15 research outputs found
On variables with few occurrences in conjunctive normal forms
We consider the question of the existence of variables with few occurrences
in boolean conjunctive normal forms (clause-sets). Let mvd(F) for a clause-set
F denote the minimal variable-degree, the minimum of the number of occurrences
of variables. Our main result is an upper bound mvd(F) <= nM(surp(F)) <=
surp(F) + 1 + log_2(surp(F)) for lean clause-sets F in dependency on the
surplus surp(F).
- Lean clause-sets, defined as having no non-trivial autarkies, generalise
minimally unsatisfiable clause-sets.
- For the surplus we have surp(F) <= delta(F) = c(F) - n(F), using the
deficiency delta(F) of clause-sets, the difference between the number of
clauses and the number of variables.
- nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of
natural numbers all numbers of the form 2^n - 1.
We conjecture that this bound is nearly precise for minimally unsatisfiable
clause-sets.
As an application of the upper bound we obtain that (arbitrary!) clause-sets
F with mvd(F) > nM(surp(F)) must have a non-trivial autarky (so clauses can be
removed satisfiability-equivalently by an assignment satisfying some clauses
and not touching the other clauses). It is open whether such an autarky can be
found in polynomial time.
As a future application we discuss the classification of minimally
unsatisfiable clause-sets depending on the deficiency.Comment: 14 pages. Revision contains more explanations, and more information
regarding the sharpness of the boun
Using Local Search to Find \MSSes and MUSes
International audienceIn this paper, a new complete technique to compute Maximal Satisfiable Subsets (MSSes) and Minimally Unsatisfiable Subformulas (MUSes) of sets of Boolean clauses is introduced. The approach improves the currently most efficient complete technique in several ways. It makes use of the powerful concept of critical clause and of a computationally inexpensive local search oracle to boost an exhaustive algorithm proposed by Liffiton and Sakallah. These features can allow exponential efficiency gains to be obtained. Accordingly, experimental studies show that this new approach outperforms the best current existing exhaustive ones
How many conflicts does it need to be unsatisfiable?
A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. Clearly, a CNF formula has to have conflicts in order to be unsatisfiable—in fact, there have to be many conflicts, and it is the goal of this paper to quantify how many.
An unsatisfiable k-CNF has at least 2^k clauses; a lower bound of 2^k for the number of conflicts follows easily. We improve on this trivial bound by showing that an unsatisfiable k-CNF formula requires Ω(2.32^k) conflicts. On the other hand there exist unsatisfiable k-CNF formulas with O((4^k(log k)^3)/k) conflicts. This improves the simple bound O(4^k) arising from the unsatisfiable k-CNF formula with the minimum number of clauses
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails
On Davis–Putnam reductions for minimally unsatisfiable clause-sets
"Minimally unsatisfiable clause-sets" are fundamental building blocks of satisfiability (SAT) theory. In order to establish a structural theory about them,elimination of certain types of degenerations via "Davis-Putnam (DP) reductions" are essential. These DP-reductions have been used at many placessince more than 50 years, and we now show that we have certain forms of confluence, that is, that the applications of DP-reductions are independent oftheir implementation, to a certain degree
Irreducible Subcube Partitions
A \emph{subcube partition} is a partition of the Boolean cube
into subcubes. A subcube partition is irreducible if the only sub-partitions
whose union is a subcube are singletons and the entire partition. A subcube
partition is tight if it "mentions" all coordinates. We study extremal
properties of tight irreducible subcube partitions: minimal size, minimal
weight, maximal number of points, maximal size, and maximal minimum dimension.
We also consider the existence of homogeneous tight irreducible subcube
partitions, in which all subcubes have the same dimensions. We additionally
study subcube partitions of , and partitions of
into affine subspaces, in both cases focusing on the minimal
size. Our constructions and computer experiments lead to several conjectures on
the extremal values of the aforementioned properties.Comment: 39 page