On the Boundedness Problem for Higher-Order Pushdown Vector Addition Systems

International audienceKarp and Miller's algorithm is a well-known decision procedure that solves the termination and boundedness problems for vector addition systems with states (VASS), or equivalently Petri nets. This procedure was later extended to a general class of models, well-structured transition systems, and, more recently, to pushdown VASS. In this paper, we extend pushdown VASS to higher-order pushdown VASS (called HOPVASS), and we investigate whether an approach à la Karp and Miller can still be used to solve termination and boundedness.We provide a decidable characterisation of runs that can be iterated arbitrarily many times, which is the main ingredient of Karp and Miller's approach. However, the resulting Karp and Miller procedure only gives a semi-algorithm for HOPVASS. In fact, we show that coverability, termination and boundedness are all undecidable for HOPVASS, even in the restricted subcase of one counter and an order 2 stack. On the bright side, we prove that this semi-algorithm is in fact an algorithm for higher-order pushdown automata

New Hardness Results for the Permanent Using Linear Optics

In 2011, Aaronson gave a striking proof, based on quantum linear optics, that the problem of computing the permanent of a matrix is #P-hard. Aaronson\u27s proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant\u27s seminal proof of the same fact in 1979. Nevertheless, it did not show #P-hardness of the permanent for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques. First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronson\u27s original proof, this implies that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents of orthogonal matrices over the finite field F_{p^4} for p != 2, 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan. Finally, we use more elementary arguments to prove #P-hardness for the permanent of a positive semidefinite matrix. This result shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly, despite the fact that they can be efficiently sampled by a classical computer

Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ? V(G) it holds that h(v) ? L(v). The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. The authors showed that the problem is polynomial-time solvable if H belongs to the class called bi-arc graphs, and for all other graphs H it is NP-complete. We are interested in the complexity of the LHom(H) problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rz??ewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H, i.e., if every vertex has a loop. In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHom(H) problem is NP-hard. For every such H we find a constant k = k(H), such that the LHom(H) problem on instances G with n vertices and treewidth t - can be solved in time k^t ? n^?(1), provided that G is given along with a tree decomposition of width t, - cannot be solved in time (k-?)^t ? n^?(1), for any ? > 0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|. Obtaining matching upper and lower bounds shows that the set of algorithmic tools that we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of the LHom(H) problem, e.g. with different parameterizations

Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity

We considerably sharpen the known connections between circuit-analysis algorithms and circuit lower bounds, show intriguing equivalences between the analysis of weak circuits and (apparently difficult) circuits, and provide strong new lower bounds for approximately computing Boolean functions with depth-two neural networks and related models. - We develop approaches to proving THR o THR lower bounds (a notorious open problem), by connecting algorithmic analysis of THR o THR to the provably weaker circuit classes THR o MAJ and MAJ o MAJ, where exponential lower bounds have long been known. More precisely, we show equivalences between algorithmic analysis of THR o THR and these weaker classes. The epsilon-error CAPP problem asks to approximate the acceptance probability of a given circuit to within additive error epsilon; it is the "canonical" derandomization problem. We show: - There is a non-trivial (2^n/n^{omega(1)} time) 1/poly(n)-error CAPP algorithm for poly(n)-size THR o THR circuits if and only if there is such an algorithm for poly(n)-size MAJ o MAJ. - There is a delta > 0 and a non-trivial SAT (delta-error CAPP) algorithm for poly(n)-size THR o THR circuits if and only if there is such an algorithm for poly(n)-size THR o MAJ. Similar results hold for depth-d linear threshold circuits and depth-d MAJORITY circuits. These equivalences are proved via new simulations of THR circuits by circuits with MAJ gates. - We strengthen the connection between non-trivial derandomization (non-trivial CAPP algorithms) for a circuit class C, and circuit lower bounds against C. Previously, [Ben-Sasson and Viola, ICALP 2014] (following [Williams, STOC 2010]) showed that for any polynomial-size class C closed under projections, non-trivial (2^{n}/n^{omega(1)} time) CAPP for OR_{poly(n)} o AND_{3} o C yields NEXP does not have C circuits. We apply Probabilistic Checkable Proofs of Proximity in a new way to show it would suffice to have a non-trivial CAPP algorithm for either XOR_2 o C, AND_2 o C or OR_2 o C. - A direct corollary of the first two bullets is that NEXP does not have THR o THR circuits would follow from either: - a non-trivial delta-error CAPP (or SAT) algorithm for poly(n)-size THR o MAJ circuits, or - a non-trivial 1/poly(n)-error CAPP algorithm for poly(n)-size MAJ o MAJ circuits. - Applying the above machinery, we extend lower bounds for depth-two neural networks and related models [R. Williams, CCC 2018] to weak approximate computations of Boolean functions. For example, for arbitrarily small epsilon > 0, we prove there are Boolean functions f computable in nondeterministic n^{log n} time such that (for infinitely many n) every polynomial-size depth-two neural network N on n inputs (with sign or ReLU activation) must satisfy max_{x in {0,1}^n}|N(x)-f(x)|>1/2-epsilon. That is, short linear combinations of ReLU gates fail miserably at computing f to within close precision. Similar results are proved for linear combinations of ACC o THR circuits, and linear combinations of low-degree F_p polynomials. These results constitute further progress towards THR o THR lower bounds