verification tool builders

Abstract: The Boolean Satisfiability (SAT) problem is a remarkably expressive computational problem, serving both as a cornerstone of complexity theory and as a unifying language for a wide range of search and reasoning tasks. While $\mathrm{NP}$-complete in general, SAT is well known to be linear-time solvable on instances with bounded treewidth. This motivates the search for "hidden" tree-like structures in formulas---for instance by detecting so-called weak $t$-treewidth backdoors. Such backdoors are sets of $k$ variables that can be assigned in such a way that the remaining instance has treewidth at most $t$ and is satisfiable whenever the original instance is. Ever since Gaspers and Szeider [ICALP 2012] showed that finding a weak $t$-treewidth backdoor for a SAT formula is $\mathrm{W}[2]$-hard with respect to $k$, tractability results have been restricted to structurally sparse formulas---most notably, a linear-time algorithm for $d$-SAT (bounded clause-length SAT) by Fomin, Lokshtanov, Misra, Ramanujan, and Saurabh [SODA 2015] and a non-uniform polynomial-time algorithm when the incidence graph of the input formula belongs to a nowhere-dense graph class (or more generally is $K_{d,d}$-free) by Lokshtanov, Panolan, and Ramanujan [ICALP 2022]. In this paper, we push the tractability boundaries of finding weak $t$-treewidth backdoors beyond sparsity and into the regime of "well-behaved" dense formulas. Aiming towards relaxing structural input restrictions from nowhere-dense to edge-stable graph classes, as featured in the theory of beyond sparsity for FO model checking, or to bounded VC-dimension set systems, we identify signed semi-ladder-freeness as the correct dense structural restriction that establishes the boundary between tractability and intractability along this line.

Abstract: The Tarski fixed-point problem asks for a fixed point of a monotone map on a $k$-dimensional grid lattice of width $n$ and has applications in game theory, economics, and verification. Substantial gaps remain between the known upper and lower bounds on its quantum query complexity. We prove two hardness bootstrapping theorems for quantum query lower bounds. Starting from a hard instance family whose instances have pair-wise incomparable unique fixed points, and a lower bound established by the nonnegative adversary method, we bootstrap this lower bound by (i) a factor of $Ω(t)$ when the dimension is multiplied by $t$ and the width remains unchanged, and (ii) a factor of $Ω(\log m)$ when the width is multiplied by $m$ and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an $Ω(k\log^2 n)$ lower bound for $k\geq 4$, improving previously best-known classical and quantum lower bounds; (ii) for every fixed constant $α>0$, there exists a constant $δ(α)>0$ such that $Q_ε(\text{Tarski}(n,k))=Ω(kn^{δ(α)})$ whenever $k>n^α$, and in particular, an $Ω(kn^{0.0072})$ lower bound for $k>n^{2/203}$, the first lower bound with simultaneous linear dependence on the dimension and polynomial dependence on the width; and (iii) a superpolynomial lower bound of $(k/\log k)^{Ω(\log k)}$ for exponentially wide grids, where $n=2^{Θ(k)}$. The last result rules out a route to polynomial-time algorithms for simple stochastic games and ARRIVAL through generic polynomial-query black-box Tarski solvers.