On Stronger Calculi for QBFs††thanks: The work was supported by the Austrian Science Foundation (FWF) under grant S11409-N23. Partial results have been announced at the QBF Workshop 2014 (http://www.easychair.org/smart-program/VSL2014/QBF-program.html).
Quantified Boolean formulas (QBFs) generalize propositional formulas by admitting quantifications over propositional variables. QBFs can be viewed as (restricted) formulas of first-order predicate logic and easy translations of QBFs into first-order formulas exist. We analyze different translations and show that first-order resolution combined with such translations can polynomially simulate well-known deduction concepts for QBFs. Furthermore, we extend QBF calculi by the possibility to instantiate a universal variable by an existential variable of smaller level. Combining such an enhanced calculus with the propositional extension rule results in a calculus with a universal quantifier rule which essentially introduces propositional formulas for universal variables. In this way, one can mimic a very general quantifier rule known from sequent systems.
Quantified Boolean formulas (QBFs) generalize propositional formulas by admitting quantifications over propositional variables. QBFs can be viewed in two different ways, namely (i) as a generalization of propositional logic and (ii) as a restriction of first-order predicate logic (where we interpret over a two element domain). A number of calculi are available for QBFs: the ones based on variants of resolution for QBFs [13, 11, 2, 3], the ones based on instantiating universal variables with truth constants combined with propositional resolution and an additional instantiation rule , and different sequent systems [7, 14, 10, 9].
In all these calculi (except the latter ones from [7, 14, 9]), the possibility to instantiate a given formula is limited. In purely resolution-based calculi, formulas (or more precisely universal variables) are never instantiated. In instantiation-based calculi, instantiation is restricted to truth constants. In contrast, sequent systems possess flexible quantifier rules, and (existential) variables as well as (propositional) formulas can be used for instantiation with tremendous speed-ups in proof complexity. This motivates why we are interested in strengthening instantiation techniques for instantiation-based calculi.
We allow to replace (some) universal variables not only by truth constants but by existential variables left of in the quantifier prefix. This approach mimics the effect of quantifier rules introducing atoms in sequent calculi from . We add a propositional extension principle (known from extended resolution ), which enables the introduction of propositional formulas for universal variables via extension variables (or names for the formula). Contrary to , where we proposed propositional extensions of the form which can be eliminated if the cut rule is available in the sequent calculus, such an elimination is not possible here for which reason we have to use (classical) extensions.
We show that such combinations can polynomially simulate Q-resolution with resolution over existential and universal variables (QU-res , Theorem 5), Q-resolution (Q-res , Corollary 1) and the instantiation-based calculus IR-calc  (Theorem 5, Corollary 2). The latter simulation provides a soundness proof for IR-calc independent from strategy extraction.
Structure. In Sect. 2 we introduce necessary definitions and notations. In Sect. 3 different translations from QBFs to (restrictions of) first-order logic  are reconsidered. In Sect. 4 different calculi based on (variants of) the resolution calculus are described. Here, we introduce our calculi generalized from IR-calc. In Sect. 5 we present our results on polynomial simulations between considered calculi and in Sect. 6 we provide exponential separations. In the last section we conclude and discuss future research possibilities.
We assume familiarity with the syntax and semantics of propositional logic, QBFs and first-order logic (see, e.g.,  for an introduction). We recapitulate some notions and notations which are important for the rest of the paper.
We consider a propositional language based on a set of Boolean variables and truth constants (true) and (false), both of which are not in . A variable or a truth constant is called atomic and connectives are from . A literal is a variable or its negation. A clause is a disjunction of literals, but sometimes we consider it as a set of literals. Tautological clauses contain a variable and its negation and the empty clause is denoted by . Propositional formulas are denoted by capital Latin letters like possibly annotated with subscripts, superscripts or primes.
