Wolfram states several phenomena are normally computationally irreducible. Computational irreducibility explains why many natural systems are hard to...

that justifies computation as an organizing principle of science. For instance, he argues that the concept of computational irreducibility (that some complex...

in terms of how model-building observers infer from measurements the computational capabilities embedded in non-linear processes. An observer's notion...

needs to generate an irreducible polynomial. For this, the common method is to take a polynomial at random and test it for irreducibility. For sake of efficiency...

Eisenstein's criterion Perron's irreducibility criterion Hilbert's irreducibility theorem Cohn's irreducibility criterion Irreducible component of a topological...

Volatility, uncertainty, complexity and ambiguity Arthur Winfree Computational irreducibility Zero-Force Evolutionary Law Project complexity Johnson, Steven...

divisor computation is needed anyway to ensure the fraction is actually irreducible. Every rational number has a unique representation as an irreducible fraction...

crises needs to take adequate account of. The other three are: computational irreducibility, emergent phenomena, and radical uncertainty.[citation needed]...

Wolfram formulated the concepts of intrinsic randomness and computational irreducibility, and suggested that rule 110 may be universal—a fact proved later...

point-to-point transitions have zero probability. In this case, φ-irreducibility generalizes irreducibility by using a reference measure φ on the measurable space...

topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and...

proof Computational algebraic geometry Computer algebra system Differential analyser Proof checker Model checker Symbolic-numeric computation Symbolic...

and it is suspected that behavior of such systems might be computationally irreducible, which means it would not be possible to even approximate the...

decades. The main computational method is homotopy continuation. This supports, for example, a model of floating-point computation for solving problems...

question of whether a given NDTM will halt is still computationally irreducible, and the computation required to find an NDTM busy beaver is significantly...

and others that are linked to here. In conjunction with computational geometry, a computational synthetic geometry has been founded, having close connection...

mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular...

tradition known as Computational theory of mind, also referred to as computationalism, which advances the thesis that neural computation explains cognition...

The hash is the remainder after the division of that polynomial by an irreducible polynomial over GF(2). It is possible to update a Rabin fingerprint using...

Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved...

theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical...

factors) into a product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials...

varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra...

irreducible polynomial. When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial"...

\square } Gauss's lemma (irreducibility)—A non-constant polynomial in Z[X] is irreducible in Z[X] if and only if it is both irreducible in Q[X] and primitive...

Review of Hidden Markov Models and Their Applications". Archives of Computational Methods in Engineering. 28 (3): 1429–1448. doi:10.1007/s11831-020-09422-4...

Classification and Computation in Cellular Automata Classes III and IV". In Zenil, Hector (ed.). Irreducibility and Computational Equivalence: 10 Years...

{s}}\mid s)}{p({\tilde {s}})}}\right)} Integrated information (φ) as the irreducibility of that cause–effect structure across the minimum information partition...

O'Shea, Donal Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Fourth edition. Undergraduate...

precision that guarantees that this method produces either a factor, or an irreducibility proof. Although this method finishes in polynomial time, it is not used...