Showing papers on "Conjunctive normal form published in 2019"

Tractable Learning and Inference for Large-Scale Probabilistic Boolean Networks

Abstract: Probabilistic Boolean networks (PBNs) have previously been proposed so as to gain insights into complex dynamical systems. However, identification of large networks and their underlying discrete Markov chain which describes their temporal evolution still remains a challenge. In this paper, we introduce an equivalent representation for PBNs, the stochastic conjunctive normal form network (SCNFN), which enables a scalable learning algorithm and helps predict long-run dynamic behavior of large-scale systems. State-of-the-art methods turn out to be 400 times slower for middle-sized networks (i.e., containing 100 nodes) and incapable of terminating for large networks (i.e., containing 1000 nodes) compared to the SCNFN-based learning, when attempting to achieve comparable accuracy. In addition, in contrast to the currently used methods which introduce strict restrictions on the structure of the learned PBNs, the hypothesis space of our training paradigm is the set of all possible PBNs. Moreover, SCNFNs enable efficient sampling so as to statistically infer multistep transition probabilities which can provide information on the activity levels of individual nodes in the long run. Extensive experimental results showcase the scalability of the proposed approach both in terms of sample and runtime complexity. In addition, we provide examples to study large and complex cell signaling networks to show the potential of our model. Finally, we suggest several directions for future research on model variations, theoretical analysis, and potential applications of SCNFNs.

A finite $$\epsilon $$ϵ-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables

Abstract: In this paper, we propose a generalized Benders decomposition-based branch and bound algorithm (GBDBAB) to solve two-stage convex mixed-binary nonlinear stochastic programs with mixed-binary variables in both first and second-stage decisions. In order to construct the convex hull of the MINLP subproblem for each scenario in closed-form, we first represent each MINLP subproblem as a generalized disjunctive program in conjunctive normal form (CNF). Second, we apply basic steps to convert the CNF of the MINLP subproblem into disjunctive normal form to obtain the convex hull of the MINLP subproblem. We prove that GBD is able to converge for the problems with pure binary variables given that the convex hull of each subproblem is constructed in closed-form. However, for problems with mixed-binary first and second-stage variables, we propose an algorithm, GBDBAB, where we may have to branch and bound on the continuous first-stage variables to obtain an optimal solution. We prove that the algorithm GBDBAB can converge to $$\epsilon $$-optimality in a finite number of steps. Since constructing the convex hull can be expensive, we propose a sequential convexification scheme that progressively applies basic steps to the CNF. Computational results on a problem with quadratic constraints, a constrained layout problem, and a planning problem, demonstrate the effectiveness of the algorithm.

Clausal Form Transformation in MaxSAT

Abstract: Some clausal form transformation algorithms used in SAT solving cannot be used in MaxSAT solving because they preserve satisfiability but do not preserve the minimum number of unsatisfied formulas. In this paper we define three different MaxSAT clausal form transformations, inspired on the transformations applied in SAT, that derive a multiset of clauses $\psi$ from a multiset of arbitrary propositional formulas $\phi$ in such a way that the minimum number of unsatisfied clauses in $\psi$ is equal to the minimum number of unsatisfied formulas in $\phi$ .

Bosphorus: Bridging ANF and CNF Solvers

Abstract: Algebraic Normal Form (ANF) and Conjunctive Normal Form (CNF) are commonly used to encode problems in Boolean algebra. ANFs are typically solved via Grobner ¨ basis algorithms, often using more memory than is feasible; while CNFs are solved using SAT solvers, which cannot exploit the algebra of polynomials naturally. We propose a paradigm that bridges between ANF and CNF solving techniques: the techniques are applied in an iterative manner to learn facts to augment the original problems. Experiments on over 1,100 benchmarks arising from four different applications domains demonstrate that learnt facts can significantly improve runtime and enable more benchmarks to be solved.

