎 are actually the correct answers of the oracle 퐶 to the queries in 푞. With given ⟨푥, 푧⟩ = ⟨푥, 푑, 푞, 푎⟩ the encoded computation of 푀 on the path 푑 can be simulated deterministically in time polynomial in |푧|. With given ⟨푥, 푧⟩, also the problem of determining whether the answers 푎 for the queries 푞 are correct P 퐶 훴푘−1 P for the oracle 퐶 is in P ⊆ P ⊆ 훴푘 . Therefore the set of all tuples ⟨푥, 푧⟩ ′ P which fulfill both properties, call it 퐶 , is in 훴푘 . By the quantifier characterization P ′ of 훴푘 it holds that ⟨푥, 푧⟩ ∈ 퐶 if and only if ∃푦1 ... a푘푦푘 : ⟨푥, 푧, 푦1, . . . , 푦푘⟩ ∈ 퐵 for some set 퐵 ∈ P and polynomially bounded, alternating quantifiers [15, 16]. But then 푥 ∈ 퐴 ⇔ ∃⟨푧, 푦1⟩∀푦2 ... a푘푦푘 : ⟨푥, 푧, 푦1, 푦2, . . . , 푦푘⟩ ∈ 퐵 quantifiers with length bounded exponentially in |푥|.

We next state the known correspondence between the classes of the exponential hierarchy (which are defined via oracle machines) and the alternating time classes by the following lemma. It can be seen as the exponential equivalent of Theorem6.

Lemma 8. For all 푘 ≥ 1:

E ⋃︁ 훴푘 = ATIME훴 (푡, 푘) , 푡∈2푛풪(1) E ⋃︁ 훱푘 = ATIME훱 (푡, 푘) . 푡∈2푛풪(1)

E E Proof. We show only the 훴푘 case as it can easily be adapted to the 훱푘 case. For “⊆”, apply the foregoing Lemma7. Use an alternating machine to guess the exponentially long quantified words and check in deterministic exponential time 푛풪(1) if the resulting word is in 퐵. Now to “⊇”. Let 푡 ∈ 2 s. t. 퐴 ∈ ATIME훴 (푡, 푘). {︀ 푡(|푥|) ⃒ }︀ P Then 퐵 := ⟨푥, 0 ⟩ ⃒ 푥 ∈ 퐴 is in 훴푘 , therefore

푡(|푥|) 푥 ∈ 퐴 ⇔ ∃푦1 ... a푘푦푘 : ⟨푥, 0 , 푦1, . . . , 푦푘⟩ ∈ 퐶 푡(|푥|) ⇔ ∃⟨푦0, 푦1⟩∀푦2 ... a푘푦푘 :(푦0 = 0 ) and ⟨푥, 푦0, 푦1, . . . , 푦푘⟩ ∈ 퐶 ′ ⇔ ∃⟨푦0, 푦1⟩∀푦2 ... a푘푦푘 : ⟨푥, 푦0, 푦1, . . . , 푦푘⟩ ∈ 퐶 for some 퐶, 퐶′ ∈ P and alternating quantifiers which are exponentially bounded E in |푥|. By the previous lemma then it holds 퐴 ∈ 훴푘 . Orponen gave a characterization of the exponential hierarchy via an indirect simulation technique [12]. He introduced it primarily due to its non-relativizing 6 M. Lück

nature (while direct simulation relativizes), however it also allows to use polynomial time machines to characterize languages with much higher complexity. Informally spoken, the whole computation of an exponential time machine is encoded into quantified oracles (instead of exponentially long words), which are then verified bit for bit, but in parallel, by an alternating oracle machine withonly polynomial runtime. Baier and Wagner [2] investigated more generally so-called type 0, type 1 and type 2 quantifiers, improving Orponen’s result. These oracle characterizations play a major role for classifying the complexity of QBSF, the logic introduced in this paper, as they translate to the quantification of Boolean functions (which are per se exponentially large objects).

퐸 Theorem 9 ([2]). Let 푄 ∈ {훴, 훱}, 푘 ∈ N. For every 퐿 ∈ 푄푘 there is a polynomial 푝 and a deterministic polynomial time oracle machine 푀 s. t.

푝(|푥|) 푝(|푥|) 푝(|푥|) 푥 ∈ 퐿 ⇔ a1퐴1 ⊆ {0, 1} ... a푘퐴푘 ⊆ {0, 1} a푘+1푦 ∈ {0, 1} s. t. 푀 accepts ⟨푥, 푦⟩ with oracle ⟨퐴1, . . . , 퐴푘⟩,

where a1 = ∃ if 푄 = 훴 and a1 = ∀ if 푄 = 훱, and a푖 ∈ {∃, ∀} ∖ {a푖−1} for 1 < 푖 ≤ 푘 + 1.

For sets 퐴1, . . . , 퐴푘 the term ⟨퐴1, . . . , 퐴푘⟩ := {⟨푖, 푥⟩ | 푥 ∈ 퐴푖, 1 ≤ 푖 ≤ 푘 } is called efficient disjoint union in this context. It allows the machine to access an arbitrary number of oracles in its computations by writing down the corresponding oracle index together with the query. The following is a variant of the above theorem where the number 푘 of alternations is not fixed but polynomial in the input size: Theorem 10 ([8]). For every set 퐿 ∈ AEXP(poly) there is a polynomial 푝 and a deterministic polynomial time oracle machine 푀 s. t.

