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There have been several recent attempts to improve the accuracy of grammar induction systems by bounding the recursive complexity of the induction model (Ponvert et al., 2011; Noji and Johnson, 2016; Shain et al., 2016; Jin et al., 2018).…
It is natural for probabilistic programs to use conditionals to express alternative substructures in models, and loops (recursion) to express repeated substructures in models. Thus, probabilistic programs with conditionals and recursion…
Incremental gradient and incremental proximal methods are a fundamental class of optimization algorithms used for solving finite sum problems, broadly studied in the literature. Yet, without strong convexity, their convergence guarantees…
In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability…
It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that…
It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a…
We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to…
Algorithms are ways of mapping problems to solutions. An algorithm is invertible precisely when this mapping is injective, such that the initial problem can be uniquely inferred from its solution. While invertible algorithms can be…
Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…
Primitive recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, primitive corecursion, is underappreciated and still seen as exotic. We aim to put them both on equal footing by giving a…
We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous…
Iterative refinement -- start with a random guess, then iteratively improve the guess -- is a useful paradigm for representation learning because it offers a way to break symmetries among equally plausible explanations for the data. This…
We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms…
We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories. Operational semantics of coinductive Horn clause resolution is cast in terms of coinductive uniform proofs; its…
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known…
We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…
Two apparently unrelated fields -- normalizing flows and causality -- have recently received considerable attention in the machine learning community. In this work, we highlight an intrinsic correspondence between a simple family of…
Tail recursive functions allow for a wider range of optimisations than general recursive functions. For this reason, much research has gone into the transformation and optimisation of this family of functions, in particular those written in…
Martin-L\"of's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive…
We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this…