N-level Modulo-Based CNF encodings of Pseudo-Boolean constraints for MaxSAT

Abstract: Many combinatorial problems in various fields can be translated to Maximum Satisfiability (MaxSAT) problems. Although the general problem is $\mathcal {N}\mathcal {P}$ -hard, more and more practical problems may be solved due to the significant effort which has been devoted to the development of efficient solvers. The art of constraints encoding is as important as the art of devising algorithms for MaxSAT. In this paper, we present several encoding methods of pseudo-Boolean constraints into Boolean satisfiability problems in Conjunctive Normal Form (CNF) formula, which are based on the idea of modular arithmetic and only generate auxiliary variables for each unique combination of weights. These techniques are efficient in encoding and solving MaxSAT problems. In particular, our solvers won the partial MaxSAT industrial category from 2010 through 2012 and ranked second in the 2017 main weighted track of the MaxSAT evaluation. We prove the correctness and the pseudo-polynomial space complexity of our encodings and also give a heuristics of the base selection for modular arithmetic. Our experimental results show that our encoding compactly encodes the constraints, and the obtained clauses are efficiently handled by a state-of-the-art SAT solver.

Read-Once Resolutions in Horn Formulas

Abstract: In this paper, we discuss the computational complexity of Read-once resolution (ROR) with respect to Horn formulas. Recall that a Horn formula is a boolean formula in conjunctive normal form (CNF), such that each clause has at most one positive literal. Horn formulas find applications in a number of domains such as program verification and logic programming. It is well-known that deduction in ProLog is based on unification, which in turn is based on resolution and instantiation. Resolution is a sound and complete procedure to check whether a boolean formula in CNF is satisfiable. Although inefficient in general, resolution has been used widely in theorem provers, on account of its simplicity and ease of implementation. This paper focuses on two variants of resolution, viz., Read-once resolution and Unit Read-once resolution (UROR). Both these variants are sound, but incomplete. In this paper, the goal is to check for the existence of proofs (refutations) of Horn formulas under these variants. We also discuss the computational complexity of determining optimal length proofs where appropriate.

A SAT Approach to Branchwidth

Abstract: Branch decomposition is a prominent method for structurally decomposing a graph, a hypergraph, or a propositional formula in conjunctive normal form. The width of a branch decomposition provides a measure of how well the object is decomposed. For many applications, it is crucial to computing a branch decomposition whose width is as small as possible. We propose an approach based on Boolean Satisfiability (SAT) to finding branch decompositions of small width. The core of our approach is an efficient SAT encoding that determines with a single SAT-call whether a given hypergraph admits a branch decomposition of a certain width. For our encoding, we propose a natural partition-based characterization of branch decompositions. The encoding size imposes a limit on the size of the given hypergraph. To break through this barrier and to scale the SAT approach to larger instances, we develop a new heuristic approach where the SAT encoding is used to locally improve a given candidate decomposition until a fixed-point is reached. This new SAT-based local improvement method scales now to instances with several thousands of vertices and edges.

Classifying quantum data by dissipation

Abstract: We investigate a general class of dissipative quantum circuit capable of computing arbitrary conjunctive normal form (CNF) Boolean formulas. In particular, the clauses in a CNF formula define a local generator of Markovian quantum dynamics which acts on a network of qubits. Fixed points of this dynamical system encode the evaluation of the CNF formula. The structure of the corresponding quantum map partitions the Hilbert space into sectors, according to decoherence-free subspaces (DFSs) associated with the dissipative dynamics. These sectors then provide a natural and consistent way to classify quantum data (i.e., quantum states). Indeed, the attractive fixed points of the network allow one to learn the sector(s) for which some particular quantum state is associated. We show how this structure can be used to dissipatively prepare quantum states (e.g., entangled states) and outline how it may be used to generalize certain classical computational learning tasks.

Propagation complete encodings of smooth DNNF theories.

Abstract: We investigate conjunctive normal form (CNF) encodings of a function represented with a smooth decomposable negation normal form (DNNF). Several encodings of DNNFs and decision diagrams were considered by (Abio et al. 2016). The authors differentiate between encodings which implement consistency or domain consistency from encodings which implement unit refutation completeness or propagation completeness (in both cases implements means by unit propagation). The difference is that in the former case we do not care about properties of the encoding with respect to the auxiliary variables while in the latter case we treat all variables (the input ones and the auxiliary ones) in the same way. The latter case is useful if a DNNF is a part of a problem containing also other constraints and a SAT solver is used to test satisfiability. The currently known encodings of smooth DNNF theories implement domain consistency. Building on this and the result of (Abio et al. 2016) on an encoding of decision diagrams which implements propagation completeness, we present a new encoding of a smooth DNNF which implements propagation completeness. This closes the gap left open in the literature on encodings of DNNFs.