푝(|푥|) 푝(|푥|) 푥 ∈ 퐿 ⇔ a1퐴1 ⊆ {0, 1} ... a푝(|푥|)퐴푝(|푥|) ⊆ {0, 1} 푝(|푥|) 푝(|푥|) a1푦1 ∈ {0, 1} ... a푝(|푥|)푦푝(|푥|) ∈ {0, 1} s. t. 푀 accepts ⟨푥, 푦1, . . . , 푦푝(|푥|)⟩ with oracle ⟨퐴1, . . . , 퐴푝(|푥|)⟩,

where a1 ... a푝(|푥|) is an alternating quantifier sequence. Obviously each quantified word can be efficiently encoded in its own additional oracle. There are only polynomially many quantified words, so in the unbounded case we can drop the word quantifiers completely. These characterizations all have tight upper bounds. Suppose that a language is characterized by such a sequence of quantified oracles. Then conversely an alternating machine can non-deterministically guess the oracle sets with runtime exponential in 푝 and then simulate 푀 including the word quantifiers in deterministic exponential time. Together with Lemma8 we obtain: Corollary 11. 퐿 ∈ AEXP(poly) if and only if there is a polynomial 푝 and an deterministic polynomial time oracle machine 푀 s. t. 푥 ∈ 퐿 iff

푝(|푥|) 푝(|푥|) a1퐴1 ⊆ {0, 1} ... a푝(|푥|)퐴푝(|푥|) ⊆ {0, 1} : Complete Problems for the Exponential Hierarchy 7

푀 accepts 푥 with oracle ⟨퐴1, . . . , 퐴푝(|푥|)⟩, where a1,..., a푝(|푥|) is an alternating sequence of quantifiers.

E Corollary 12. For all 푘 ≥ 1, 퐿 ∈ 훴푘 if and only if there is a polynomial 푝 and a deterministic polynomial time oracle machine 푀 s. t. 푥 ∈ 퐿 iff

푝(|푥|) 푝(|푥|) 푝(|푥|) ∃퐴1 ⊆ {0, 1} ... a푘퐴푘 ⊆ {0, 1} a푘+1푦 ∈ {0, 1} : 푀 accepts ⟨푥, 푦⟩ with oracle ⟨퐴1, . . . , 퐴푘⟩, where a1 = ∃,..., a푘+1 is an alternating sequence of quantifiers. E Corollary 13. For all 푘 ≥ 1, 퐿 ∈ 훱푘 if and only if there is a polynomial 푝 and a deterministic polynomial time oracle machine 푀 s. t. 푥 ∈ 퐿 iff

푝(|푥|) 푝(|푥|) 푝(|푥|) ∀퐴1 ⊆ {0, 1} ... a푘퐴푘 ⊆ {0, 1} a푘+1푦 ∈ {0, 1} : 푀 accepts ⟨푥, 푦⟩ with oracle ⟨퐴1, . . . , 퐴푘⟩, where a1 = ∀,..., a푘+1 is an alternating sequence of quantifiers.

3 Second-order QBF

In this section the logic QBSF is introduced formally. It is a straightforward generalization of usual QBF to include function variables; it could be interpreted as a “second-order” extension: It behaves similarly to plain QBF as second-order logic behaves to first-order logic.

Definition 14 (Syntax of QBSF). The constants 1 and 0 are quantified Boolean second-order formulas (qbsfs) . If 푓 푛 is a function symbol of arity 푛 ≥ 0 and 푛 푛 휙1, . . . , 휙푛 are qbsfs, then 푓 (휙1, . . . , 휙푛), 휙1 ∧ 휙2, ¬휙1 and ∃푓 휙1 are all qbsfs. Abbreviations like 휙 ∨ 휓, 휙 → 휓, 휙 ↔ 휓 and ∀푓 푛 휓 can be defined from this as usual. In this setting, propositions can be understood as functions of arity zero. If the arity of a symbol is clear or does not matter we drop the indicator from now on. Furthermore a sequence 푥1, . . . , 푥푠 of variables can be abbreviated as 푥 if the number 푠 does not matter, ∃푥 meaning ∃푥1 ... ∃푥푛 and so on. For practical reasons we transfer the terms first-order variable and second-order variable to the Boolean realm when referring to functions of arity zero resp. greater than zero.

Definition 15 (Semantics of QBSF). An interpretation ℐ is a map from function variables 푓 푛 to 푛-ary Boolean functions. A function variable occurs freely if it is not in the scope of a matching quantifier. Write Fr(휙) for all free variables in the qbsf 휙. For ℐ that are defined on Fr(휙), write 휙 ℐ for the valuation of 휙 in ℐ, which is defined as J K

푐 ℐ := 푐 for 푐 ∈ {0, 1} J K 휙 ∧ 휓 ℐ := 휙 ℐ · 휓 ℐ J K J K J K 8 M. Lück

¬휙 ℐ := 1 − 휙 ℐ J 푛 K 푛J K 푓 (휙1, . . . , 휙푛) ℐ := ℐ(푓 )( 휙1 ℐ ,..., 휙푛 ℐ ) J 푛 K {︀ J K J⃒ K 푛 }︀ ∃푓 휙 ℐ := max 휙 ℐ[푓 푛↦→퐹 ] ⃒ 퐹 : {0, 1} → {0, 1} J K J K where ℐ[푓 푛 ↦→ 퐹 ] is the interpretation s. t. ℐ[푓 푛 ↦→ 퐹 ](푓 푛) = 퐹 and ℐ[푓 푛 ↦→ 퐹 ](푔푚) = ℐ(푔푚) for 푔푚 ̸= 푓 푛.

Lemma 16. The set of interpretations satisfying a qbsf 휙 is invariant under substitution of equivalent subformulas in 휙.

