Our main contributions are as follows.

Responsibility has long been a subject of study in law and philosophy. More recently, it became a focus of AI literature. The article investigates the computational complexity of two important properties of responsibility in collective decision-making: diffusion and gap. It shows that the sets of diffusion-free and gap-free decision-making mechanisms are $Π_2$-complete and $Π_3$-complete, respectively. At the same time, the intersection of these classes is $Π_2$-complete.

In this manuscript I present an analysis on the performance of OpenAI O1-preview model in solving random K-SAT instances for K$\in {2,3,4}$ as a function of $\alpha=M/N$ where $M$ is the number of clauses and $N$ is the number of variables of the satisfiable problem. I show that the model can call an external SAT solver to solve the instances, rather than solving them directly. Despite using external solvers, the model reports incorrect assignments as output. Moreover, I propose and present an analysis to quantify whether the OpenAI O1-preview model demonstrates a spark of intelligence or merely makes random guesses when outputting an assignment for a Boolean satisfiability problem.

In procedural content generation, modeling the generation task as a constraint satisfaction problem lets us define local and global constraints on the generated output. However, a generator's perceived quality often involves statistics rather than just hard constraints. For example, we may desire that generated outputs use design elements with a similar distribution to that of reference designs. However, such statistical properties cannot be expressed directly as a hard constraint on the generation of any one output. In contrast, methods which do not use a general-purpose constraint solver, such as Gumin's implementation of the WaveFunctionCollapse (WFC) algorithm, can control output statistics but have limited constraint propagation ability and cannot express non-local constraints. In this paper, we introduce You-Only-Randomize-Once (YORO) pre-rolling, a method for crafting a decision variable ordering for a constraint solver that encodes desired statistics in a constraint-based generator. Using a solver-based WFC as an example, we show that this technique effectively controls the statistics of tile-grid outputs generated by several off-the-shelf SAT solvers, while still enforcing global constraints on the outputs.1 Our approach is immediately applicable to WFC-like generation problems and it offers a conceptual starting point for controlling the design element statistics in other constraint-based generators.

The remarkable achievements of machine learning techniques in analyzing discrete structures have drawn significant attention towards their integration into combinatorial optimization algorithms. Typically, these methodologies improve existing solvers by injecting learned models within the solving loop to enhance the efficiency of the search process. In this work, we derive a single differentiable function capable of approximating solutions for the Maximum Satisfiability Problem (MaxSAT). Then, we present a novel neural network architecture to model our differentiable function, and progressively solve MaxSAT using backpropagation. This approach eliminates the need for labeled data or a neural network training phase, as the training process functions as the solving algorithm. Additionally, we leverage the computational power of GPUs to accelerate these computations. Experimental results on challenging MaxSAT instances show that our proposed methodology outperforms two existing MaxSAT solvers, and is on par with another in terms of solution cost, without necessitating any training or access to an underlying SAT solver. Given that numerous NP-hard problems can be reduced to MaxSAT, our novel technique paves the way for a new generation of solvers poised to benefit from neural network GPU acceleration.

In this work we face a challenging puzzle video game: A Good Snowman is Hard to Build. The objective of the game is to build snowmen by moving and stacking snowballs on a discrete grid. For the sake of player engagement with the game, it is interesting to avoid that a player finds a much easier solution than the one the designer expected. Therefore, having tools that are able to certify the optimality of solutions is crucial. Although the game can be stated as a planning problem and can be naturally modelled in PDDL, we show that a direct translation to SAT clearly outperforms off-the-shelf state-of-the-art planners. As we show, this is mainly due to the fact that reachability properties can be easily modelled in SAT, allowing for shorter plans, whereas using axioms to express a reachability derived predicate in PDDL does not result in any significant reduction of solving time with the considered planners. We deal with a set of 51 levels, both original and crafted, solving 43 and with 8 challenging instances still remaining to be solved.

