Related papers: Formal Probabilistic Methods for Combinatorial Str…
We present a formalization of basics related to infinite words in the generic proof assistant Isabelle/HOL. Furthermore, we present a formalization of purely morphic and morphic languages. Finally, we present a formalized definition of…
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…
Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about…
An old result by Shearer relates the Lov\'asz Local Lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard core lattice gas on graphs. We use this…
Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…
The constructive Lov\'{a}sz Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the…
Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical…
Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…
In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral…
We try to bring to light some combinatorial structure underlying formal proofs in logic. We do this through the study of the Craig Interpolation Theorem which is properly a statement about the structure of formal derivations. We show that…
Mechanized theorem proving is becoming the basis of reliable systems programming and rigorous mathematics. Despite decades of progress in proof automation, writing mechanized proofs still requires engineers' expertise and remains labor…
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep…
Recently, a growing number of researchers have applied machine learning to assist users of interactive theorem provers. However, the expressive nature of underlying logics and esoteric structures of proof documents impede machine learning…
We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…
Probabilistic programming languages (PPLs) allow programmers to construct statistical models and then simulate data or perform inference over them. Many PPLs restrict models to a particular instance of simulation or inference, limiting…
Probabilistic puzzles can be confusing, partly because they are formulated in natural languages - full of unclarities and ambiguities - and partly because there is no widely accepted and intuitive formal language to express them. We propose…
In probabilistic reasoning, the traditionally discrete domain has been elevated to the hybrid domain encompassing additionally continuous random variables. Inference in the hybrid domain, however, usually necessitates to condone trade-offs…
The Quantum Lov\'asz Local Lemma (QLLL) [AKS12] establishes non-constructively that any quantum system constrained by a local Hamiltonian has a zero-energy ground state, if the local Hamiltonian terms overlap only in a certain restricted…
The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which…
Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such…