41 research outputs found
The combinatorics of minimal unsatisfiability: connecting to graph theory
Minimally Unsatisfiable CNFs (MUs) are unsatisfiable CNFs where removing any clause destroys unsatisfiability. MUs are the building blocks of unsatisfia-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisfiability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and flipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the first part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisfiable 2-CNFs), and characterising MUs with deficiency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have efficient algorithms for many interesting problems, namely that we provide a full classification of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classifi-cation of 2-MUs to the classification of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisfiable Sub-clause-sets (MUSs)
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
Hunting for Tractable Languages for Judgment Aggregation
Judgment aggregation is a general framework for collective decision making
that can be used to model many different settings. Due to its general nature,
the worst case complexity of essentially all relevant problems in this
framework is very high. However, these intractability results are mainly due to
the fact that the language to represent the aggregation domain is overly
expressive. We initiate an investigation of representation languages for
judgment aggregation that strike a balance between (1) being limited enough to
yield computational tractability results and (2) being expressive enough to
model relevant applications. In particular, we consider the languages of Krom
formulas, (definite) Horn formulas, and Boolean circuits in decomposable
negation normal form (DNNF). We illustrate the use of the positive complexity
results that we obtain for these languages with a concrete application: voting
on how to spend a budget (i.e., participatory budgeting).Comment: To appear in the Proceedings of the 16th International Conference on
Principles of Knowledge Representation and Reasoning (KR 2018
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
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
Range-Restricted Interpolation through Clausal Tableaux
We show how variations of range-restriction and also the Horn property can be
passed from inputs to outputs of Craig interpolation in first-order logic. The
proof system is clausal tableaux, which stems from first-order ATP. Our results
are induced by a restriction of the clausal tableau structure, which can be
achieved in general by a proof transformation, also if the source proof is by
resolution/paramodulation. Primarily addressed applications are query synthesis
and reformulation with interpolation. Our methodical approach combines
operations on proof structures with the immediate perspective of feasible
implementation through incorporating highly optimized first-order provers