“The pigeonhole principle: ‘If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole’ is an obvious yet fundamental priniciple of Nature as it captures the very essence of counting. Here however, we show that in quantum mechanics, this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box.”

The pigeonhole principle states that there is no one-to-one function from a set of size n to a set of size n − 1. In other words, if n pigeons are put into n − 1 holes, then at least one hole will be occupied by more than one pigeon. This simple fact has an astonishing variety of applications in mathematics. It also corresponds to a tautology that has been used extensively in the study of the c...

We consider a new way of looking at multisets, involving linear rather than multiplicative notation and operations which mirror those on natural numbers. The motivation is to create a framework for mathematics which more closely connects with scientific usage in which repetition and relative concepts of equality play an important role. The tropical calculus is introduced as an important tool. a...

We investigate the proof complexity of analytic subsystems of the deep inference proof system SKSg (the calculus of structures). Exploiting the fact that the cut rule (i↑) of SKSg corresponds to the ¬-left rule in the sequent calculus, we establish that the “analytic” system KSg+c↑ has essentially the same complexity as the monotone Gentzen calculus MLK . In particular, KSg + c↑ quasipolynomial...

We prove that every property of 3CNF formulas that implies unsatisfiability and is expressible in Datalog has asymptotic probability zero when formulas are randomly generated by taking 6n non-trivial clauses of exactly three literals uniformly and independently. Our result is a consequence of designing a winning strategy for Duplicator in the existential kpebble game on the structure that encod...

We introduce a new way of composing proofs in rule-based proof systems that generalizes tree-like and dag-like proofs. In the new definition, proofs are directed graphs of derived formulas, in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs circular. We show that, for all sets of standard inference rules, ci...

Abstract We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths the full binary tree. show that resulting principle equivalent $$\Sigma ^0_2$$ Σ 2 0 -ind...

By now Artificial Intelligence (AI), Theoretical Computer Science (CS theory) and Operations Research (OR) have investigated a variety of search and optimization problems. However, methods from these scientific areas use different problem descriptions, models, and tools. They also address problems with particular efficiency requirements. For example, approaches from CS theory are mainly concern...

LA is a simple and natural logical system for reasoning about matrices. We show that LA, over finite fields, proves a host of matrix identities (so called “hard matrix identities”) from the matrix form of the pigeonhole principle. LAP is LA with matrix powering; we show that LAP extended with quantification over permutations is strong enough to prove fundamental theorems of linear algebra (such...