Minimal trap spaces (MTSs) capture subspaces in which the Boolean dynamics is trapped, whatever the update mode. They correspond to the attractors of the most permissive mode. Due to their versatility, the computation of MTSs has recently gained traction, essentially by focusing on their enumeration. In this paper, we address the logical reasoning on universal properties of MTSs in the scope of two problems: the reprogramming of Boolean networks for identifying the permanent freeze of Boolean variables that enforce a given property on all the MTSs, and the synthesis of Boolean networks from universal properties on their MTSs. Both problems reduce to solving the satisfiability of quantified propositional logic formula with 3 levels of quantifiers ($\exists\forall\exists$). In this paper, we introduce a Counter-Example Guided Refinement Abstraction (CEGAR) to efficiently solve these problems by coupling the resolution of two simpler formulas. We provide a prototype relying on Answer-Set Programming for each formula and show its tractability on a wide range of Boolean models of biological networks.

We consider a weakly supervised learning scenario where the supervision signal is generated by a transition function $\sigma$ of labels associated with multiple input instances. We formulate this problem as \emph{multi-instance Partial Label Learning (multi-instance PLL)}, which is an extension to the standard PLL problem. Our problem is met in different fields, including latent structural learning and neuro-symbolic integration. Despite the existence of many learning techniques, limited theoretical analysis has been dedicated to this problem. In this paper, we provide the first theoretical study of multi-instance PLL with possibly an unknown transition $\sigma$. Our main contributions are as follows. Firstly, we propose a necessary and sufficient condition for the learnability of the problem. This condition non-trivially generalizes and relaxes the existing small ambiguity degree in the PLL literature, since we allow the transition to be deterministic. Secondly, we derive Rademacher-style error bounds based on a top-$k$ surrogate loss that is widely used in the neuro-symbolic literature. Furthermore, we conclude with empirical experiments for learning under unknown transitions. The empirical results align with our theoretical findings; however, they also expose the issue of scalability in the weak supervision literature.

We observed that Conjunctive Normal Form (CNF) encodings of structured SAT instances often have a set of consecutive clauses defined over a small number of Boolean variables. To exploit the pattern, we propose a transformation of CNF to an alternative representation, Conjunctive Minterm Canonical Form (CMCF). The transformation is a two-step process: CNF clauses are first partitioned into disjoint subsets such that each subset contains CNF clauses with shared Boolean variables. CNF clauses in each subset are then replaced by Minterm Canonical Form (i.e., partial solutions), which is found by enumeration. We show empirically that a simple Stochastic Local Search (SLS) solver based on CMCF can consistently achieve a higher success rate using fewer evaluations than the SLS solver WalkSAT on two representative classes of structured SAT problems.

When solving a combinatorial problem using propositional satisfiability (SAT), the encoding of the problem is of vital importance. We study encodings of Pseudo-Boolean (PB) constraints, a common type of arithmetic constraint that appears in a wide variety of combinatorial problems such as timetabling, scheduling, and resource allocation. In some cases PB constraints occur together with at-most-one (AMO) constraints over subsets of their variables (forming PB(AMO) constraints). Recent work has shown that taking account of AMOs when encoding PB constraints using decision diagrams can produce a dramatic improvement in solver efficiency. In this paper we extend the approach to other state-of-the-art encodings of PB constraints, developing several new encodings for PB(AMO) constraints. Also, we present a more compact and efficient version of the popular Generalized Totalizer encoding, named Reduced Generalized Totalizer. This new encoding is also adapted for PB(AMO) constraints for a further gain. Our experiments show that the encodings of PB(AMO) constraints can be substantially smaller than those of PB constraints. PB(AMO) encodings allow many more instances to be solved within a time limit, and solving time is improved by more than one order of magnitude in some cases. We also observed that there is no single overall winner among the considered encodings, but efficiency of each encoding may depend on PB(AMO) characteristics such as the magnitude of coefficient values.