Abstract Envy-free cake-cutting protocols procedurally divide an infinitely divisible good among a set of agents so that no agent prefers another’s allocation to their own. These protocols are highly complex and difficult to prove correct. Recently, Bertram, Levinson, and Hsu introduced a language called Slice for describing and verifying cake-cutting protocols. Slice programs can be translated to formulas encoding envy-freeness, which are solved by SMT. While Slice works well on smaller protocols, it has difficulty scaling to more complex cake-cutting protocols. We improve Slice in two ways. First, we show any protocol execution in Slice can be replicated using piecewise uniform valuations. We then reduce Slice’s constraint formulas to formulas within the theory of linear real arithmetic, showing that verifying envy-freeness is efficiently decidable. Second, we design and implement a linear type system which enforces that no two agents receive the same part of the good. We implement our methods and verify a range of challenging examples, including the first nontrivial four-agent protocol.

Cache replacement policies typically use some form of statistics on past access behavior. As a common limitation, however, the extent of the history being recorded is limited to either just the data in cache or, more recently, a larger but still finite-length window of accesses, because the cost of keeping a long history can easily outweigh its benefit.

The fair division literature in economics considers how to divide resources between multiple agents such that the allocation is envy-free: each agent receives their favorite piece. Researchers have developed a variety of fair division protocols for the most standard setting, where the agents want to split a single item, however, the protocols are highly intricate and the proofs of envy-freeness involve tedious case analysis. We propose Slice, a domain specific language for fair-division. Programs in our language can be converted to logical formulas encoding envy-freeness and other target properties. Then, the constraints can be dispatched to automated solvers. We prove that our constraint generation procedure is sound and complete. We also report on a prototype implementation of Slice, which we have used to automatically check envy-freeness for several protocols from the fair division literature.

Analyzing patterns in a sequence of events has applications in text analysis, computer programming, and genomics research. In this paper, we consider the all-window-length analysis model which analyzes a sequence of events with respect to windows of all lengths. We study the exact co-occurrence counting problem for the all-window-length analysis model. Our first algorithm is an offline algorithm that counts all-window-length co-occurrences by performing multiple passes over a sequence and computing single-window-length co-occurrences. This algorithm has the time complexity O(n) for each window length and thus a total complexity of O(n²) and the space complexity O(|I|) for a sequence of size n and an itemset of size |I|. We propose AWLCO, an online algorithm that computes all-window-length co-occurrences in a single pass with the time complexity of O(n) and space complexity of O(√{n|I|}), assuming perfect hashing. Following this, we generalize our use case to patterns in which we propose an algorithm that computes all-window-length co-occurrence with time complexity O(n|I|), assuming perfect hashing, with an additional pre-processing step and space complexity O(√{n|I|}+|I|), plus the overhead of the Aho-Corasick algorithm [Aho and Corasick, 1975].