Proof. Proven by simple induction.

Write QBSF for the set of all qbsfs 휙 for which ∅ |= 휙 holds, i.e., 휙 is satisfied by the empty interpretation. If in a formula 휙 all quantifiers are at the beginning of 휙, then it is in prenex form. A second-order qbf is simple if all function symbols have only propositions as arguments. It is in conjunctive normal form (CNF) if it is in prenex form, simple and the matrix is in propositional CNF. Analogously define disjunctive normal form (DNF).

Theorem 17. QBSF ∈ AEXP(poly).

Proof. First transform 휙 into prenex form 휙′ in polynomial time. Evaluate 휙′ by alternating between existentially and universally states for each quantifier alternation, and guess and write down the truth tables for the quantified Boolean functions. These functions have arity at most |휙|, thus this whole step requires time 2|휙| · |휙′|. Evaluate the matrix in deterministic exponential time by looking up the truth tables and accept if and only if it is true.

log Theorem 18. QBSF in CNF or DNF is ≤m -hard for AEXP(poly). Proof. Let 퐿 ∈ AEXP(poly), where 퐿 ⊆ 훴* for some alphabet 훴. Let bin(퐿) := { bin(푥) | 푥 ∈ 퐿 }, where bin(·) efficiently encodes words from 훴* over {0, 1}. As log 퐿 ≤m bin(퐿), we only need to consider languages 퐿 over {0, 1}. By Corollary 11 there is a polynomial 푝 and a deterministic oracle Turing machine 푀 with polynomial runtime such that

푝(|푥|) 푝(|푥|) 푥 ∈ 퐿 ⇔ a1퐴1 ⊆ {0, 1} ... a푝(|푥|)퐴푝(|푥|) ⊆ {0, 1} s. t. 푀 accepts 푥 with oracle ⟨퐴1, . . . , 퐴푝(|푥|)⟩,

for an alternating quantifier sequence a1,..., a푝(|푥|). Let ℓ = 푝(|푥|). In this reduction we will represent the oracles 퐴푖 as their characteristic Boolean functions, call them 푐푖. Translate 푀 and 푥 into a formula ∃푧휙푥(푧), Complete Problems for the Exponential Hierarchy 9

where |푧| and |휙푥| both are polynomial in |푥|, and where 휙푥 is in CNF. The encoding can be done as in [6] and is possible in logspace (iterate over each possible timestep, tape, position and transition). In an oracle-free setting we could now claim that ∃푧휙푥(푧) is true if and only if 푀 accepts 푥. The oracle questions queried in transitions to the state 푞? however require special handling. Assume that 푀 uses only 푟 + 푚 tape cells for the oracle questions, 푟 ∈ 풪(log ℓ), 푚 ∈ 풪(ℓ), i.e., it writes the index of the oracle and the 푡 concrete query always on the same cells. Let the proposition 푧푝 ∈ 푧 mean that at timestep 푡 on position 푝 of the oracle tape there is a one. Then modify 휙푥 as follows. Let a clause 퐶 of 휙푥 encode a possible transition from some state 푡 푡 푞 to the state 푞? in timestep 푡. If the correct oracle answer is 푞+, then 푧1 . . . 푧푟 푡 푡 must represent some number 푖 in binary, 1 ≤ 푖 ≤ ℓ, and 푐푖(푧푟+1, . . . , 푧푟+푚) = 1 푡 푡 must hold. For the answer 푞− analogously 푐푖(푧푟+1, . . . , 푧푟+푚) = 0. Therefore any transition to 푞? at a timestep 푡 must be encoded not in 퐶 but instead in + + − − + − the new clauses 퐶1 , . . . , 퐶ℓ , 퐶1 , . . . , 퐶ℓ . Every such clause 퐶푖 /퐶푖 contains the same literals as 퐶, but additionally says that the oracle number is 푖, and contains a single second-order atom of the form 푐푖(...) or ¬푐푖(...). The new state of the transition is then obviously changed to 푞+/푞− instead of 푞?. As ℓ is polynomial in |푥|, there are also only polynomially many cases for the oracle number. The number of arguments of the characteristic functions is exactly 푚 which is again polynomial in |푥|. The logspace-computability of the new clauses 푚 푚 is straightforward. Altogether the second-order qbf a1푐1 ... aℓ푐ℓ ∃푧 휙푥 is true if and only if 푥 ∈ 퐿. Let us now consider the DNF case. As 푀 is deterministic, there is another deterministic oracle machine 푀 ′ with identical runtime which simulates 푀 including the oracle calls, but then rejects any word that is accepted by 푀 and ′ ′ vice versa. Let the formula 휙푥 be the translation of (푀 , 푥) as explained before. ′ ′ Then 푥 ∈ 퐿 iff a1푐1 ... aℓ푐ℓ ¬∃푧 휙푥 is true iff a1푐1 ... aℓ푐ℓ ∀푧 휙̂︁푥 is true, where ′ ′ 휙̂︁푥 is the dual formula of 휙푥 (i.e., the negation normal form of its negation) and thus in DNF.

log Corollary 19. QBSF in CNF or DNF is ≤m -complete for AEXP(poly).

