Approximate Problems for Finite Transducers

Finite (word) state transducers extend finite state automata by defining a binary relation over finite words, called rational relation. If the rational relation is the graph of a function, this function is said to be rational. The class of sequential functions is a strict subclass of rational functions, defined as the functions recognised by input-deterministic finite state transducers. The class membership problems between those classes are known to be decidable. We consider approximate versions of these problems and show they are decidable as well. This includes the approximate functionality problem, which asks whether given a rational relation (by a transducer), is it close to a rational function, and the approximate determinisation problem, which asks whether a given rational function is close to a sequential function. We prove decidability results for several classical distances, including Hamming and Levenshtein edit distance. Finally, we investigate the approximate uniformisation problem, which asks, given a rational relation R, whether there exists a sequential function that is close to some function uniformising R. As its exact version, we prove that this problem is undecidable

The Ultimate Signs of Second-Order Holonomic Sequences

A real-valued sequence f = {f(n)}_{n ∈ ℕ} is said to be second-order holonomic if it satisfies a linear recurrence f (n + 2) = P (n) f (n + 1) + Q (n) f (n) for all sufficiently large n, where P, Q ∈ ℝ(x) are rational functions. We study the ultimate sign of such a sequence, i.e., the repeated pattern that the signs of f (n) follow for sufficiently large n. For each P, Q we determine all ultimate signs that f can have, and show how they partition the space of initial values of f. This completes the prior work by Neumann, Ouaknine and Worrell, who have settled some restricted cases. As a corollary, it follows that when P, Q have rational coefficients, f either has an ultimate sign of length 1, 2, 3, 4, 6, 8 or 12, or never falls into a repeated sign pattern. We also give a partial algorithm that finds the ultimate sign of f (or tells that there is none) in almost all cases

Type-Safe and Portable Support for Packed Data (Experience Paper)

When components of a system exchange data, they need to serialise the data so that it can be sent over the network. Then, the recipient has to deserialise the data in order to be able to process it. These steps take time and have an impact on the overall system’s performance. A solution to this is to use packed data, which has a unified representation between the memory and the network, removing the need for any marshalling steps. Additionally, using this data representation can improve the program’s performance thanks to the data locality enabled by the compact representation of the data in memory. Unfortunately, no mainstream programming languages support packed data, whether it’s out-of-the-box or through a compiler extension. We present packed-data, a Haskell library that allows for type safe building and reading of packed data in a functional style. The library does not rely on compiler modifications, making it portable, and leverages meta-programming to allow programmers to pack their own data types effortlessly. We evaluate the usability and performance of the library, and conclude that it allows traversing packed data up to 60% faster than unpacked data. We also reflect on how to enhance the performance of library-based support for packed data. Our implementation approach is general and can easily be used with any programming languages that support higher-kinded types

RacerF: Lightweight Static Data Race Detection for C Code (Experience Paper)

We present RacerF, a novel static analyser for thread-modular data race detection. The approach behind RacerF exploits static analysis of sequential program behaviour whose results are generalised for multi-threaded programs using a combination of lightweight under- and over-approximating methods. The tool is implemented as a plugin of the Frama-C platform and can leverage several analysis backends, most notably the Frama-C’s abstract interpreter EVA. Although our methods are mostly heuristic without providing formal guarantees, our experimental evaluation shows that even for intricate programs, RacerF can provide very precise results competitive with more heavyweight approaches while being faster than them

Decomposing Multiparameter Persistence Modules

Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation

Nearest Neighbor Searching in a Dynamic Simple Polygon

In the nearest neighbor problem, we are given a set S of point sites that we want to store such that we can find the nearest neighbor of a (new) query point efficiently. In the dynamic version of the problem, the goal is to design a data structure that supports both efficient queries and updates, i.e. insertions and deletions in S. This problem has been widely studied in various settings, ranging from points in the plane to more general distance measures and even points within simple polygons. When the sites do not live in the plane but in some domain, another dynamic problem arises: what happens if not the sites, but the domain itself is subject to updates? Updating sites often results in local changes to the solution or data structure, while updating the domain may incur many global changes. For example, in the closest pair problem, inserting a point only requires us to check if this point is in the new closest pair, while updating the domain might change the distances between most pairs of points in our set. Presumably, this is the reason that this form of dynamization has received much less attention. Only some basic problems, such as shortest paths and ray shooting, have been studied in this setting. Here, we tackle the nearest neighbor problem in a dynamic simple polygon. We allow insertions into both the set of sites and the polygon. An insertion in the polygon is the addition of a line segment starting at the boundary of the polygon. We present a near-linear size -in both the number of sites and the complexity of the polygon- data structure with sublinear update and query time. This is the first nearest neighbor data structure that allows for updates to the domain

Computing Geomorphologically Salient Networks via Discrete Morse Theory

Rivers, estuaries, intertidal zones, and other hydrological systems often give rise to complex networks of interconnected channels. Even today, such networks are typically drawn manually by domain experts. Traditional watershed methods for automating this process, where water flows are assumed to follow steepest descent, fail to capture behavior particular to low-relief terrains. At SoCG 2017, Kleinhans et al. proposed a method to construct a network of source-to-sink paths separated by sufficient sediment volume. However, this method is unstable with respect to minor changes of the input terrain, and constructs only channels that flow from one side of the terrain to the other, thereby failing to detect the dead-end channels ("fingers") that characterize intertidal zones. We show how to compute geomorphologically salient networks that avoid these issues. After extending elevation data to a discrete Morse function on the terrain, we identify channels that flow through saddles and have sufficient volume of sediment on both sides. We then detect fingers, which follow the boundary of "spurs" that have sufficient volume of sediment above a particular height. The main challenge here lies in meaningfully modeling salient spurs and determining suitable heights to measure volume. We implemented our method and applied it to real-world data. Our expert users have validated the mathematical modeling by confirming that the resulting (finger) channels indeed constitute a geomorphologically salient network

Violating Constant Degree Hypothesis Requires Breaking Symmetry

The Constant Degree Hypothesis was introduced by Barrington et. al. [David A. Mix Barrington et al., 1990] to study some extensions of q-groups by nilpotent groups and the power of these groups in a computation model called NuDFA (non-uniform DFA). In its simplest formulation, it establishes exponential lower bounds for MOD_q∘MOD_m∘AND_d circuits computing AND of unbounded arity n (for constant integers d,m and a prime q). While it has been proved in some special cases (including d = 1), it remains wide open in its general form for over 30 years. In this paper we prove that the hypothesis holds when we restrict our attention to symmetric circuits with m being a prime. While we build upon techniques by Grolmusz and Tardos [Vince Grolmusz and Gábor Tardos, 2000], we have to prove a new symmetric version of their Degree Decreasing Lemma and use it to simplify circuits in a symmetry-preserving way. Moreover, to establish the result, we perform a careful analysis of automorphism groups of MOD_m∘AND_d subcircuits and study the periodic behaviour of the computed functions. Our methods also yield lower bounds when d is treated as a function of n. Finally, we present a construction of symmetric MOD_q∘MOD_m∘AND_d circuits that almost matches our lower bound and conclude that a symmetric function f can be computed by symmetric MOD_q∘MOD_p∘AND_d circuits of quasipolynomial size if and only if f has periods of polylogarithmic length of the form p^k q^

Front Matter, Table of Contents, Preface, Conference Organization

Front Matter, Table of Contents, Preface, Conference Organizatio