Backdoor DNFs

We introduce backdoor DNFs, as a tool to measure the theoretical hardness of CNF formulas. Like backdoor sets and backdoor trees, backdoor DNFs are defined relative to a tractable class of CNF formulas. Each conjunctive term of a backdoor DNF defines a partial assignment that moves the input CNF formula into the base class. Backdoor DNFs are more expressive and potentially smaller than their predecessors backdoor sets and backdoor trees. We establish the fixed-parameter tractability of the backdoor DNF detection problem. Our results hold for the fundamental base classes Horn and 2CNF, and their combination. We complement our theoretical findings by an empirical study. Our experiments show that backdoor DNFs provide a significant improvement over their predecessors.

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick’s constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples.

Bounds on the size of PC and URC formulas

In this paper, we investigate CNF encodings, for which unit propagation is strong enough to derive a contradiction if the encoding is not consistent with a partial assignment of the variables (unit refutation complete or URC encoding) or additionally to derive all implied literals if the encoding is consistent with the partial assignment (propagation complete or PC encoding). We prove an exponential separation between the sizes of PC and URC encodings without auxiliary variables and strengthen the known results on their relationship to the PC and URC encodings that can use auxiliary variables. Besides of this, we prove that the sizes of any two irredundant PC formulas representing the same function differ at most by a factor polynomial in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.

On Reducing Clause DataBase in Glucose

Modern CDCL SAT solvers generally save the variable value when backtracking. We propose a new measure called nbSAT based on the saved assignment to predict the usefulness of a learnt clause when reducing clause database in Glucose 3.0. Roughly speaking, The nbSAT value of a learnt clause is equal to the number of literals satised by the current partial assignment plus the number of other literals that would be satised by the saved assignment. We study the nbSAT measure by empirically show that it may be more accurate than the LBD measure originally used in Glucose. Based on this study, we implement an improvement in Glucose 3.0 to remove half of learnt clauses with large nbSAT values instead of half of clauses with large LBD values. This improvement, together with a resolution method to keep the learnt clauses or resolvents produced using a learnt clause that subsume an original clause, makes Glucose 3.0 more eective for the application and crafted instances from the SAT 2014 competition.

Ch.9 Assignment of rights, transfer of obligations, assignment of contracts, s.1: Assignment of rights, Art.9.1.4

This commentary analyses Article 9.1.4 of the UNIDROIT Principles of International Commercial Contracts (PICC) concerning partial assignment. Art 9.1.4 recognizes the economic need for assigning part of a right. Taking into account the underlying rationale of Section 9.1 of the PICC that the assignment should not burden the obligor, Art 9.1.4 draws a distinction between a right to the payment of a monetary sum and a right to other performance. According to this provision, a right to the payment of a monetary sum may be assigned partially. A right to other performance may be assigned partially only if it is divisible, and the assignment does not render the obligation significantly more burdensome. This commentary discusses the right to the payment of a monetary sum, right to other performance, the divisibility of a right, the additional burden that a partial assignment may put on the obligor, and the effects of partial assignment on the obligee and obligor.