We extend the propositional language by Boolean quantifiers. Universal () and existential () quantification is allowed within a QBF. The superscript is used to distinguish Boolean quantifiers from first-order quantifiers introduced later. QBFs are denoted by Greek letters. Observe that we allow non-prenex formulas, i.e., quantifiers may occur deeply in a QBF. An example for a non-prenex QBF is , where , , and are variables. Moreover, free variables (like ) are allowed, i.e., there might be occurrences of variables in the formula for which we have no quantification. Formulas without free variables are called closed; otherwise they are called open. The universal (existential) closure of is (), for which we often write () if is the set of all free variables in . A formula in prenex conjunctive normal form (PCNF) has the form , where is the quantifier prefix, and is the (propositional) matrix which is in CNF. Often we write a QBF as ( for all and the elements of are pairwise disjoint). We define the level of a literal , , as the index such that the variable of occurs in . The logical complexity of a formula , , is the number of occurrences of connectives and quantifiers.
We use a first-order language consisting of (objects) variables, function symbols (FSs), predicate symbols (PSs), together with the truth constants and connectives mentioned above. Quantifiers and bind object variables. Terms and formulas are defined according to the usual formation rules. We identify -ary PSs with propositional atoms, and -ary FSs with constants. Clauses, tautological clauses and the empty clause are defined as in the propositional case.
Let be the set of first-order variables and be the set of terms. A substitution is a mapping of type such that only for finitely many variables . We represent by a finite set of the form . The domain of , , is the set . The range of , , is the set . We call a variable substitution if . The empty substitution is denoted by . We often write substitutions post-fix, e.g., we use instead of . Algebraically, substitutions define a monoid with being the neutral element under the usual composition of substitutions.
Substitutions are extended to terms and formulas in the usual way, e.g., , , and , where is an -place FS, is an -place PS, are terms, and are (quantifier-free) formulas and is a binary connective. For substitutions and , is more general than if there is a substitution such that . A substitution is called a permutation if is one-one and a variable substitution. A permutation is called a renaming (substitution) of an expression (i.e., is a term or a quantifier-free formula) if , where is the set of all variables occurring in . For an expression , is a variant of provided is a renaming substitution.
Let be a non-empty set of expressions. A substitution is called a unifier of if . Unifier is called most general unifier (mgu), if for every unifier of , is more general than .
Let and be two proof systems. polynomially simulates (p-simulates) if there is a polynomial such that, for every natural number and every formula , the following holds. If there is a proof of in of size , then there is a proof of (or a suitable translation of it) in whose size is less than .
3 Different translations of QBFs to first-order logic
We introduce different translations of (closed) QBFs to (closed) formulas in (restrictions of) first-order logic. We start with the basic translation from  in Fig. 1. Obviously, the QBF and the first-order formula enjoy a very similar structure. Especially the variable dependencies expressed by the quantifier prefix are exactly the same.
propQBFIsomorphic Let be a (closed) QBF and let be its (closed) first-order translation. Then , i.e., and are isomorphic. The proof in the appendix is by induction on the logical complexity of .
The basic translations from Fig. 1 can be extended to generating a skolemized version of . We restrict our attention here to QBFs in PCNF.
Let be a closed QBF in PCNF with matrix and let be its closed first-order translation. For any existential variable in the quantifier prefix of , let be the sequence of universal variables left of (in exactly the same order in which they occur in the prefix). Let be the Skolem function symbol associated to . We call the skolemized form of and denote it by , where the substitution is as follows.
Traditionally, is denoted as a quantifier-free formula with the assumption that all free variables are (implicitly) universally quantified.
The number of universal variables a Skolem function depends on can be optimized, e.g., by using miniscoping or dependency schemes . As we will see later on, most of our results do not depend on such optimizations.
Let be a closed QBF in PCNF with matrix and let be its closed first-order translation. Let be the skolemized form of . Then .
Since we interpret over a two-element domain, proper Skolem function symbols (i.e., the arity is greater than ) can be eliminated by introducing new predicate symbols. The resulting formula belongs to EPR (Effectively PRopositional logic or more traditionally it belongs to the Bernays-Schoenfinkel class).