Weighted Model Counting with Algebraic Decision Diagrams

Abstract: We present an algorithm to compute exact literal-weighted model counts of Boolean formulas in Conjunctive Normal Form Our algorithm employs dynamic programming and uses Algebraic Decision Diagrams as the primary data structure We implement this technique in ADDMC, a new model counter We empirically evaluate various heuristics that can be used with ADDMC We then compare ADDMC to state-of-the-art exact weighted model counters (Cachet, c2d, d4, and miniC2D) on 1914 standard model counting benchmarks and show that ADDMC significantly improves the virtual best solver

Wombit: A Portfolio Bit-Vector Solver Using Word-Level Propagation

Abstract: We develop an idea originally proposed by Michel and Van Hentenryck of how to perform bit-vector constraint propagation on the word level. Most operations are propagated in constant time, assuming the bit-vector fits in a machine word. In contrast, bit-vector SMT solvers usually solve bit-vector problems by (ultimately) bit-blasting, that is, mapping the resulting operations to conjunctive normal form clauses, and using SAT technology to solve them. Bit-blasting generates intermediate variables which can be an advantage, as these can be searched on and learnt about. As each approach has advantages, it makes sense to try to combine them. In this paper, we describe an approach to bit-vector solving using word-level propagation with learning. We have designed alternative word-level propagators to Michel and Van Hentenryck’s, and evaluated different variants of the approach. We have also experimented with different approaches to learning and back-jumping in the solver. Based on the insights gained, we have built a portfolio solver, Wombit, which essentially extends the STP bit-vector solver. Using machine learning techniques, the solver makes a judicious up-front decision about whether to use word-level propagation or fall back on bit-blasting.

Computing Shortest Resolution Proofs

Abstract: Propositional resolution is a powerful proof system for unsatisfiable propositional formulas in conjunctive normal form. Resolution proofs represent useful explanations of infeasibility, with important applications. This motivates the challenge of computing shortest resolution proofs, i.e. those with the smallest number of inference steps. This paper proposes a SAT-based approach for this problem. Concretely, the paper investigates new propositional encodings for computing shortest resolution proofs and devises a number of optimizations, including symmetry breaking, additional constraints on the structure of proofs, as well as exploiting related concepts in infeasibility analysis, such as minimal correction subsets. Experimental results show the suitability of the proposed approach.

Application of Probabilistic Spin Logic (PSL) in Detecting Satisfiability of a Boolean Function

Abstract: Probabilistic Spin Logic (PSL) is a computing model that can be implemented using stochastic units (called p-bits) such as low-barrier nanomagnets. Besides computing a given Boolean function, PSL can also compute the inverse of the function. In this paper, we propose a methodology to exploit the invertibility of PSL in detecting the satisfiability (SAT) of a given Boolean function. First, we propose an algorithm to derive a PSL implementation for any Boolean function given in Conjunctive Normal Form (CNF). For a given SAT problem in CNF, we realize the given function in PSL using the proposed algorithm, clamp only the output to logic ‘1’ and detect the satisfiability of the given function by observing the probability distribution of the possible states. We demonstrate that PSL is highly selective of the satisfiable input patterns even for tough problems where only one out of one million input patterns is satisfiable. A key parameter deciding the effectiveness of the proposed technique is the time that is allowed for PSL computation, which would ultimately depend on the characteristics of the PSL hardware implementation.

Fixed-point Elimination in the Intuitionistic Propositional Calculus

Abstract: It follows from known results in the literature that least and greatest fixed-points of monotone polynomials on Heyting algebras—that is, the algebraic models of the Intuitionistic Propositional Calculus—always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the μ-calculus based on intuitionistic logic is trivial, every μ-formula being equivalent to a fixed-point free formula. In the first part of this article, we give an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given μ-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene’s iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. The axiomatization yields a decision procedure for the μ-calculus based on propositional intuitionistic logic. The second part of the article deals with closure ordinals of monotone polynomials on Heyting algebras and of intuitionistic monotone formulas; these are the least numbers of iterations needed for a polynomial/formula to converge to its least fixed-point. Mirroring the elimination procedure, we show how to compute upper bounds for closure ordinals of arbitrary intuitionistic formulas. For some classes of formulas, we provide tighter upper bounds that, in some cases, we prove exact.

