We will present a system of Intensional Higher-Order Intuitionistic Logic with Higher-Order Saraswat Constraints, hoI(C). The Logic and its variants were developed by Artalejo, Nieva, Leach (Madrid), and subsequently by Lipton, Nieva and Hermant, the original aim being to add constraints to a lambda-Prolog style language.

If constraints are unrestricted, cut-elimination is easily shown to fail, because of the way the system handles equality rules. Our work studies conditions on the constraint system C that guarantee cut-elimination. The techniques used are based on Takahashi, Prawitz ande Andrews' work in classical type theory, Lipton and DeMarco for intuitionistic type theory, with ideas developed by Hermant for Logique Modulo, Hermant and Lipton for Logic with sequent axioms, as well as cut-elimination techniques due to Okada and Maehara.

We will briefly discuss Kripke and HA semantics for these calculi, with some remarks on fibrational semantics developed by the speaker with Gianluca Amato (Pescara, Italy) if time permits!

This will be a blackboard talk (with maybe a few slides) and will include work in progress!

Many computer systems are reactive: they execute in continuous interaction with their environment. From the perspective of functional programming, reactive systems can be modeled and programmed as stream functions. Several lines of research have exploited this point of view. Works based on Nakano’s guarded recursion use type-theoretical modalities to guarantee productivity. Works based on synchronous programming use more specialized methods but guarantee incremental processing of stream fragments. In this paper, we contribute to a recent synthesis between the two approaches by describing how a language with a family of type-theoretical modalities can be given an incremental semantics in the synchronous style. We prove that this semantics coincides with a more usual big-step semantics.

I will introduce a multicore algorithm for the parallel detection of Strongly Connected Components. The algorithm uses a concurrent Union-Find forest to share partial SCCs. As a consequence, we get parallel speedup even if the graph consists of a single SCC. The algorithm works on-the-fly, so it is useful for parallel LTL model checking.

Since parallel graph algorithms are error prone, and this one is used in verification tools, we would like to verify it. We will report a first step, modeling the algorithm in TLA+ and model checking it for instances with a few workers and small graphs.

We study a generalization of the classic Online Joint Replenishment Problem (JRP) with Delays that we call the Online Weighted Cardinality JRP with Delays. The JRP is an extensively studied inventory management problem wherein requests for different item types arrive at various points in time. A request is served by ordering its corresponding item type. The cost of serving a set of requests depends on the item types ordered. Furthermore, each request incurs a delay penalty while it is left unserved. The objective is to minimise the total service and delay costs. In the Weighted Cardinality JRP, each item type has a positive weight and the cost of ordering is a non-decreasing, concave function of the total weight of the item types ordered. This problem was first considered in the offline setting by Cheung et al.~(2015) but nothing is known in the online setting. Our main result is a deterministic, constant competitive algorithm for this problem.

With the world as connected as ever and data being created at an unprecedented rate, there has been an increasing need for algorithms that efficiently process massive graphs. Graph streaming is one such memory efficient setting where the edges of a graph are presented as a sequence of edges and an algorithm's goal is to store a sublinear number of edges needed to solve a graph problem exactly or approximately. In particular, this talk discusses the insertion-only, dynamic (or insertion-deletion) and sliding window graph streaming models of computation, highlighting the well-studied maximum matching and sister minimum vertex cover problems, and focusing on a recent space optimal (up to constant factors) algorithm for minimum vertex cover in the dynamic graph streaming model.

Let $b \ge 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any $\times b$-invariant, non-dense subset $A$ of $[0,1)$, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained. This is joint work with Bing Li and Yufeng Wu.