Related papers: Notes on proof by dichotomy
Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…
We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs…
We discuss an "operational" approach to testing convex composite hypotheses when the underlying distributions are heavy-tailed. It relies upon Euclidean separation of convex sets and can be seen as an extension of the approach to testing by…
We study how to perform tests on samples of pairs of observations and predictions in order to assess whether or not the predictions are prudent. Prudence requires that that the mean of the difference of the observation-prediction pairs can…
We present a continuation method that enables one to track or continue branches of periodic orbits directly in an experiment when a parameter is changed. A control-based setup in combination with Newton iterations ensures that the periodic…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
Because of several technological limitations of traditional silicon based computing, for past few years a paradigm shift, from silicon to carbon, is occurring in computational world. DNA computing has been considered to be quite promising…
Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect…
We present a novel technique for proving program termination which introduces a new dimension of modularity. Existing techniques use the program to incrementally construct a termination proof. While the proof keeps changing, the program…
Proofs are traditionally syntactic, inductively generated objects. This paper reformulates first-order logic (predicate calculus) with proofs which are graph-theoretic rather than syntactic. It defines a combinatorial proof of a formula…
Many versions of the Stokes theorem are known. More advanced of them require complicated mathematical machinery to be formulated which discourages the users. Our theorem is sufficiently simple to suit the handbooks and yet it is pretty…
We state a construction theorem for specifications starting from single-site conditional probabilities (singleton part). We consider general single-site spaces and kernels that are absolutely continuous with respect to a chosen product…
Understanding and creating mathematics using natural mathematical language - the mixture of symbolic and natural language used by humans - is a challenging and important problem for driving progress in machine learning. As a step in this…
We encode/decode Prolog terms as unique natural numbers. Our encodings have the following properties: a) are bijective b) natural numbers always decode to syntactically valid terms c) they work in low polynomial time in the bitsize of the…
Dependent Object Types (DOT) is intended to be a core calculus for modelling Scala. Its distinguishing feature is abstract type members, fields in objects that hold types rather than values. Proving soundness of DOT has been surprisingly…
The $\epsilon$-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory $ID_1$ using a variant of the cut-elimination…
We have developed an alternative approach to teaching computer science students how to prove. First, students are taught how to prove theorems with the Coq proof assistant. In a second, more difficult, step students will transfer their…
Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers…
We present a novel verification technique to prove interesting properties of a class of array programs with a symbolic parameter N denoting the size of arrays. The technique relies on constructing two slightly different versions of the same…
Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$ \sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)}…