4 Fragments with bounded quantifier alternation

E In Orponen’s original characterization of the 훴푘 classes a language 퐴 is expressed by a sequence of 푘 alternatingly quantified oracles, the input being verified by P a 훱푘 oracle machine [12]. Baier and Wagner improved this to a single word quantifier in the “first-order” suffix of the characterization insteadof 푘 word quantifiers (see Theorem9). In this section we use a different strategy to reduce the first-order quantifier alternations directly on the level of formulas. The difference to Baier’s and Wagner’s result is that we obtain CNF formulas where previously only DNF formulas could be obtained and vice versa. We define the following restricted problem of QBSF: 10 M. Lück

Definition 20. Let 푛, 푚, 푘, ℓ be non-negative integers, or 휔, and 푃, 푄 ∈ {훴, 훱}. 푛 ℓ Write QBSF(푃푚푄푘) for the restriction of QBSF to (prenex) formulas of the form

′ 0 ′ 0 a1푓1 ... a푝푓푝 a1푔1 ... a푞푔푞 퐻

where 퐻 is quantifier-free, 푓푖 are functions of arbitrary arities, 푔푖 are functions of arity zero (i.e., propositional variables), 푝 ≤ 푛, 푞 ≤ ℓ, the quantifiers a1 ... a푝 ′ ′ alternate at most 푚 − 1 times, the quantifiers a1 ... aℓ alternate at most 푘 − 1 ′ ′ times, a1 = ∃ iff 푄 = 훴, and a1 = ∃ iff 푄 = 훴.

1 2 Example. The formula ∃푓 ∀푥 ∀푦 (푥 ∧ 푦 ↔ 푓(푥, 푦)) is in QBSF(훴1 훱1 ) and 휔 휔 1 휔 1 1 QBSF(훴1 , 훱1 ), but not in QBSF(훱1 훴휔 ) or in QBSF(훴1 훱1 ). The following theorem demonstrates the reduction of propositional quantifier alternations. The idea behind this is that the truth of the whole first-order part can be encoded in a single Boolean function. Denote dual quantifiers as ∃ := ∀ and ∀ := ∃.

0 0 Theorem 21 (First-order alternation reduction). Let 휙 = a푓a1푥1 ... a푘푥푘퐻 be a qbsf such that 퐻 is quantifier-free and in CNF (a = ∃) resp. in DNF (a = ∀). 0 0 ′ Then there is an equivalent qbsf 휉 = a푔a푥1 ... a푥푘퐻 computable in logarithmic space, where 퐻′ is in CNF (a = ∃) resp. in DNF (a = ∀). Proof. Let 푓 have arity 푚, this will be important later on. The first-order part of ′ 휙 is 휙 := a1푥1 ... a푘푥푘퐻 (we drop the arities from now on) — as all quantified variables are merely propositions, it is an ordinary qbf, except that function atoms occur in 퐻. Let ℐ be some interpretation of 푓. To verify ℐ |= 휙′ we can use a set 푆 which models an assignment tree of 휙′. 푆 should have the following properties: It contains the empty assignment; further if 푆 contains some partial assignment 푠 to 푥1, . . . , 푥푗−1 and a푗 = ∃ (∀), then it must also contain 푠 ∪ {푥푗 ↦→ 0} or (and) 푠 ∪ {푥푗 ↦→ 1}. For any total assignment 푠 ∈ 푆, i.e., which is defined on all 푥1, . . . , 푥푘, the interpretation ℐ ∪ 푠 must satisfy 퐻. It is clear that, by the semantics of QBSF, such 푆 exists iff ℐ |= 휙′. This set 푆 is encoded in a new quantified function 푔 with arity 푚* := max{푚, 2푘}. We define how 푔 represents each (possibly partial) assignment 푠 ∈ 푆. For each propositional variable 푥푖 we use two bits, the first one tells if 푠(푥푖) is defined (1 if yes, 0 if no), and the second one tells thevalue 푠(푥푖) ∈ {0, 1} (and, say, 0 if undefined). It is 푚* ≥ 2푘, so all bits will fit into the arguments of 푔, and if 푔 has larger arity than 2푘 then the trailing bits are assumed constant 0. Write ⟨푠⟩ for the binary vector of length 푚* which encodes 푠, then 푔(⟨푠⟩) = 1 iff 푠 ∈ 푆. For the actual reduction of the quantifier rank consider two cases. In the ⋀︀푛 case a = ∃ it is 퐻 in CNF, say 퐻 := 푖=1 퐶푖 for clauses 퐶푖. The conditions of the set 푆 encoded by a given 푔 are verified by the following formula in CNF:

푛 ⋀︁ (︁ )︁ 휗휙′ (푔) :=∀푥1 ... ∀푥푘 푔(0) ∧ 푔(1, 푥1,..., 1, 푥푘, 0) → 퐶푖 ∧ 푖=1 Complete Problems for the Exponential Hierarchy 11

푘−1 (︃ ⋀︁ 푔(1, 푥1,..., 1, 푥푖, 0) → 푖=1 a푖=∃ )︃ (︀ )︀ 푔(1, 푥1,..., 1, 푥푖, 1, 0, 0) ∨ 푔(1, 푥1,..., 1, 푥푖, 1, 1, 0) ∧

푘−1 (︃ )︃ ⋀︁ 푔(1, 푥1,..., 1, 푥푖, 0) → 푔(1, 푥1,..., 1, 푥푖, 1, 1, 0) ∧ 푖=1 a푖=∀ (︃ )︃ 푔(1, 푥1,..., 1, 푥푖, 0) → 푔(1, 푥1,..., 1, 푥푖, 1, 0, 0)

′ ′ 휗휙′ (푔) is logspace-computable from 휙 . In 휙 now replace 휙 with ∃푔 휗휙′ . To see the correctness of this step assume that ℐ is an interpretation of 푥1, . . . , 푥푘. Since ′ ℐ |= 휙 ⇔ ℐ |= ∃푔 휗휙′ , as explained above, we can apply Lemma 16. ′ ′ For the case a = ∀ it is 휙 is in DNF. Consider 휗휓 where 휓 is the dual of 휙 . Note that 휓 itself has a matrix in CNF and 휗휓 thus can be constructed as above. Further it holds

′ ℐ |= 휙 ⇔ ℐ ̸|= 휓 ⇔ ℐ ̸|= ∃푔 휗휓 ⇔ ℐ |= ∀푔 ¬휗휓.