Let be a closed QBF in PCNF with matrix and let be its closed first-order translation. Let the skolemized form of . Replace any occurrence of a predicate of the form by where is a proper function symbol and is a non-empty list of universal variables. The formula resulting after all possible replacements is the EPR formula .
We will see later that the first-order and the EPR translation have different proof-theoretical properties because some resolutions are blocked by different predicate symbols. Proposition 3 is Lemma 1 in  (stated without a proof).
propSatEquiv Let be a closed QBF. Then
|is satisfiable||iff||is satisfiable.|
A proof can be found in the appendix.
|\prfbyaxiomAxiomC||\prftree[r]Res x ∨C_1¬x∨C_2C_1∨C_2||\prftree[r]Fac C∨ℓ∨ℓC∨ℓ||\prftree[r]R D∨mD|
is a non-tautological clause from the matrix. If then . Variable is existential (Q-res) and existential or universal (QU-res), is a literal and is a universal literal. If is existential, then holds.
is a non-tautological clause from the matrix , where is a shorthand for if and if .
|\prftree[r]Res x^τ ∨C_1¬x^τ∨C_2C_1∨C_2||\prftree[r]Fac C∨ℓ^τ ∨ℓ^τC∨ℓ^τ||\prftree[r]InstC|
is an assignment to universal variables and .
4 Different calculi based on resolution
We introduce different calculi used in this paper. We start with two resolution calculi, Q-res and QU-res, for QBFs in Fig. 2. Observe that the consequence of each rule is non-tautological. We continue with the calculus in Fig. 3, where we use the same presentation as in . is the quantifier prefix and is the quantifier-free matrix in CNF. In the following instantiation-based calculi, inference rules do not work on usual clauses but on annotated clauses based on extended assignments. An extended assignment is a partial mapping from the Boolean variables to . An annotated clause consists of annotated literals of the form , where is an extended assignment to universal variables and with . Composition of extended assignments is defined using completion. The expression is called the completion of by . Then , the completion of by , is defined as follows.
The function allows instantiations of clauses; it computes for an extended assignment and an annotated clause . Later on, we will clarify the relation between annotations and substitutions in first-order logic.
We extend by the possibility to instantiate universal variables by existential ones. Technically the instantiation is performed by a global substitution . If a universal variable is replaced by some existential variable , i.e., , then must hold. We name the calculus equipped with the substitution and depict the rules in Fig. 4.
is a non-tautological clause from the matrix .
, Res, Fac and Inst are the same as in .
It is immediately apparent that this calculus is sound and complete. We get completeness, when we use the empty substitution as because then, reduces to which is sound and complete . Soundness follows from the validity of QBFs of the form
If the right formula has an refutation, then it is false and therefore the left formula has to be false.
is a non-tautological clause from the matrix or from .
, , , Res, Fac and Inst are the same as in .
If then and by construction.
We further enhance by the possibility to use propositional extensions [19, 6]. This extension operation is a generalization of the well-known structure-preserving translation to (conjunctive) normal form in propositional logic. For presentational reasons, we require to have all extensions at the very beginning of the deduction in order to allow extension variables as replacements for universal variables. Figure 5 shows the inference rules of this calculus , where is a sequence of (clausal representations of) extensions of the form with being of the form or of the form () and is a variable neither occurring in nor in nor in . The variables are existential. The quantification extends the quantifier prefix such that for all variables occurring in and is minimal. Due to the requirements on the extension variables and the placement of , the resulting calculus is sound. Completeness is not an issue here, because we can use an empty .
The usual formalization of clauses and resolvents as sets of literals can be simulated in our formalizations by the factoring rule Fac. We assume in the following that Fac is applied as soon as possible.
We finally introduce first-order resolution. Let be a clause and let and be two distinct literals in both of which are either negated or unnegated. If there is an mgu of and , then the clause is called a factor of . The clause is called the premise of the factoring operation.
Let and be two clauses and let be a variant of which has no variable in common with . A clause is a resolvent of the parent clauses and if the following conditions hold:
and are literals of opposite sign whose atoms are unifiable by an mgu .
