arXiv subjects
Publications and source records attributed to Petr Savický.
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of sequences of directed graphs (digraphs). Our results address their decomposition into strongly connected components, periodicity, spectral properties, and asymptotic behaviour of their large powers.
We introduce an inhomogeneous variant of random 2-SAT. Each variable $v_1,\ldots,v_n$ is assigned a type from a state space $Λ$, independently at random. Clause inclusion is governed by a symmetric measurable kernel $W$ on $(Λ\times \{+,-\})^2$, in analogy with the inhomogeneous random graph model of Bollobás, Janson, and Riordan: given literals $\ell_i\in\{v_i,\neg v_i\}$ and $\ell_j\in\{v_j,\neg v_j\}$, the clause $\{\ell_i,\ell_j\}$ appears with probability $W(\mathrm{type}(\ell_i),\mathrm{type}(\ell_j))/(2n)$. In particular, for a variable $v_i$ of type $x\inΛ$, the slices $W((+,x),\cdot)$ and $W((-,x),\cdot)$ describe how $v_i$ and $\neg v_i$ interact with other literals. We identify a parameter $ρ^*(W)$, defined as the spectral radius of an integral operator derived from $W$, and show that $ρ^*(W)<1$ and $ρ^*(W)>1$ correspond to asymptotically almost surely satisfiable and unsatisfiable instances, respectively. The satisfiability threshold for homogeneous random 2-SAT is well-established, occurring when the ratio of clauses to variables is $1$. This corresponds to a weight function of $W \equiv 1$ and a clause density of $1/(2n)$. Our result extends this classical result to a broad class of models controlled by types of variables.
Two CNF formulas are called ucp-equivalent, if they behave in the same way with respect to the unit clause propagation (UCP). A formula is called ucp-irredundant, if removing any clause leads to a formula which is not ucp-equivalent to the original one. As a consequence of known results, the ratio of the size of a ucp-irredundant formula and the size of a smallest ucp-equivalent formula is at most $n^2$, where $n$ is the number of the variables. We demonstrate an example of a ucp-irredundant formula for a symmetric definite Horn function which is larger than a smallest ucp-equivalent formula by a factor $Ω(n/\ln n)$. Consequently, a general upper bound on the above ratio cannot be smaller than this.
We investigate conjunctive normal form (CNF) encodings of a function represented with a 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 by unit propagation from encodings which are unit refutation complete or propagation complete. The difference is that in the former case we do not care about propagation strength of the encoding with respect to the auxiliary variables while in the latter case we treat all variables (the main and the auxiliary ones) in the same way. The currently known encodings of DNNF theories implement domain consistency. Building on these encodings we generalize the result of (Abio et al. 2016) on a propagation complete encoding of decision diagrams and present a propagation complete encoding of a DNNF and its generalization for variables with finite domains.
A distance magic labeling of an $n$-dimensional hypercube is a labeling of its vertices by natural numbers from $\{0, \ldots, 2^n-1\}$, such that for all vertices $v$ the sum of the labels of the neighbors of $v$ is the same. Such a labeling is called neighbor-balanced, if, moreover, for each vertex $v$ and an index $i=0,\ldots,n-1$, exactly half of the neighbors of $v$ have digit $1$ at $i$-th position of the binary representation of their label. We demonstrate examples of non-neighbor-balanced distance magic labelings of $6$-dimensional hypercube obtained by a SAT solver.
We describe a compilation language of backdoor decomposable monotone circuits (BDMCs) which generalizes several concepts appearing in the literature, e.g. DNNFs and backdoor trees. A $\mathcal{C}$-BDMC sentence is a monotone circuit which satisfies decomposability property (such as in DNNF) in which the inputs (or leaves) are associated with CNF encodings from a given base class $\mathcal{C}$. We consider the class of propagation complete (PC) encodings as a base class and we show that PC-BDMCs are polynomially equivalent to PC encodings. Additionally, we use this to determine the properties of PC-BDMCs and PC encodings with respect to the knowledge compilation map including the list of efficient operations on the languages.
In this paper we investigate CNF formulas, for which the unit propagation is strong enough to derive a contradiction if the formula together with a partial assignment of the variables is unsatisfiable (unit refutation complete or URC formulas) or additionally to derive all implied literals if the formula is satisfiable (propagation complete or PC formulas). If a formula represents a function using existentially quantified auxiliary variables, it is called an encoding of the function. We prove several results on the sizes of PC and URC formulas and encodings. One of them are separations between the sizes of formulas of different types. Namely, we prove an exponential separation between the size of URC formulas and PC formulas and between the size of PC encodings using auxiliary variables and URC formulas. Besides of this, we prove that the sizes of any two irredundant PC formulas for 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.
Constraint "at most one" is a basic cardinality constraint which requires that at most one of its $n$ boolean inputs is set to $1$. This constraint is widely used when translating a problem into a conjunctive normal form (CNF) and we investigate its CNF encodings suitable for this purpose. An encoding differs from a CNF representation of a function in that it can use auxiliary variables. We are especially interested in propagation complete encodings which have the property that unit propagation is strong enough to enforce consistency on input variables. We show a lower bound on the number of clauses in any propagation complete encoding of the "at most one" constraint. The lower bound almost matches the size of the best known encodings. We also study an important case of 2-CNF encodings where we show a slightly better lower bound. The lower bound holds also for a related "exactly one" constraint.
FL$_\mathrm{ew}$-algebras form the algebraic semantics of the full Lambek calculus with exchange and weakening. We investigate two relations, called satisfiability and positive satisfiability, between FL$_\mathrm{ew}$-terms and FL$_\mathrm{ew}$-algebras. For each FL$_\mathrm{ew}$-algebra, the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory; we identify and investigate the complements as fragments of its universal theory. We offer characterizations of those algebras that (positively) satisfy just those terms that are satisfiable in the two-element Boolean algebra providing its semantics to classical propositional logic. In case of positive satisfiability, these algebras are just the nontrivial weakly contractive FL$_\mathrm{ew}$-algebras. In case of satisfiability, we give a characterization by means of another property of the algebra, the existence of a two-element congruence. Further, we argue that (positive) satisfiability problems in FL$_\mathrm{ew}$-algebras are computationally hard. Some previous results in the area of term satisfiability in MV-algebras or BL-algebras are thus brought to a common footing with known facts on satisfiability in Heyting algebras.