′ Therefore replace 휙 now with ∀푔 휗̂︀휓, where 휗̂︀휓 is the dual of 휗휓, and hence again in DNF. Finally replace all occurrences of 푓(푎1, . . . , 푎푚) by 푓(푎1, . . . , 푎푚, 0,..., 0), i.e., pad any possible interpretation of 푓 with zeros up to arity 푚*. The functions 푔 and 푓 have then the same arity and identical quantifier type a. Hence we can merge them into a single function: Replace a푓a푔 by aℎ, and as well each expression 푓(푎1, . . . , 푎푚* ) in the matrix with ℎ(0, 푎1, . . . , 푎푚* ) and likewise 푔(푎1, . . . , 푎푚* ) with ℎ(1, 푎1, . . . , 푎푚* ). It is easy to see that the matrix then holds for some (all) interpretation(s) of ℎ if and only if it holds for some (all) interpretation(s) of 푓 and 푔. This concludes the proof.

log Theorem 22. The following problems restricted to CNF or DNF are ≤m - complete:

푘 휔 E 푘 휔 E – If 푘 is even, then QBSF(훴푘 훴1 ) for 훴푘 and QBSF(훱푘 훱1 ) for 훱푘 . 푘 휔 E 푘 휔 E – If 푘 is odd, then QBSF(훴푘 훱1 ) for 훴푘 and QBSF(훱푘 훴1 ) for 훱푘 . 휔 휔 휔 휔 – QBSF(훴휔 훴1 ) and QBSF(훴휔 훱1 ) for AEXP(poly). Proof. The upper bounds work as in Theorem 17 by guessing the truth tables of the quantified functions. For the lower bound consider a reduction similar to the proof of Theorem 18. By Theorem9 we can already correctly choose the number and quantifier type of E E the functions 푐1, . . . , 푐푘 when reducing from a 훴푘 or 훱푘 language. The first-order part can be constructed accordingly as in Theorem 18, but, due to the single ′ word quantifier introduced in Theorem9, has now the form ∃푦 ∃푧 휙푥(푦, 푧) in 12 M. Lück

′ CNF or ∀푦 ∀푧 휙푥(푦, 푧) in DNF. Note that we can represent the computation of ′ ′ the deterministic machine on input 푥 arbitrarily as ∃푧 휙푥 in CNF or ∀푧 휙푥 in DNF, therefore choose the quantifiers matching as stated above. 휔 By this construction, the hardness for the CNF cases with 훴1 first-order part 휔 and the DNF cases with 훱1 first-order part is shown. In the remaining cases apply Theorem 21 to obtain an equivalent formula but with CNF matrix after 휔 휔 an 훱1 first-order prefix resp. DNF matrix after an 훴1 first-order prefix.

5 Fragments with Skolem functions

In the previous sections we considered the QBSF problem where function atoms could occur multiple times in a formula, in particular with different arguments. A Skolem function of a proposition 푥 however is a Boolean function that depends only on certain other propositions 푦1, . . . , 푦푛, the so-called dependencies of 푥. Hence, to connect QBSF to the Dependency QBF problem [13] and other logics of independence, we now focus on formulas where all quantified functions are Skolem functions: Definition 23. Let 푛, 푚, 푘, ℓ be non-negative integers or 휔. Let 푃, 푄 ∈ {훴, 훱}. uniq 푛 ℓ 푛 ℓ Write QBSF (푃푚푄푘) for the restriction of QBSF(푃푚푄푘) to formulas in which for all function symbols 푓 it holds that 푓 always occurs with the same arguments. In contrast, DQBF is defined as follows:

Definition 24. Every formula of the form ∀푥 ∃푦1(푧1) ... ∃푦푛(푧푛) 퐻 is called a dqbf, where the matrix 퐻 is a quantifier-free propositional formula and 푧푖 ⊆ 푥 f. a. 푖 = 1, . . . , 푛. A dqbf of this form is true if for all 푖 = 1, . . . , 푛 there is a Skolem function 푌푖 of 푦푖 depending only on 푧푖 s. t. for all assignments to 푥 the matrix 퐻 evaluates to true, provided the values of 푌푖 are assigned to 푦푖. As a decision problem, DQBF is defined as the set of all true dqbfs.