Estimating the Density of States of Boolean Satisfiability Problems on Classical and Quantum Computing Platforms

Abstract: Given a Boolean formula $\phi(x)$ in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly $e$ clauses, for all values of $e$. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the \emph{hardness} of the problem instance. Consequently, in real-world scenarios, this problem is typically infeasible even when using state-of-the-art algorithms. While finding an exact answer to this problem is a computationally intensive task, we propose a novel approach for estimating density of states based on the concentration of measure inequalities. The methodology results in a quadratic unconstrained binary optimization (QUBO), which is particularly amenable to quantum annealing-based solutions. We present the overall approach and compare results from the D-Wave quantum annealer against the best-known classical algorithms such as the Hamze-de Freitas-Selby (HFS) algorithm and satisfiability modulo theory (SMT) solvers.

Formalizing CNF SAT Symmetry Breaking in PVS.

Abstract: The Boolean satisfiability problem (SAT) remains a central problem to theoretical as well as practical computer science. Recently, the need to trust the results obtained by SAT solvers has led to research in formalizing these. Nevertheless, tools in the ecosystem of SAT problems (such as preprocessors, model counters, etc.) would need to be verified as well in order for the results to be trusted. In this paper we explore a step towards a formalized symmetry breaking tool for SAT: formalizing SAT symmetry breaking for formulas in conjunctive normal form (CNF) using the Prototype Verification System (PVS).

Realizing Boolean Functions Using Probabilistic Spin Logic (PSL)

Abstract: Probabilistic Spin Logic (PSL) is a novel computing model that can be implemented using stochastic units (called p-bits) such as low-barrier nanomagnets. A PSL can exhibit accuracy which is comparable to a conventional digital circuit. Remarkably, a PSL can also be exploited to compute the inverse of a function. In this paper, using simulations, we examine the application of PSL in realizing Boolean functions. We propose a methodology to implement any Boolean function given in Conjunctive Normal Form (CNF) using PSL. Our methodology is based on synthesizing a given function in terms of NOT/AND/OR gates and deriving appropriate interconnections between p-bits. Further, we demonstrate the application of PSL in computing the inverse of a given function.

Fill-a-Pix Puzzle as a SAT Problem

Abstract: Fill-a-Pix Puzzle is a Picture Logic Puzzle that has not been solved as a SAT Problem as well as there is no SAT Conjunctive Normal Form (CNF) Encoding Method to solve this puzzle yet. There are several practical SAT problems in various fields such as Artificial Intelligence (AI), Automatic Theorem Proving, Circuit Design, etc. Fill-a-Pix puzzle is also one of the SAT problems. This research proposes the SAT CNF Encoding Method to solve Fill-a-Pix Puzzle as a SAT Problem using SAT Solvers. The proposed SAT CNF Encoding Method will be executed on different standard SAT solvers – MiniSAT, CryptoMiniSAT and RSAT. The evaluation is presented regarding the CPU Execution Times of each solver for executing the proposed SAT CNF Encoding, the Number of Variables and Clauses produced by the proposed SAT CNF Encoding as well as the Comparison of Fill-a-Pix Puzzle with the other Similar Puzzles such as Sudoku and Slitherlink based on the Number of Variables and Clauses produced by the proposed SAT CNF Encoding when executing Puzzle Sizes above 50 × 50.

Measure and Conquer for Max Hamming Distance XSAT

Abstract: XSAT is defined as the following: Given a propositional formula in conjunctive normal form, can one find an assignment to variables such that there is exactly only 1 literal that is true in every clause, while the other literals are false. The decision problem XSAT is known to be NP-complete. Crescenzi and Rossi [Pierluigi Crescenzi and Gianluca Rossi, 2002] introduced the variant where one searches for a pair of two solutions of an X3SAT instance with maximal Hamming Distance among them, that is, one wants to identify the largest number k such that there are two solutions of the instance with Hamming Distance k. Dahllof [Vilhelm Dahllof, 2005; Vilhelm Dahllof, 2006] provided an algorithm using branch and bound method for Max Hamming Distance XSAT in O(1.8348^n); Fu, Zhou and Yin [Linlu Fu and Minghao Yin, 2012] worked on a more specific problem, the Max Hamming Distance X3SAT, and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an exact exponential algorithm to solve the Max Hamming Distance XSAT problem in O(1.4983^n) time. Like all of them, we will use the branch and bound technique alongside a newly defined measure to improve the analysis of the algorithm.

