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Implicit computational complexity (ICC) is a subfield of computational complexity theory that characterizes programs by constraints on the way in which
Implicit computational complexity
Implicit_computational_complexity
Class of problems solvable in polynomial time
In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can
P_(complexity)
Analysis of computer programs without executing them
resource certification have been Static Analysis (SA) and Implicit Computational Complexity (ICC). SA is algorithmic in nature: it focuses on a broad
Static_program_analysis
Amount of resources to perform an algorithm
computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation
Computational_complexity
Problem a computer might be able to solve
science. In computational complexity theory, it is usually implicitly assumed that any string in {0, 1}* represents an instance of the computational problem
Computational_problem
Relation between deterministic and nondeterministic space complexity
In computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic
Savitch's_theorem
Property whereby no efficient algorithm can distinguish two distributions
In computational complexity and cryptography, two families of distributions are computationally indistinguishable if no efficient algorithm can tell the
Computational indistinguishability
Computational_indistinguishability
Set of problems in computational complexity theory
In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly
Complexity_class
Algorithm that employs a degree of randomness as part of its logic or procedure
Papadimitriou, Christos (1993), Computational Complexity (1st ed.), Addison Wesley, ISBN 978-0-201-53082-7 Chapter 11: Randomized computation, pp. 241–278. Rabin
Randomized_algorithm
Unintentional learning of complex information
unconscious (implicit) previous experiences. Implicit memory Implicit cognition Sun, Ron (2008). The Cambridge handbook of computational psychology. Cambridge:
Implicit_learning
Algorithmically defined graph
all of its states. In computational complexity theory, several complexity classes have been defined in connection with implicit graphs, defined as above
Implicit_graph
Processing of natural language by a computer
revolution changes (computational) linguistics. Proceedings of the EACL 2009 Workshop on the Interaction between Linguistics and Computational Linguistics. Resnik
Natural_language_processing
Book by Stephen Wolfram
discusses how facts about the computational universe inform evolutionary theory, SETI, free will, computational complexity theory, and philosophical fields
A_New_Kind_of_Science
Logarithm to the base of the mathematical constant e
of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x)
Natural_logarithm
Problem in math and computer science
is a fundamental problem which can be formulated as follows: Given a computational system with a set of allowed rules or transformations, decide whether
Reachability_problem
Analysis and solving of problems that involve fluid flows
Computational fluid dynamics (CFD) is a branch of both fluid mechanics and computational physics that uses numerical analysis and data structures to analyze
Computational_fluid_dynamics
Subfield of mathematical optimization
optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including
Combinatorial_optimization
Mathematical operation in linear algebra
product of two square n×n matrices. Its computational complexity is therefore O(n3), in a model of computation for which the scalar operations take constant
Matrix_multiplication
Mathematical construct in computer algebra
mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular
Gröbner_basis
Type of artificial neural network
In machine learning, a neural field (also known as implicit neural representation, neural implicit, or coordinate-based neural network), is a mathematical
Neural_field
Multidisciplinary endeavour
cognitive psychology, philosophy, and the arts (e.g., computational art as part of computational culture). Is the application of computer systems to emulate
Computational_creativity
In computational complexity theory, DLOGTIME is the complexity class of all computational problems solvable in a logarithmic amount of computation time
DLOGTIME
Process of compacting a lexicon while retaining semantics
leads to a reduction of dimensionality, which results in less computational complexity and a positive influence on efficiency. Semantic compression is
Semantic_compression
Branch of physics
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic
Computational electromagnetics
Computational_electromagnetics
Complexity class
In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution
PLS_(complexity)
Subdivision of space into cells
Venkatakrishnan, V; Mavriplis, D. J (May 1991). "Implicit solvers for unstructured meshes". Journal of Computational Physics. 105 (1): 23. doi:10.1006/jcph.1993
Mesh_generation
Algorithm to search the nodes of a graph
Succession to the throne shared by the Commonwealth realms. The computational complexity of DFS was investigated by John Reif. More precisely, given a graph
Depth-first_search
Algorithm to multiply matrices
