Related papers: On the logical depth function
This paper studies codes that correct bursts of deletions. Namely, a code will be called a $b$-burst-deletion-correcting code if it can correct a deletion of any $b$ consecutive bits. While the lower bound on the redundancy of such codes…
Deep inference is a proof theoretic methodology that generalizes the standard notion of inference of the sequent calculus, whereby inference rules become applicable at any depth inside logical expressions. Deep inference provides more…
The depth of a link measures the minimum height of a resolving tree for the link whose leaves are all unlinks. We show that the depth of the closure of a strictly positive braid word is the length of the word minus the number of distinct…
This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with…
Recently, there has been an increasing interest in the bottom-up evaluation of the semantics of logic programs with complex terms. The presence of function symbols in the program may render the ground instantiation infinite, and finiteness…
We apply to logic programming some recently emerging ideas from the field of reduction-based communicating systems, with the aim of giving evidence of the hidden interactions and the coordination mechanisms that rule the operational…
This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width…
The possibility of translating logic programs into functional ones has long been a subject of investigation. Common to the many approaches is that the original logic program, in order to be translated, needs to be well-moded and this has…
In this paper, we study arbitrary infinite binary information systems each of which consists of an infinite set called universe and an infinite set of two-valued functions (attributes) defined on the universe. We consider the notion of a…
Lemma 1 from the paper [N.E. Gretsky, J.M. Ostroy, W.R. Zame, Subdifferentiability and the duality gap, Positivity 6: 261--274, 2002] asserts that the value function $v$ of an infinite dimensional linear programming problem in standard form…
Logic programming, as exemplified by datalog, defines the meaning of a program as its unique smallest model: the deductive closure of its inference rules. However, many problems call for an enumeration of models that vary along some set of…
A binary string of length $2^k$ induces the Boolean function of $k$ variables whose Shannon expansion is the given binary string. This Boolean function then is representable via a unique reduced ordered binary decision diagram (ROBDD). The…
LogicWeb has traditionally lacked devices for dealing with intractable queries. We address this limitation by adopting length-bounded inference, a form of approximate reasoning. A length-bounded inference is of the form $prov(P,G,n)$ which…
Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual…
We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously…
Recent theoretical results show transformers cannot express sequential reasoning problems over long inputs, intuitively because their computational depth is bounded. However, prior work treats the depth as a constant, leaving it unclear to…
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
The depth-bounded fragment of the pi-calculus is an expressive class of systems enjoying decidability of some important verification problems. Unfortunately membership of the fragment is undecidable. We propose a novel type system,…
Given a numerical function $h:\mathbb Z_{\geq 0}\to\mathbb Z_{\geq 0}$ with $h(0)>0$, the Hilbert depth of $h$ is $\operatorname{hdepth}(h)=\max\{d\;:\;\sum\limits_{j=0}^k (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$; see…