In this section we will prove that the restricted problem QBSFuniq is complete for the same complexity classes as the general case, i.e., the number of function quantifier alternations again determines the level in the exponential hierarchy. Orponen’s characterization via alternating oracle quantification allows polynomial time machines to recognize languages with exponential time complexity [12]. Hannula, Kontinen, Virtema and Vollmer generalized this to handle a polynomial number of oracles [8]. In the following we use the notion of tableaus to adapt these characterization to our needs. Call an oracle machine single-query machine if it asks at most one oracle question. An indirect simulation with a single-query machine allows, since the oracle tape cells directly correspond to the arguments of the quantified functions, an encoding in QBSFuniq as follows. If 푀 is an ATM with state set 푄 and tape alphabet 훤 , then a configuration of 푀 is a finite sequence 퐶 ∈ (푄 ∪ 훤 )* which contains exactly one state. A tableau Complete Problems for the Exponential Hierarchy 13

푇 of 푀 is a finite sequence 퐶1, . . . , 퐶푛 of equally long configurations of 푀 such that each 퐶푖+1 results from 퐶푖 by a transition of 푀. A tableau is pure if all states assumed in the tableau except the last one have the same alternation type, i.e., all existential or all universal. A pure tableau 퐶1, . . . , 퐶푛 is alternating if 푛 ≥ 2 and 푞푛 has a different alternation type than 푞1, . . . , 푞푛−1, where 푞푖 is the ′ ′ state of 퐶푖. If 푇, 푇 are tableaus of 푀, then 푇 is a successor tableau of 푇 if its first configuration is equal to 푇 ’s final configuration. Let 푇 be a pure tableau that assumes 푞 as its last state. Say that 푇 is 푘-accepting if – either 푘 = 1 and 푞 is an accepting state of the machine 푀, – or 푘 > 1 and 푇 is alternating and ∙ 푞 is existential and 푇 has a pure (푘 − 1)-accepting successor tableau, ∙ or 푞 is universal and all pure successor tableaus of 푇 are (푘 − 1)-accepting. Theorem 25 (Single-query indirect simulation). For every 푄 ∈ {훴, 훱} and 퐸 P every 푄푔(푛)-machine 푀 there is a polynomial ℎ and a single-query oracle 훴4 - machine 푁 such that 푀 accepts 푥 if and only if

a1퐴1 a2퐴2 ... a푔(|푥|)퐴푔(|푥|) : 푁 accepts 푥 with oracle ⟨퐴1, . . . , 퐴푔(|푥|)⟩,

where a1,... are alternating quantifiers starting with ∃ (푄 = 훴) resp. ∀ (푄 = 훱), ℎ(|푥|) and further 퐴푖 ⊆ {0, 1} f. a. 푖 = 1, . . . , 푔(|푥|). Proof. Let 푀 have runtime 푓, and let 푚 := 푔(|푥|) and 푛 := 푓(|푥|). W.l.o.g. we can assume the following: 푚 ≤ 푛, 2 ≤ 푛, 푀 does always exactly 푚 alternations before it accepts or rejects, and in each alternation phase it does exactly 푛 steps before it alternates, rejects or accepts. These properties imply that 푀 accepts 푥 if and only if all resp. some of its pure tableau starting with the initial configuration are 푚-accepting. We further assume for simplicity that 푀 uses only one tape.

The idea is to encode the tableaus 푇1, . . . , 푇푚 of the alternation phases of the computation of 푀 in the oracles 퐴1, . . . , 퐴푚 as follows. Words of 퐴푖 are indexed cells 푤 = (푐, 푡, 푝). Here 푝 denotes the position of the cell on the tape, 푡 is the current timestep, and 푐 ∈ 푄 ∪ 훤 is either the symbol written at position 푝, or the state of 푀 if 푝 happens to be the head position at timestep 푡. Let ℎ be the polynomial size of such a window 푤 encoded over {0, 1} (by binary encoding ℎ(|푥|) of 푡 and 푝). A set 퐴 ⊆ {0, 1} then represents a tableau 푇 = 퐶1, . . . , 퐶푛 in the sense that (푐, 푡, 푝) ∈ 퐴 if and only if the cell at tape position 푝 contains 푐 at timestep 푡. We now translate the acceptance condition of 푀 into a formal expression, using 푘-acceptance of tableaus: {︃ ∃퐴 ⊆ {0, 1}ℎ(|푥|) (Val ∧ Init ∧ Alt ∧ Acc ) if = ∃ : 푖 푖 푖 푖 푖+1 a푖 Acc푖 = ℎ(|푥|) ∀퐴푖 ⊆ {0, 1} (Val푖 ∧ Init푖) → (Alt푖 ∧ Acc푖+1) if a푖 = ∀

for 1 ≤ 푖 ≤ 푚. The semantics is that Val푖 is true if 퐴푖 encodes a pure tableau of 푀, Init1 is true if there is the initial configuration of 푀 on 푥 encoded in 퐴1, Init푖 14 M. Lück for 푖 > 1 is true if the first configuration of 퐴푖 is equal to the last configuration of 퐴푖−1, (i.e., 퐴푖 is a successor tableau of 퐴푖−1), Alt푖 for 푖 < 푚 is true if the tableau encoded in 퐴푖 is end-alternating, Alt푚 is true if the tableau encoded in 퐴푚 is 1-accepting, and Acc푚+1 is always true.

By the above definitions 푀 accepts 푥 if and only if the predicate Acc1 is true, as it states that 푀 has an 푚-accepting pure initial tableau. The formula can be written in prenex form, i.e., Acc1 holds if and only if a1퐴1 ⊆ ℎ(|푥|) ℎ(|푥|) {0, 1} ... a푚퐴푚 ⊆ {0, 1} 푉1 holds, where the predicate 푉푖 is defined as {︃ (Val푖 ∧ Init푖 ∧ Alt푖 ∧ 푉푖+1) if a푖 = ∃ 푉푖 := (Val푖 ∧ Init푖) → (Alt푖 ∧ 푉푖+1) if a푖 = ∀ for 1 ≤ 푖 ≤ 푚, and 푉푚+1 = 1. To prove the theorem we give now a single-query oracle ATM 푁 with polynomial runtime and 4 alternations which accepts if and only if 푉1 is true. For a predicate 푃 write 푃 for its complement. Group the predicates above as follows:

풯푖 := { Val푗, Init푗, Alt푗 | 1 ≤ 푗 < 푖 } for 푖 = 1, . . . , 푚 + 1, 0 {︀ }︀ 1 {︀ }︀ 0 1 ℱ푖 := Val푖 , ℱ푖 := Init푖 for 푖 = 1, . . . , 푚, and ℱ푚+1 := ∅, ℱ푚+1 := ∅.