the computational complexity of matrix multiplication) remains unknown. As of September 2025[update], the best bound on the asymptotic complexity of a
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Application of computational algorithms, methods and programs to phylogenetic analyses
Computational phylogenetics, phylogeny inference, or phylogenetic inference focuses on computational and optimization algorithms, heuristics, and approaches
Computational_phylogenetics
In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes that is an exponential time analogue of the polynomial
Exponential_hierarchy
Thesis on the nature of computability
originally called computational complexity-theoretic Church–Turing thesis by Ethan Bernstein and Umesh Vazirani (1997). The complexity-theoretic Church–Turing
Church–Turing_thesis
Algorithm for solving systems of linear equations
the size of the inputs) of O(n3). This complexity is a good measure of the time needed for the whole computation when the time for each arithmetic operation
Gaussian_elimination
Study of discrete mathematical structures
and has close ties to logic, while complexity studies the time, space, and other resources taken by computations. Automata theory and formal language
Discrete_mathematics
Unsolved problem on graph query complexity
each edge is a pair of distinct vertices. Graphs are assumed to have an implicit representation in which each vertex has a unique identifier or label and
Aanderaa–Karp–Rosenberg conjecture
Aanderaa–Karp–Rosenberg_conjecture
Unpredictable phenomenon in complex systems
organized. The descriptive power of the observer's chosen (or implicit) computational model class, for example, can be an overwhelming determinant in
Emergence
Numerical calculations carrying along derivatives
be used depends on the sweep count. The computational complexity of one sweep is proportional to the complexity of the original code. Forward accumulation
Automatic_differentiation
Description of a system using mathematical concepts and language
added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems
Mathematical_model
Automated planner
and B. Nebel (1995). Complexity results for SAS+ planning. Computational Intelligence, 11:625-656. T. Bylander (1991). Complexity results for planning
Stanford Research Institute Problem Solver
Stanford_Research_Institute_Problem_Solver
Interdisciplinary field studying perception, cognition, and characteristics of art
2005.22.4.583. Zhang, K.; Harrell, S.; Ji, X. (2012). "Computational Aesthetics: On the Complexity of Computer-Generated Paintings". Leonardo. 45 (3): 243–248
Psychology_of_art
Israeli mathematician and computer scientist (1931–2026)
with Dana Scott, of the 1976 ACM Turing Award for their work on computational complexity. Rabin was born in 1931 in Breslau, Lower Silesia, Prussia, Germany
Michael_O._Rabin
Computer graphics techniques for generating realistic animations of fluids
animation differs from computational fluid dynamics (CFD) in that fluid animation is used primarily for visual effects, whereas computational fluid dynamics is
Fluid_animation
Multidimensional search tree for points in k dimensional space
node is a k-dimensional point. Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts
K-d_tree
Soviet American mathematician
of the City University of New York. Victor Pan is an expert in computational complexity and has developed a number of new algorithms. One of his notable
Victor_Pan
Hypothesis in neuroscience
explanation of data, under complexity costs; cf. Occam's razor and more formal treatments of computational costs. Here, complexity is the divergence between
Free_energy_principle
Describes approximate behavior of a function
the symbol Omega. The digit zero should not be used. Asymptotic computational complexity Asymptotic expansion: Approximation of functions by a series, generalizing
Big_O_notation
Search problem in quantum mechanics
Bernstein–Vazirani problem. In the Bernstein–Vazirani problem, the hidden function is implicitly specified in an oracle; while in the 2D hidden linear function problem
Hidden linear function problem
Hidden_linear_function_problem
Measuring the Semantic Complexity of Schema-Agnostic Queries", In Proceedings of the 11th International Conference on Computational Semantics (IWCS), London
Schema-agnostic_databases
Thought experiment on artificial intelligence
computationalism: Mental states are computational states (which is why computers can have mental states and help to explain the mind); Computational states
Chinese_room
theorem (computational complexity theory, structural complexity theory) Cook's theorem (computational complexity theory) Fagin's theorem (computational complexity
List_of_theorems
Rendering method
rendering algorithms for generating digital images. On a spectrum of computational cost and visual fidelity, ray tracing-based rendering techniques, such
Ray_tracing_(graphics)
Software programming optimization technique
algorithms has a specific name in computing: computational complexity. All functions have a computational complexity in time (i.e. they take time to execute)
Memoization
Hypothetical consciousness in artificial systems
distributed memory. MIT press, 1988. "Implicit Learning and Consciousness: An Empirical, Philosophical and Computational Consensus in the Making". Routledge
