We show that, surprisingly, the notion of optimal finite-time regret is not a uniquely defined property in this context and that, in general, it is decoupled from the asymptotic rate. We discuss alternative choices and propose a notion of finite-time optimality that we argue is meaningful .

The Heaviside activation function is for instance used in Hopfield networks [ 1 ] that have recently seen a resurge due to their connections t o attention layers [ 2, 3 ] and the 2024 Nobel Prize in Physics that was partially award ed for their development. Moreover, the Heaviside activation function is closely related to quantized neural networks [ 4, 5 ], playing a key role in enabling energy efficient deployment o f large language models (LLMs) [ 6, 7 ]. We refer to neural networks with several hidden layers and th e Heaviside activation function as deep Heaviside (neural) networks (DHNs). These networks are also known as (linear) threshold networks. The Heaviside activation function can be traced back to the fi rst attempts to build an artificial counterpart of a biological neuron. In the brain, the inputs of a neuron contribute to its membrane potential and the neuron discharges/fires if th e membrane potential exceeds a certain threshold.

We consider the problem of assigning items to platforms while satisfying group and individual fairness constraints. Each item is associated with certain groups and has a preference ordering over platforms. Each platform enforces group fairness by specifying an upper and a lower bound on the number of items that can be matched to it from each group. Although there may be multiple optimal solutions that satisfy the group fairness constraints, we aim to achieve `probabilistic individual fairness' by computing a distribution over `group fair' matchings such that each item has a reasonable probability of being matched to one of its top choices. When each item can belong to multiple groups, the problem of finding a maximum size group-fair matching is NP-hard even when all the group lower bounds are 0, and there are no individual fairness constraints. Given a total of $n$ items, we achieve a $O(\Delta \log n)$ approximation algorithm when an item can belong to at most $\Delta$ groups, and all the group lower bounds are 0. We also provide two approximation algorithms in terms of the total number of groups that have items in the neighborhood of a platform. When each item belongs to a single group, we provide a polynomial-time algorithm that computes a probabilistic individually fair distribution over group fair matching. We further extend our model and algorithms to address the following notions of fairness: `maxmin group fairness', which maximizes the representation of the worst-off groups, and `mindom group fairness', which minimizes the representation of the most dominant groups.

Answer set programming (ASP) is a well-established knowledge representation formalism that grew from the observation that stable models [33] of a logic program can be used to encode search problems [59, 62, 49]. ASP is rapidly gaining adoption, with applications in domains such as decision support for the Space Shuttle [63], product configuration [75], phylogenetic inference [45, 11], knowledge management [37], e-Tourism [65], biology [32], robotics [5], and machine learning [41, 12]. The success of ASP can, to a large extend, be explained by two factors. The first factor is a rich, first-order language, ASP-Core2 [13], to express knowledge in, with an easy-to-understand modeling methodology known as generate-define-and-test. The second factor is the availability of a large number of reliable tools -- grounders [31, 46] and solvers [28, 3, 16] -- that allow to efficiently compute stable models of a given logic program. Throughout its history, ASP has always benefited from progress in other domains of combinatorial search. For instance, the addition of conflict-driven clause learning (CDCL) [60] to Boolean satisfiability (SAT) solvers is often recognized as one of the most important leaps forward in SAT solving; this technique was very quickly adopted in ASP.

Answer Set Programming (ASP) is a well-known paradigm of declarative programming with roots in logic programming and non-monotonic reasoning. Similar to other closely related problem-solving technologies, such as SAT/SMT, QBF, Planning and Scheduling, advancements in ASP solving are assessed in competition events. In this paper, we report about the design and results of the Sixth ASP Competition, which was jointly organized by the University of Calabria (Italy), Aalto University (Finland), and the University of Genoa (Italy), in affiliation with the 13th International Conference on Logic Programming and Non-Monotonic Reasoning. This edition maintained some of the design decisions introduced in 2014, e.g., the conception of sub-tracks, the scoring scheme, and the adherence to a fixed modeling language in order to push the adoption of the ASP-Core-2 standard. On the other hand, it featured also some novelties, like a benchmark selection stage classifying instances according to their empirical hardness, and a "Marathon" track where the top-performing systems are given more time for solving hard benchmarks.

Lifted inference has been proposed for various probabilistic logical frameworks in order to compute the probability of queries in a time that depends on the size of the domains of the random variables rather than the number of instances. Even if various authors have underlined its importance for probabilistic logic programming (PLP), lifted inference has been applied up to now only to relational languages outside of logic programming. In this paper we adapt Generalized Counting First Order Variable Elimination (GC-FOVE) to the problem of computing the probability of queries to probabilistic logic programs under the distribution semantics. In particular, we extend the Prolog Factor Language (PFL) to include two new types of factors that are needed for representing ProbLog programs. These factors take into account the existing causal independence relationships among random variables and are managed by the extension to variable elimination proposed by Zhang and Poole for dealing with convergent variables and heterogeneous factors. Two new operators are added to GC-FOVE for treating heterogeneous factors. The resulting algorithm, called LP$^2$ for Lifted Probabilistic Logic Programming, has been implemented by modifying the PFL implementation of GC-FOVE and tested on three benchmarks for lifted inference. A comparison with PITA and ProbLog2 shows the potential of the approach.