Let be a set of clauses. A sequence is called deduction (first-order resolution deduction) of a clause from if and for all , one of the following conditions hold.
is an input clause from .
is a factor of a for .
is a resolvent of and for .
An refutation of is an deduction of the empty clause from . The size of a deduction is given by , where is the number of character occurrences in . An deduction has tree form if every occurrence of a clause is used at most once as a premise in a factoring operation or as a parent clause in a resolution operation.
Next we introduce the subsumption rule taken from Definition 2.3.4 in . Contrary to the usual use of subsumption in automated deduction as a deletion rule, here we add clauses which are (factors of) instantiations of clauses.
If and are clauses, then subsumes or is subsumed by , if there is a substitution such that . A set of clauses is obtained from a set by subsumption if where is subsumed by a clause of .
Resolution can be extended by the subsumption rule (Definition 3.2.3 in ).
By a derivation of a set of clauses from a set of clauses by plus subsumption, we mean a sequence of clause such that the following conditions are fulfilled.
For all there is a clause subsuming the clause or there exist clauses such that is subsumed by a resolvent of and .
polynomially simulates plus subsumption.
The subsumption rule is not necessary but makes proofs of polynomial simulation results much more convenient. It allows instantiated deductions for which eventually the lifting theorem provides a deduction “on the most general level”.
5 Polynomial simulations of calculi
In this section we show that together with a suitable translation (denoted by ) polynomially simulates QU-res, Q-res and .
thmRfoPsimQU polynomially simulates QU-res. The proof is by induction on the number of clauses in the QU-res deduction. It can be found in the appendix. It shows that first-order literals obtained from universal literals in the QBF and eliminated by R are eliminated by resolutions with and without instantiating the first-order resolvent.
The following results are immediate consequences of Theorem 5.
polynomially simulates QU-res.
as well as polynomially simulates Q-res.
We present a soundness proof of independent from strategy extraction by a polynomial simulation of by .
Let and be two substitutions. The composition of and , , is obtained from
by deleting all for which holds.
Let and be two substitutions as defined in Definition 5, where are universal variables and . Then is the composition .
Let be the completion of by defined in (1). Since as well as is a subset of the set of universal variables and as well as is a subset of , and therefore for all . Hence, the completion of the two substitutions and is exactly their composition .
In the following, we deal with annotated clauses of the form where any is an existential literal and any is the restriction of assignment to exactly those universal variables for which holds. We denote the sequence of all universal variables with by where we assume the same order as in the quantifier prefix. A first-order clause corresponding to is constructed as follows
where is the isomorphic counterpart of (cf. the remark after Proposition 1). Using together with mimics the effect of ; the difference is the explicit notation of all universal variables left of and not only the variables in .
thmRfoPsimIRcalc polynomially simulates .
In the proof, we construct by induction on the number of derived clauses in the IR-calc deduction stepwisely a deduction in plus subsumption. We consider the sequence of first-order clauses obtained from the original clauses as a skeleton for the final proof. Since the clauses in the skeleton do not follow by a single application of an inference rule, we have to provide a short deduction of the clauses.
We utilize Proposition 2 and allow subsumption in the simulation. The proof is by strong mathematical induction on the number of derived clauses in the IR-calc deduction. Let denote the statement “Given a IR-calc deduction from a QBF and a sequence of first-order clauses , the clause has a short deduction in plus subsumption from ”.
Base: . is a consequence of the axiom rule using clause from the matrix . Let be the assignment induced by . Then we have a clause from which we can derive by resolution steps using and . The number of these steps is equal to the number of universal variables in .
IH: Suppose hold for some .
Step: We have to show . Consider and .
Case 1: is derived by the axiom rule. Then proceed like in the base case.
Case 2: is a consequence of the rule Inst with premise (for some with ) and assignment . By IH and Remark 1, we have a short plus subsumption deduction of . is of the form . By Lemma 1, for any universal variable with . Therefore is of the form . Now and can be derived by subsumption.
Case 3: is a consequence of the rule Fac with premise (for some with ). By IH, we have a short plus subsumption deduction of , where is of the form . We generate a factor of simply by omitting one of the duplicates.
