Related papers: P is not equal to NP by Modus Tollens
We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable…
Parallel tempering (PT), also known as replica exchange, is the go-to workhorse for simulations of multi-modal distributions. The key to the success of PT is to adopt efficient swap schemes. The popular deterministic even-odd (DEO) scheme…
Scheduling on related machines ($Q||C_{\max}$) is one of the most important problems in the field of Algorithmic Mechanism Design. Each machine is controlled by a selfish agent and her valuation can be expressed via a single parameter, her…
Just recently, complementarity relations (CRs) have been derived from the basic rules of Quantum Mechanics. The complete CRs are equalities involving quantum coherence, $C$, quantum entanglement, and predictability, $P$. While the first two…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…
The main purpose of this paper is to study the NP-complete subset-sum problem, not in the usual context of time-complexity-based classification of the algorithms (exponential/polynomial), but through a new kind of algorithmic classification…
A common practice of ML systems development concerns the training of the same model under different data sets, and the use of the same (training and test) sets for different learning models. The first case is a desirable practice for…
We present new algorithms for a posteriori verification of neural networks (NNs) approximating solutions to PDEs. These verification algorithms compute accurate estimates of $L^p$ norms of NNs and their derivatives. When combined with…
We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a…
Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
We use machine learning to provide a tractable measure of the amount of predictable variation in the data that a theory captures, which we call its "completeness." We apply this measure to three problems: assigning certain equivalents to…
The Church-Turing thesis is one of the pillars of computer science; it postulates that every classical system has equivalent computability power to the so-called Turing machine. While this thesis is crucial for our understanding of…
We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system $P$, we prove that the following statements are equivalent, 1.…
We introduce a new type of generalized Turing machines (GTMs), which are intended as a tool for the mathematician who studies computability in Analysis. In a single tape cell a GTM can store a symbol, a real number, a continuous real…
Classical computability theory tells us that self-modifying code (SMC) on a deterministic universal Turing machine can be simulated by non-SMC code on the same model. That abstraction, however, omits the external timing inputs, concurrency,…
We discuss the problem of construction of inference procedures which can manipulate with uncertainties measured in ordinal scales and fulfill to the property of strict monotonicity of conclusion. The class of A-valuations of plausibility is…
This article expands our work in [Ca16]. By its reliance on Turing computability, the classical theory of effectivity, along with effective reducibility and Weihrauch reducibility, is only applicable to objects that are either countable or…
There is an important and interesting open question in computational complexity on the relation between the complexity classes $\mathcal{NP}$ and $\mathcal{PSPACE}$. It is a widespread belief that $\mathcal{NP}\ne\mathcal{PSPACE}$. In this…
We investigate modal logical aspects of provability predicates $\mathrm{Pr}_T(x)$ satisfying the following condition: $\mathbf{M}$: If $T \vdash \varphi \to \psi$, then $T \vdash \mathrm{Pr}_T(\ulcorner \varphi \urcorner) \to…