Třídy Booleovských formulí s efektivně řešitelným SATem

The thesis studies classes of Boolean formulae for which the well-known satisfiability problem is solvable in polynomially bounded time. It focusses on classes based on unit resolution; it describe classes of unit refutation complete formulae, unit propagation complete formulae and focuses on the class of SLUR formulae. It presents properties of SLUR formulae as well as the recently obtained results. The main result is the coNP-completness of membership testing. Finally, several hierarchies are built over the SLUR class and their properties and mutual relations are studied. Powered by TCPDF (www.tcpdf.org)Práce studuje třídy booleovkských formulí pro které je problém splnitelnosti řešitelný v polynomiálním čase. Zaměřuje se na třídy založené jednotkové rezoluci; popisuje třídy unit refutation complete formulí, unit propagation complete formulí a specialně se zaměřuje na třídu SLUR. Shrnuje její vlastnosti a poslední výsledky dosažené v této oblasti. Hlavním výsledkem je coNP-úplnost testování zda daná formule patří do třídy SLUR. V závěru je třída SLUR rozvinuta do několika různých hierarchií a jsou studovány jejich vlastnosti a vzájemný vztah vzhledem k inkluzi. Powered by TCPDF (www.tcpdf.org)Katedra teoretické informatiky a matematické logikyDepartment of Theoretical Computer Science and Mathematical LogicFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Linear and Second-order Optical Response of the III-V Mono-layer Superlattices

We report the first fully self-consistent calculations of the nonlinear optical properties of superlattices. The materials investigated are mono-layer superlattices with GaP grown on the the top of InP, AlP and GaAs (110) substrates. We use the full-potential linearized augmented plane wave method within the generalized gradient approximation to obtain the frequency dependent dielectric tensor and the second-harmonic-generation susceptibility. The effect of lattice relaxations on the linear optical properties are studied. Our calculations show that the major anisotropy in the optical properties is the result of strain in GaP. This anisotropy is maximum for the superlattice with maximum lattice mismatch between the constituent materials. In order to differentiate the superlattice features from the bulk-like transitions an improvement over the existing effective medium model is proposed. The superlattice features are found to be more pronounced for the second-order than the linear optical response indicating the need for full supercell calculations in determining the correct second-order response.Comment: 9 pages, 4 figures, submitted to Phy. Rev.

Trading inference effort versus size in CNF Knowledge Compilation

Knowledge Compilation (KC) studies compilation of boolean functions f into some formalism F, which allows to answer all queries of a certain kind in polynomial time. Due to its relevance for SAT solving, we concentrate on the query type "clausal entailment" (CE), i.e., whether a clause C follows from f or not, and we consider subclasses of CNF, i.e., clause-sets F with special properties. In this report we do not allow auxiliary variables (except of the Outlook), and thus F needs to be equivalent to f. We consider the hierarchies UC_k <= WC_k, which were introduced by the authors in 2012. Each level allows CE queries. The first two levels are well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in KC, that is, f is represented by the set of all prime implicates, while UC_1 = WC_1 is the same as UC, the class of unit-refutation complete clause-sets introduced by del Val 1994. We show that for each k there are (sequences of) boolean functions with polysize representations in UC_{k+1}, but with an exponential lower bound on representations in WC_k. Such a separation was previously only know for k=0. We also consider PC < UC, the class of propagation-complete clause-sets. We show that there are (sequences of) boolean functions with polysize representations in UC, while there is an exponential lower bound for representations in PC. These separations are steps towards a general conjecture determining the representation power of the hierarchies PC_k < UC_k <= WC_k. The strong form of this conjecture also allows auxiliary variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the separation results from the discontinued arXiv:1302.442

On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201

Generalising unit-refutation completeness and SLUR via nested input resolution

We introduce two hierarchies of clause-sets, SLUR_k and UC_k, based on the classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC (Unit refutation Complete), introduced in 1994. The class SLUR, introduced in [Annexstein et al, 1995], is the class of clause-sets for which unit-clause-propagation (denoted by r_1) detects unsatisfiability, or where otherwise iterative assignment, avoiding obviously false assignments by look-ahead, always yields a satisfying assignment. It is natural to consider how to form a hierarchy based on SLUR. Such investigations were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we consider the "limit hierarchy" SLUR_k, based on generalising r_1 by r_k, that is, using generalised unit-clause-propagation introduced in [Kullmann, 1999, 2004]. The class UC, studied in [Del Val, 1994], is the class of Unit refutation Complete clause-sets, that is, those clause-sets for which unsatisfiability is decidable by r_1 under any falsifying assignment. For unsatisfiable clause-sets F, the minimum k such that r_k determines unsatisfiability of F is exactly the "hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k such that after application of any falsifying partial assignments, r_k determines unsatisfiability. The class UC_k is given by the clause-sets which have hardness <= k. We observe that UC_1 is exactly UC. UC_k has a proof-theoretic character, due to the relations between hardness and tree-resolution, while SLUR_k has an algorithmic character. The correspondence between r_k and k-times nested input resolution (or tree resolution using clause-space k+1) means that r_k has a dual nature: both algorithmic and proof theoretic. This corresponds to a basic result of this paper, namely SLUR_k = UC_k.Comment: 41 pages; second version improved formulations and added examples, and more details regarding future directions, third version further examples, improved and extended explanations, and more on SLUR, fourth version various additional remarks and editorial improvements, fifth version more explanations and references, typos corrected, improved wordin

Hardness measures and resolution lower bounds

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable).Comment: 43 pages, preliminary version (yet the application part is only sketched, with proofs missing