Artificial_consciousness
Computer approximation for real numbers
or comma) there. If the radix point is not specified, then the string implicitly represents an integer and the unstated radix point would be off the right-hand
Floating-point_arithmetic
Distance from a point to the boundary of a set
is not perfect as it runs in raster space in order to avoid the computational complexity of solving the problem in the (continuous) vector space. The rendered
Signed_distance_function
Type of computational algorithm
In computational complexity theory, a log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space. Conceptually
Log-space_reduction
High-area shapes can shift to hold many grid points
with a family of sets. A computational problem related to Blichfeldt's theorem has been shown to be complete for the PPP complexity class, and therefore unlikely
Blichfeldt's_theorem
Creating a complex 3D surface or object by combining primitive objects
Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry
Constructive_solid_geometry
Type of database query
databases and is in the realm of database theory. For the study of the computational complexity of evaluating conjunctive queries, two problems have to be distinguished
Conjunctive_query
Differential equation exhibiting high rate of dissipation
{\mathrm {Re} }\,h\lambda \ll -1} . Because implicit Runge-Kutta methods have high computational complexity, special efficiency requirements are also considered
Stiff_equation
Standard form of Boolean function
contradictions, as well as valid CNFs. An important set of problems in computational complexity involves finding assignments to the variables of a Boolean formula
Conjunctive_normal_form
Data structure for storing non-overlapping sets
Bernard A. Galler and Michael J. Fischer in 1964. In 1973, their time complexity was bounded to O ( log ∗ ( n ) ) {\displaystyle O(\log ^{*}(n))} , the
Disjoint-set_data_structure
Every graph has evenly many odd vertices
commonly arise. The complexity class PPA encapsulates the difficulty of finding a second odd vertex, given one such vertex in a large implicitly-defined graph
Handshaking_lemma
Benefit lost by a choice between options
It incorporates all associated costs of a decision, both explicit and implicit. Thus, opportunity costs are not restricted to monetary or financial costs:
Opportunity_cost
Formula for systems of linear equations
the same (up to a constant factor independent of n) computational complexity as the computation of a single determinant. Moreover, the Bareiss algorithm
Cramer's_rule
Branch of mathematics
and Macaulay in 1913 further investigated the ideals of polynomial rings implicit in E. Noether's work. Lasker proved a special case of the Lasker-Noether
Abstract_algebra
Extension of Datalog with integer arithmetic
limit-closed: if a max (resp. min) limit fact holds for an integer bound, it implicitly holds for all larger (resp. smaller) bounds. Fact entailment remains undecidable
DatalogZ
Consumption of resources that is indirectly required to achieve a goal
fixing them is not worth the cost, because of the overhead. For example, an implicit data structure or succinct data structure may provide low space overhead
Overhead_(computing)
Theoretical model
This use of abstract machines is fundamental to the field of computational complexity theory, such as with finite state machines, Mealy machines, push-down
Abstract_machine
Type of functional equation (mathematics)
differential equation comprising differential and algebraic terms, given in implicit form. Differential equations are closely related to difference equations
Differential_equation
Class of algorithms for pattern analysis
kernel functions, which enable them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that
Kernel_method
In mathematics, invariant of square matrices
(2016), "Faster geometric algorithms via dynamic determinant computation", Computational Geometry, 54: 1–16, arXiv:1206.7067, doi:10.1016/j.comgeo.2015
Determinant
Class in computational complexity theory
In computational complexity theory, a decision problem is P-complete (complete for the complexity class P) if it is in P and every problem in P can be
P-complete
Set of polygons to define the surface of a 3D model
while polygon meshes only explicitly represent the surface (the volume is implicit). Objects created with polygon meshes must store different types of elements
Polygon_mesh
Surface representing points of constant value within a volume
using computer graphics, and are used as data visualization methods in computational fluid dynamics (CFD), allowing engineers to study features of a fluid
Isosurface
Type of climate model
Geophysical Fluid Dynamics Laboratory AOGCMs represent the pinnacle of complexity in climate models and internalise as many processes as possible. However
General_circulation_model
Area of mathematics
Neuroarchitectures“. Computational Cognitive Science. 2: 3. https://doi.org/10.1186/s40469-016-0010-8 Larsen-Freeman, D. (1997). "Chaos/Complexity Science and
Dynamical_systems_theory
List of concepts in artificial intelligence
day. computational chemistry A branch of chemistry that uses computer simulation to assist in solving chemical problems. computational complexity theory
