Visual data such as images and videos are typically modeled as discretizations of inherently continuous, multidimensional signals. Existing continuous-signal models attempt to exploit this fact by modeling the underlying signals of visual (e.g., image) data directly. However, these models have not yet been able to achieve competitive performance on practical vision tasks such as large-scale image and video classification. Building on a recent line of work on deep state space models (SSMs), we propose \method, a new multidimensional SSM layer that extends the continuous-signal modeling ability of SSMs to multidimensional data including images and videos. We show that S4ND can model large-scale visual data in $1$D, $2$D, and $3$D as continuous multidimensional signals and demonstrates strong performance by simply swapping Conv2D and self-attention layers with \method\ layers in existing state-of-the-art models. On ImageNet-1k, \method\ exceeds the performance of a Vision Transformer baseline by $1.5\%$ when training with a $1$D sequence of patches, and matches ConvNeXt when modeling images in $2$D. For videos, S4ND improves on an inflated $3$D ConvNeXt in activity classification on HMDB-51 by $4\%$. S4ND implicitly learns global, continuous convolutional kernels that are resolution invariant by construction, providing an inductive bias that enables generalization across multiple resolutions. By developing a simple bandlimiting modification to S4 to overcome aliasing, S4ND achieves strong zero-shot (unseen at training time) resolution performance, outperforming a baseline Conv2D by $40\%$ on CIFAR-10 when trained on $8 \times 8$ and tested on $32 \times 32$ images. When trained with progressive resizing, S4ND comes within $\sim 1\%$ of a high-resolution model while training $22\%$ faster.

Boolean Skolem function synthesis concerns syn¬thesizing outputs as Boolean functions of inputs such that a relational specification between inputs and outputs is satisfied. This problem, also known as Boolean functional synthesis, has several applications, including design of safe controllers for autonomous systems, certified QBF solving, cryptanalysis etc. Recently, complexity theoretic hardness results have been shown for the problem, although several algorithms proposed in the literature are known to work well in practice. This dichotomy between theoretical hardness and practical efficacy has motivated research on normal forms of specification representation that guarantee efficient synthesis, thus partially explaining the efficacy of some of these algorithms.In this paper we go one step further and ask if there exists a normal form representation of the specification that precisely characterizes "efficient" synthesis. We present a normal form called SAUNF that answers this question affirmatively. Specifically, a specification is polynomial time synthesizable iff it can be compiled to SAUNF in polynomial time. Additionally, a specification admits a polynomial-sized functional solution iff there exists a semantically equivalent polynomial-sized SAUNF representation. SAUNF is exponentially more succinct than well- established normal forms like BDDs and DNNFs, used in the context of AI problems, and strictly subsumes other more recently proposed forms like SynNNF. It enjoys compositional properties that are similar to those of DNNF. Thus, SAUNF provides the right trade-off in knowledge representation for Boolean functional synthesis.

In recent years, programming has witnessed a shift towards using standard libraries as a black box. However, there has not been a synchronous development of tools that can help demonstrate the working of such libraries in general programs, which poses an impediment to improved learning outcomes and makes debugging exasperating. We introduce Eye, an interactive pedagogical tool that visualizes a program's execution as it runs. It demonstrates properties and usage of data structures in a general environment, thereby helping in learning, logical debugging, and code comprehension. Eye provides a comprehensive overview at each stage during run time including the execution stack and the state of data structures. The modular implementation allows for extension to other languages and modification of the graphics as desired.

Functional languages manage heap data through garbage collection. Since static analysis of heap data is difficult, garbage collectors conservatively approximate the liveness of heap objects by reachability i.e. every object that is reachable from the root set is considered live. Consequently, a large amount of memory that is reachable but not used further during execution is left uncollected by the collector.

Abstract Unordered data Petri nets (UDPN) are an extension of classical Petri nets with tokens that carry data from an infinite domain and where transitions may check equality and disequality of tokens. UDPN are well-structured, so the coverability and termination problems are decidable, but with higher complexity than for Petri nets. On the other hand, the problem of reachability for UDPN is surprisingly complex, and its decidability status remains open. In this paper, we consider the continuous reachability problem for UDPN, which can be seen as an over-approximation of the reachability problem. Our main result is a characterization of continuous reachability for UDPN and polynomial time algorithm for solving it. This is a consequence of a combinatorial argument, which shows that if continuous reachability holds then there exists a run using only polynomially many data values.