By its definition 푉1 is true if and only if ∃푖 ∈ {1, . . . , 푚}, ∃푑 ∈ {0, 1} s. t. all 푑 푑 predicates in 풮푖 := 풯푖 ∪ ℱ푖 are true and further a푖 = ∀ or 푖 > 푚. Hence the machine 푁 is defined to work as follows:

1. In time 풪(log 푔) existentially guess 푖 and 푑, 푑 2. In time 풪(log 푔) universally branch on every predicate 푃 in 풮푖 , 3. Verify that 푃 is true.

푑 It only remains to verify that every predicate 푃 (and accordingly 푃 ) in 풮푖 can be checked in polynomial time, with only one oracle query, and at most two additional alternations. We sketch the required alternating procedures, where quantifier symbols ∃, ∀ always imply branching. If (푐, 푡, 푝) is a cell, then 푤 ∈ 퐴푖 means that 퐴푖 contains the encoding of 푤. The available timesteps 푡 in each tableau range over {0, . . . , 푛}. The available positions 푝 in the configurations are {0, . . . , 푛, 푛 + 1,..., 2푛 + 1}, where the input word is placed on positions 푛 + 1, . . . , 푛 + |푥| and the initial state of 푀 is given on position 푛. The predicates are checked as follows:

– Val푖: (check in parallel) ℎ ∙∀ 푤 ∈ {0, 1} : if 푤 is no valid encoded cell then 푤∈ / 퐴푖, ∙∀ 푡 ∀푝 ∃푐 ∈ 푄 ∪ 훤 :(푐, 푡, 푝) ∈ 퐴푖, ′ ′ ∙∀ 푤0 = (푐, 푡, 푝) ∀푤1 = (푐 , 푡, 푝): if 푐 ̸= 푐 then ∃푗 ∈ {0, 1} s. t. 푤푗 ∈/ 퐴푖, Complete Problems for the Exponential Hierarchy 15

∙∀ 푤0 = (푐0, 푡, 푝 − 1) ∀푤1 = (푐1, 푡, 푝) ∀푤2 = (푐2, 푡, 푝 + 1) ∀푤3 = (푐3, 푡 + 1, 푝 − 1) ∀푤4 = (푐4, 푡 + 1, 푝) ∀푤5 = (푐5, 푡 + 1, 푝 + 1) : if 푀 has no transition from (푤0, 푤1, 푤2) to (푤3, 푤4, 푤5) then ∃푗 ∈ {0,..., 5} s. t. 푤푗 ∈/ 퐴푖, ′ ′ ′ ′ ′ ∙∀ 푤0 = (푐, 푡, 푝) ∀푤1 = (푐 , 푡 , 푝 ): if 푡 < 푡 < 푛 and 푐, 푐 are states with different alternation types then ∃푗 ∈ {0, 1} s. t. 푤푗 ∈/ 퐴푖, ′ ′ – Alt푖, 푖 < 푚: ∃푤0 = (푐, 푛 − 1, 푝) ∃푤1 = (푐 , 푛, 푝 ) s. t. 푤0 and 푤1 contain states with different alternation types and ∀푗 ∈ {0, 1} : 푤푗 ∈ 퐴푖, – Alt푚: ∃푤 = (푞, 푛, 푝) : 푞 is an accepting state of 푀 and 푤 ∈ 퐴푚 – Init1: (check in parallel) ∙∀ 푖 ∈ {1,..., |푥|} ∃푤 = (푐, 0, 푛 + 푖) s. t. 푐 is the 푖-th letter of 푥 and 푤 ∈ 퐴1, ∙ (푞0, 0, 푛) ∈ 퐴1 where 푞0 is the initial sate of 푀, ∙∀ 푖∈ / {푛, . . . , 푛 + |푥|} ∃푤 = (, 0, 푖) ∈ 퐴1, ′ ′ – Init푖, 푖 > 1: ∀푤0 = (푐, 0, 푝) ∃푤1 = (푐 , 푛, 푝) : if 푐 ̸= 푐 then ∃푗 ∈ {0, 1} s. t. 푤푗 ∈/ 퐴푖−푗.

log Theorem 26. For 푘 ≥ 1, the following problems restricted to DNF are ≤m - complete:

uniq 푘 휔 E – QBSF (훴푘 훴4 ) for 훴푘 , uniq 푘 휔 E – QBSF (훱푘 훴4 ) for 훱푘 , uniq 휔 휔 – QBSF (훴휔 훴4 ) for AEXP(poly). Proof. The proof of the upper bounds is essentially the same as for Theorem 22. The lower bound proof is again similar to Theorem 18. It holds that the translation P from 훴푘 machines to deterministic machines with word quantifiers relativizes (see Baker and Selman [3, Lem. 1.1]) and additionally preserves the single-query property. As only one oracle question is asked by the encoded machine (at timestep, say, 푡), every function symbol occurs only with a fixed argument set, which describes the content of the oracle tape (modulo the oracle index) at timestep 푡. The method of single-query indirect simulation can be applied to obtain an alternative proof for the hardness of DQBF. Peterson, Reif and Azhar [13] state that every dqbf has an equivalent functional form which is in essence a QBSF uniq 휔 휔 formula with implicit function symbols. Similarly, all QBSF (훴1 훴휔 ) formulas are equivalent to the functional form of a DQBF formula with a straightforward efficient translation.

log E Corollary 27. DQBF is ≤m -complete for 훴1 .