Case 4: is a consequence of the resolution rule with parent clauses (for some with ). By IH, we have two clauses
We use of the form as a renaming of the variables in such that does not share any variable with . The resolvent is where is the mgu of the form . We add by subsumption, where maps all remaining variables to their counterpart.
polynomially simulates .
When we inspect the translation of (axiom) clauses, we observe that a universal variable is translated to an atom of the form . With the subsumption rule we can instantiate the clause by a substitution of the form for a term . This observation was the trigger to introduce the stronger calculus , where universal variables cannot be replaced only by or but also by any existential variable with .
6 Exponential separation of resolution calculi
We constructed in  a family of short closed QBFs in PCNF for which any Q-res refutation of is superpolynomial. We recapitulate the construction here. The formula is
is the pigeon hole formula for holes and pigeons in conjunctive normal form and denoted over the variables . Variable is intended to denote that pigeon is sitting in hole . is
The number of clauses in is and is . The formula is obtained from the pigeon hole formula in disjunctive normal form, , by a structure-preserving polarity-sensitive translation to clause form . The formula is simply the negation of where negation has been pushed in front of atoms and double-negation elimination has been applied.
We use new variables of the form for disjuncts in . For the first disjuncts of the form with , we use variables . For the second part, for any and the disjuncts, we use
The set of these variables for is denoted by . Due to this construction, we can speak about the conjunction corresponding to the variable .
We construct the conjunctive normal form of as follows. First, we take the clause over all variables in . The formula for the first disjuncts of is of the form
For the remaining disjuncts of , we have the formula
Then is and is . It is easy to check that is valid.
Let us modify the quantifier prefix of . By quantifier shifting rules we get, in an “antiprenexing” step, the equivalent formula . Prenexing yields the equivalent QBF
which has only one quantifier alternation instead of two. In  we showed that and have short cut-free tree proofs in a sequent system , where weak quantifiers introduce atoms. The following extends Proposition 3 in .
The proof is based on the fact that (i) the two conjuncts belong to languages with different alphabets and (ii) that the alphabets cannot be made identical by instantiation of quantifiers in Q-res. Therefore we have to refute either or under the given quantifier prefix. Since is true, there is no Q-res refutation and we have to turn to . But then, we essentially have to refute with propositional resolution and consequently, by Haken’s famous result , any Q-res refutation of is superpolynomial in .
Since QU-res, LDQ-res, LDQU-res, LDQU-res, and Q(D)-resolution (Q(D)-res)  are based on the same quantifier-handling mechanism as Q-res, the following corollary is obvious.
For the situation is not better. Since universal literals are only replaced by , no unification of the two alphabets can happen.
The quantifier prefix is unfortunate if one expects being false. Actually, the initial universal quantifier block prevents any non-empty and consequently, any refutation of reduces to an refutation of .
Any refutation of from (4) in has size superpolynomial in .
In the following we show that has a short refutation in . We use to denote the Skolem function symbol corresponding to and to denote the Skolem function symbols corresponding to . All the Skolem function symbols have arity . Let denote the formula under the first-order translation. We have
The refutation of is constructed as follows.
We use together with the first clauses from to derive (for all ). The deduction consists of clauses and applies resolution and factoring. The substitution is , where is a variable renaming from the variant generation in resolution.
We use together with the binary clauses from to derive (for all and with ). The deduction consists of clauses and applies resolution and factoring. Then is . Again is a variable renaming like above.
We use together with the derived instance of to derive by resolution. Since any variable is assigned to a variant of for all and all , all resolution steps are possible. The deduction consists of clauses.
The formula can be refuted in a similar fashion in by replacing variants of the form by Skolem constants .
Let and be the families of closed QBFs defined above. Then and have short tree refutations in consisting of clauses. Moreover the size of the refutation is .
thmNoPsimRfo The calculi QU-res, LDQ-res, LDQU-res, LDQU-res, Q(D)-res, , and IRM-calc cannot polynomially simulate tree or