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We present a probabilistic version of PCF, a well-known simply typed universal functional language. The type hierarchy is based on a single ground type of natural numbers. Even if the language is globally call-by-name, we allow a…
As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in…
We propose a novel method for fast and scalable evaluation of periodic solutions of systems of ordinary differential equations for a given set of parameter values and initial conditions. The equations governing the system dynamics are…
In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by…
We develop a novel approach for confidently accelerating inference in the large and expensive multilayer Transformers that are now ubiquitous in natural language processing (NLP). Amortized or approximate computational methods increase…
Representation determines how we can reason about a specific problem. Sometimes one representation helps us find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is,…
Deep learning models for semantics are generally evaluated using naturalistic corpora. Adversarial methods, in which models are evaluated on new examples with known semantic properties, have begun to reveal that good performance at these…
Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further…
An inductive inference system for proving validity of formulas in the initial algebra $T_{\mathcal{E}}$ of an order-sorted equational theory $\mathcal{E}$ is presented. It has 20 inference rules, but only 9 of them require user interaction;…
The elementary affine lambda-calculus was introduced as a polyvalent setting for implicit computational complexity, allowing for characterizations of polynomial time and hyperexponential time predicates. But these results rely on type…
We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations'…
There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a…
Substitutions play a crucial role in a wide range of contexts, from analyzing the dynamics of social opinions and conducting mathematical computations to engaging in game-theoretical analysis. For many situations, considering one-step…
Guaranteeing the correctness and factuality of language model (LM) outputs is a major open problem. In this work, we propose conformal factuality, a framework that can ensure high probability correctness guarantees for LMs by connecting…
In various applications the search for certificates for certain properties (e.g., stability of dynamical systems, program termination) can be formulated as a quantified constraint solving problem with quantifier prefix exists-forall. In…
We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of…
We present an extensive mechanization of the meta-theory of Martin-L\"of Type Theory (MLTT) in the Coq proof assistant. Our development builds on pre-existing work in Agda to show not only the decidability of conversion, but also the…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
Reliable verification of proofs remains a bottleneck for training and evaluating AI systems on hard mathematical reasoning. Fully formal proofs, in languages like Lean, are easy to verify because they are unambiguous and modular. Most…
We describe a Martin-L\"of-style dependent type theory, called Cocon, that allows us to mix the intensional function space that is used to represent higher-order abstract syntax (HOAS) trees with the extensional function space that…