Differentiable Set Operations for Algebraic Expressions.

Abstract: Basic principles of set theory have been applied in the context of probability and binary computation. Applying the same principles on inequalities is less common but can be extremely beneficial in a variety of fields. This paper formulates a novel approach to directly apply set operations on inequalities to produce resultant inequalities with differentiable boundaries. The suggested approach uses inequalities of the form Ei: fi(x1,x2,..,xn) and an expression of set operations in terms of Ei like, (E1 and E2) or E3, or can be in any standard form like the Conjunctive Normal Form (CNF) to produce an inequality F(x1,x2,..,xn)<=1 which represents the resulting bounded region from the expressions and has a differentiable boundary. To ensure differentiability of the solution, a trade-off between representation accuracy and curvature at borders (especially corners) is made. A set of parameters is introduced which can be fine-tuned to improve the accuracy of this approach. The various applications of the suggested approach have also been discussed which range from computer graphics to modern machine learning systems to fascinating demonstrations for educational purposes (current use). A python script to parse such expressions is also provided.

SAT to SAT-hard clause translator: work-in-progress

Abstract: Logic obfuscation emerged as an efficient solution to strengthen the security of integrated circuits (ICs) from multiple threats including reverse engineering and intellectual property (IP) theft. Emergence of Boolean Satisfiability (SAT) attacks and its variants have shown to circumvent the security mechanisms such as obfuscation and a plethora of its variants. Considering the size of ICs and the amount of time it takes to validate a defense i.e., obfuscation against SAT attack could range from few ms to days. In contrast, our current work focuses on devising an iterative, dynamic and intelligent SAT-hard clause generator for a given SAT-prone problem. The proposed Machine Learning (ML)-based SAT to unSAT clause translator is a SAT-hard clause generator that utilizes a bipartite propagation based neural network model. The utilized model comprises multiple layers of artificial neural networks to extract the dependencies of literals and variables, followed by long short term memory (LSTM) networks to validate the SAT hardness. The proposed ML-based SAT to unSAT clause translator is trained with conjunctive normal form (CNF) of the IC netlist that are both SAT solvable and SAT-hard. Further, the model is also trained to convert a CNF from satisfiable (SAT) to unsatisfiable (unSAT) form with minor perturbation (which translates to minor overheads) so that the SAT-attack cannot decrypt the keys. To the best of our knowledge, no previous work has been reported on neural network based SAT-hard clause or CNF translator for circuit obfuscation. We evaluate our proposed models's empirical performance against MiniSAT with 300 CNFs.

Finding Maximal Independent Elements of Products of Partial Orders (The Case of Chains)

Abstract: We consider one of the central intractable problems of logical data analysis – finding maximal independent elements of partial orders (dualization of a product of partial orders). An important particular case is considered with each order a chain. If each chain is of cardinality 2, the problem involves the construction of a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form. An asymptotic expression is obtained for a typical number of maximal independent elements of products for a large number of finite chains. The derivation of such asymptotic bounds is a technically complex problem, but it is necessary for the proof of existence of asymptotically optimal algorithms for the monotone dualization problem and the generalization of this problem to chains of higher cardinality. An asymptotically optimal algorithm is described for the problem of dualization of a product of finite chains.