6 Conclusion

The presented completeness results for the exponential hierarchy are in analogy to the results known for QBF; still they differ in subtle points. One difference isthat the “휔-jump” of QBSF is complete for AEXP(poly) and not for EXPSPACE. The reason for this is that any given input of length 푛 with explicit quantifiers, 16 REFERENCES

like in the QBF style, can only express 푛 alternations. This differs from decision problems which are defined via exponentially many alternations, e.g., certain games. It may be that the class AEXP(poly) is perhaps a more natural analogy to AP than AEXP, at least in the cases where the number of quantifiers is bounded by the input itself, e.g., logical operators. P Other differences are with regard to normal forms: The 훴푘 -hardness of QBF푘 already holds for CNF — but only if the rightmost quantifier happens to be existential, i.e., 푘 is odd. If it is universal, i.e., 푘 is even, then CNF-QBF푘 is in P 훴푘−1, i.e., it is supposedly easier. On the other hand, DNF establishes hardness P for 훴푘 only for even 푘. This collapse however does not occur in QBSF. This peculiar robustness can be explained if one remembers that function symbols occur in formulas. While the innermost existential guessing can be avoided in propositional formula in DNF (just scan for a single non-contradicting conjunction), this is not possible here: Is the conjunction 푓(푥1, 푥2) ∧ 푔(푥2, 푥3) self-contradicting or not? The hardness results are a hint that the structure of formulas with Boolean second-order variables is unlikely to exhibit such shortcuts as in propositional logic. On the other hand for QBSFuniq no such symmetry of CNF and DNF could be established, as Theorem 21 does not preserve the QBSFuniq condition. Similarly, the proof of Theorem 21 can at best produce 3CNF (and non-Horn) formulas, even if the matrix of the formula was already in 2CNF and Horn form. How does the complexity of QBSF change if 2CNF or 2DNF is considered, or horn formulas? How do CNF and DNF influence the complexity of the QBSFuniq P fragment? Can the single-query indirect simulation be done by a 훴푘 machine with 푘 < 4? Can the problem DQBF be generalized to incorporate universal quantification of Skolem functions?

Acknowledgement The author thanks Heribert Vollmer for helpful discussions and hints as well as the anonymous referees for spotting errors and improving the clarity of this paper.

References

1 Eric Allender, Dhiraj Holden and Valentine Kabanets. The Minimum Oracle Circuit Size Problem. 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Ed. by Ernst W. Mayr and Nicolas Ollinger. Vol. 30. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2015, pp. 21–33. 2 Herbert Baier and Klaus W. Wagner. The Analytic Polynomial-Time Hier- archy. Mathematical Logic Quarterly 44 (1998), 4, pp. 529–544. 3 Theodore P. Baker and Alan L. Selman. A second step toward the polynomial hierarchy. Theoretical Computer Science 8 (1979), 2, pp. 177–187. REFERENCES 17

4 Andreas Blass and Yuri Gurevich. Henkin quantifiers and complete problems. Annals of Pure and Applied Logic 32 (1986), pp. 1–16. 5 Ashok K. Chandra, Dexter C. Kozen and Larry J. Stockmeyer. Alternation. J. ACM 28 (Jan. 1981), 1, pp. 114–133. 6 Stephen A. Cook. The complexity of theorem-proving procedures. ACM Press, 1971, pp. 151–158. 7 Georg Gottlob, Nicola Leone and Helmut Veith. Second order logic and the weak exponential hierarchies. Mathematical Foundations of Computer Science 1995. Ed. by Gerhard Goos et al. Vol. 969. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995, pp. 66–81. 8 Miika Hannula et al. Complexity of Propositional Independence and Inclu- sion Logic. Mathematical Foundations of Computer Science 2015. Ed. by Giuseppe Italiano, Giovanni Pighizzini and Donald T. Sannella. Vol. 9234. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015, pp. 269–280. 9 Jaakko Hintikka and Gabriel Sandu. Informational Independence as a Se- mantical Phenomenon. Studies in Logic and the Foundations of Mathematics. Vol. 126. Elsevier, 1989, pp. 571–589. 10 Albert R. Meyer and Larry J. Stockmeyer. The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space. 13th Annual Symposium on Switching and Automata Theory, College Park, Maryland, USA, October 25-27, 1972. 1972, pp. 125–129. 11 Sarah E. Mocas. Separating classes in the exponential-time hierarchy from classes in PH. Theoretical Computer Science 158 (1996), 1–2, pp. 221 –231. 12 Pekka Orponen. Complexity classes of alternating machines with oracles. Automata, Languages and Programming. Ed. by Josep Diaz. Vol. 154. Berlin/Heidelberg: Springer-Verlag, 1983, pp. 573–584. 13 Gary L. Peterson, John H. Reif and Salman Azhar. Lower bounds for multiplayer noncooperative games of incomplete information. Computers & Mathematics with Applications 41 (Apr. 2001), 7-8, pp. 957–992. 14 Janos Simon. On Some Central Problems in Computational Complexity. Tech. rep. Ithaca, NY, USA, 1975. 15 Larry J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science 3 (1976), 1, pp. 1 –22. 16 Celia Wrathall. Complete sets and the polynomial-time hierarchy. Theoretical Computer Science 3 (1976), 1, pp. 23 –33.