Number Of Quantifier Alternations Research Articles

Two SAT solvers for solving quantified Boolean formulas with an arbitrary number of quantifier alternations

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) for obtaining a propositional abstraction of the QBF. If this formula is false, the truth value of the QBF is decided, otherwise further refinement steps are necessary. Classically, expansion-based solvers process the given formula quantifier-block wise and use one SAT solver per quantifier block. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided and only two incremental SAT solvers are required. While our algorithm is naturally based on the forall Exp+Res calculus that is the formal foundation of expansion-based solving, it is conceptually simpler than present recursive approaches. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques.

Succinctness of Order-Invariant Logics on Depth-Bounded Structures

We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on structures of bounded tree-depth. Order-invariance is undecidable in general and, thus, one strives for logics with a decidable syntax that have the same expressive power as order-invariant sentences. We show that on structures of bounded tree-depth, order-invariant FO has the same expressive power as FO. Our proof technique allows for a fine-grained analysis of the succinctness of this translation. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. We obtain similar results for MSO. It is known that the expressive power of MSO and FO coincide on structures of bounded tree-depth. We provide a translation from MSO to FO and we show that this translation is essentially optimal regarding the formula size. As a further result, we show that order-invariant MSO has the same expressive power as FO with modulo-counting quantifiers on bounded tree-depth structures.

Quantifier Reordering for QBF

State-of-the-art procedures for evaluating quantified Boolean formulas often expect input formulas in prenex conjunctive normal form (PCNF). We study dependency schemes as a means of reordering the quantifier prefix of a PCNF formula while preserving its truth value. Dependency schemes associate each formula with a binary relation on its variables (the dependency relation) that imposes constraints on certain operations manipulating the formula's quantifier prefix. We prove that known dependency schemes support a stronger reordering operation than was previously known. We present an algorithm that, given a formula and its dependency relation, computes a compatible reordering with a minimum number of quantifier alternations. In combination with a dependency scheme that can be computed in polynomial time, this yields a polynomial time heuristic for reducing the number of quantifier alternations of an input formula. The resolution-path dependency scheme is the most general dependency scheme introduced so far. Using an interpretation of resolution paths as directed paths in a formula's implication graph, we prove that the resolution-path dependency relation can be computed in polynomial time.

On logical hierarchies within FO^2-definable languages

We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of quantifier alternations yields an infinite hierarchy whose levels are varieties of languages (and hence admit an algebraic characterization). Using this algebraic approach, we show that the quantifier alternation hierarchy inside FO^{2}[<] is decidable within one unit. For this purpose, we relate each level of the hierarchy with decidable varieties of languages, which can be defined in terms of iterated deterministic and co-deterministic products. A crucial notion in this process is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle languages of Schwentick, Th\'erien and Vollmer.

The Complexity of Problems for Quantified Constraints

In this paper we are interested in quantified propositional formulas in conjunctive normal form with “clauses” of arbitrary shapes. i.e., consisting of applying arbitrary relations to variables. We study the complexity of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for such formulas, both with and without a bound on the number of quantifier alternations. For each of these computational goals we get full complexity classifications: We determine the complexity of each of these problems depending on the set of relations allowed in the input formulas. Thus, on the one hand we exhibit syntactic restrictions of the original problems that are still computationally hard, and on the other hand we identify non-trivial subcases that admit efficient algorithms.

Pebble Games and Subroutines in Least Fixed Point Logic

We characterize the Least Fixed Point logic (FO + LFP) as a class of pebble game programs. Such a program can be characterized unambiguously as a structure of recursive subroutines. These programs can be described as flowchart-like digraphs. Concentrating on fixed points of a bounded number of quantifier alternations, we partition FO + LFP into an infinite nonlinear hierarchy of queries, composed of expressibility classes characterized by alternations within recursive subroutines.

Complexity of deciding Tarski algebra

Let a formula of Tarski algebra contain k atomic subformulas of the kind ( f i ⩾ 0), 1 ⩽ t ⩽ k , where the polynomials f i ∈ ℤ [ X 1 ,..., X n ] have degrees deg ( f i ) < d , let 2 M be an upper bound for the absolute value of every coeffieient of the polynomials f i , 1 ⩽ i ⩽ k , let a ⩽ n be the number of quantifier alternations in the prenex form of the formula. A decision method for Tarski algebra is described with the running time polynomial in M ( k d ) ( O ( n ) ) 4 n − 2 . Previously known decision procedures have a time complexity polynomial in ( M k d ) 2 O ( n ) .

Implicational complexity in intuitionistic arithmetic

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].