FourierSAT: A Fourier Expansion-Based Algebraic Framework for Solving Hybrid Boolean Constraints

Abstract: The Boolean SATisfiability problem (SAT) is of central importance in computer science. Although SAT is known to be NP-complete, progress on the engineering side, especially that of Conflict-Driven Clause Learning (CDCL) and Local Search SAT solvers, has been remarkable. Yet, while SAT solvers aimed at solving industrial-scale benchmarks in Conjunctive Normal Form (CNF) have become quite mature, SAT solvers that are effective on other types of constraints, e.g., cardinality constraints and XORs, are less well studied; a general approach to handling non-CNF constraints is still lacking. In addition, previous work indicated that for specific classes of benchmarks, the running time of extant SAT solvers depends heavily on properties of the formula and details of encoding, instead of the scale of the benchmarks, which adds uncertainty to expectations of running time. To address the issues above, we design FourierSAT, an incomplete SAT solver based on Fourier analysis of Boolean functions, a technique to represent Boolean functions by multilinear polynomials. By such a reduction to continuous optimization, we propose an algebraic framework for solving systems consisting of different types of constraints. The idea is to leverage gradient information to guide the search process in the direction of local improvements. Empirical results demonstrate that FourierSAT is more robust than other solvers on certain classes of benchmarks.

Necessary and sufficient conditions for Boolean satisfiability.

Abstract: The study in this article seeks to find conditions that are necessary and sufficient for the satisfiability of a Boolean function. We will use the concept of special covering of a set introduced in [9] and study the relation of this concept with the satisfiability of Boolean functions. We show that the problem of existence of a special covering of a set is equivalent to the Boolean satisfiability problem. Thus, an important result is the proof of the existence of necessary and sufficient conditions for the existence of special covering of the set. This result allows us to formulate the necessary and sufficient conditions for Boolean satisfiability, considering the function in conjunctive normal form as a set of clauses. To formulate the same result in term of Boolean function we introduce the concept of proportional conjunctive normal form of a function, which is a conjunctive normal form of a function with the condition that each clause contains a negative literal or each clause contains a positive literal. Thus, we obtain that the satisfiability of a Boolean function represented in conjunctive normal form is equivalent to the possibility of converting it into a function in proportional conjunctive normal form by literal inversion.To prove these results, some algorithmic procedures are used. As a result of these procedures, in parallel, we obtain the Boolean values for the variables that provide the satisfiability of the function.Estimates of the complexity of these algorithmic procedures will be presented in the next article. Generally accepted terminology on set theory, Boolean functions, and graph theory is consistent with the terminology found in the relevant works included in the bibliography [1],[2],[3]. The newly introduced terms are not found in use by other authors and do not contradict to other terms.

Predicative proof theory of PDL and basic applications

Abstract: Propositional dynamic logic (PDL) is presented in Schutte-style mode as one-sided semiformal tree-like sequent calculus Seq pdl ω with standard cut rule and the omega-rule with principal formulas [P * ]A. The omega-rule-free derivations in Seq pdl ω are finite (trees) and sequents deducible by these finite derivations are valid in PDL. Moreover the cut-elimination theorem for Seq pdl ω is provable in Peano Arithmetic (PA) extended by transfinite induction up to Veblen's ordinal ϕ ω (0). Hence (by the cutfree subformula property) such predicative extension of PA proves that any given [P * ]-free sequent is valid in PDL iff it is deducible in Seq pdl ω by a finite cut-and omega-rule-free derivation, while PDL-validity of arbitrary star-free sequents is decidable in polynomial space. The former also implies a Herbrand-style conclusion that e.g. a given formula S = P * A ∨ Z for star-free A and Z is valid in PDL iff there is a k ≥ 0 and a cut-and omega-rule-free derivation of sequent A, P 1 A, · · · , P k A, B where P i A is an abbreviation for P · · · P i times A. This eventually leads to PSPACE-decidability of PDL-validity of S, provided that P is atomic and A is in a suitable basic conjunctive normal form. Furthermore we consider star-free formulas A in dual basic disjunctive normal form, and corresponding expansions S = P * A ∨ Z whose PDL-validity problem is known to be EXPTIME-complete. We show that cutfree-derivability in Seq pdl ω (hence PDL-validity) of such S is equivalent to plain validity of a suitable "transparent" quantified boolean formula S. Hence EXPTIME = PSPACE holds true iff the validity problem for any S involved is solvable by a polynomial-space deterministic TM. This may reduce the former problem to a more transparent complexity problem in quantified boolean logic. The whole proof can be formalized in PA extended by transfinite induction along ϕ ω (0)-actually in the corresponding primitive recursive weakening, PRA ϕ ω (0).