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
American computer scientist (born 1984)
The New Yorker, he stated: “I’m extremely worried that the C.M.F. is implicitly advocating for certain groups of people to be pushed away from rigorous
Jelani_Nelson
oracle for determining element orders. Implicit graph Matroid oracle Babai, L.; Szemeredi, E. (1984). "On the Complexity of Matrix Group Problems I". 25th
Black_box_group
allow for implicit model specification, including the κ-calculus, BioNetGen, the Allosteric Network Compiler and others. To tackle the computation problem
Multi-state modeling of biomolecules
Multi-state_modeling_of_biomolecules
Logic problem, AND of pairwise ORs
recovering shapes from information about their cross-sections. In computational complexity theory, 2-satisfiability provides an example of an NL-complete
2-satisfiability
Computational method in Bayesian statistics
Approximate Bayesian computation (ABC) constitutes a class of computational methods rooted in Bayesian statistics that can be used to estimate the posterior
Approximate Bayesian computation
Approximate_Bayesian_computation
Executing several computations during overlapping time periods
programs, such as implementing each computational execution as an operating system process, or implementing the computational processes as a set of threads
Concurrent_computing
Mathematical modelling alogorithm
external criterion. This process builds feedforward networks of optimal complexity, adapting to the noise level in the data and minimising overfitting, ensuring
Group_method_of_data_handling
Measure of algorithm performance for large inputs
(Strassen-type bilinear identities with lambda-computation). Element uniqueness problem Asymptotic computational complexity Brodnik, Andrej; Carlsson, Svante; Sedgewick
Asymptotically optimal algorithm
Asymptotically_optimal_algorithm
Multidimensional search tree for spatial coordinates
s/K Nearest neighbor queries: O(log n) k-d tree implicit k-d tree, a k-d tree defined by an implicit splitting function rather than an explicitly-stored
Relaxed_k-d_tree
Branch of elementary mathematics
on the field of combinatorics, computational number theory, which approaches number-theoretic problems with computational methods, and applied number theory
Arithmetic
Mathematical representation of economic system
for cyber security investments Economic methodology Computational economics Agent-based computational economics Endogeneity Financial model Moffatt, Mike
Economic_model
Canadian computer scientist (1944–2019)
considered to be the father of computational geometry in Canada. He did research on various aspects of computational geometry, discrete geometry, and
Godfried_Toussaint
Use of classical mechanics to model molecular systems
physical chemistry and classical mechanics, molecular mechanics is a computational method used to model molecular systems. The Born–Oppenheimer approximation
Molecular_mechanics
2010 video game
American studio Evolutionary Games in association with the Evolutionary Complexity Research Group at UCF (EPlex). GAR is a hybrid space shooter and Action
Galactic_Arms_Race
Generalization of linear programming
class of convex programs with applications to computational geometry", Proc. 8th Symposium on Computational Geometry (SCG '92), Berlin, Germany, pp. 9–15
LP-type_problem
Type of logical formula
(sub)goals). Propositional Horn clauses are also of interest in computational complexity. The problem of finding truth-value assignments to make a conjunction
Horn_clause
convergence Superconvergence Discretization Difference quotient Complexity: Computational complexity of mathematical operations Smoothed analysis — measuring
List of numerical analysis topics
List_of_numerical_analysis_topics
Technique to make a model more generalizable and transferable
cost on the optimization function to make the optimal solution unique. Implicit regularization is all other forms of regularization. This includes, for
Regularization_(mathematics)
Compact encoding of digital data
algorithms present a space–time complexity trade-off between the bytes needed to store or transmit information, and the computational resources needed to perform
Data_compression
Programming paradigm based on asynchronous data streams
dependencies are maintained implicitly within an event loop. Registration of explicit callbacks then results in creating implicit dependencies. Thus, control
Reactive_programming
Artificial intelligence model paradigm
58th Annual Meeting of the Association for Computational Linguistics. Online: Association for Computational Linguistics: 5210–5217. arXiv:2005.00955. doi:10
Foundation_model
Theory that attempts to blend economics and ergodic theory
interpreted as an implicit assumption of ergodicity. Ergodicity economics explores what aspects of economics can be informed by avoiding this implicit assumption
Ergodicity_economics
Language for controlling a computer
languages, types are implicit; one form of this is when the compiler can infer types based on context. The downside of implicit typing is the potential
Programming_language
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