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Theorem in computability theory
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the
Rice's_theorem
Generalization of Rice's theorem
In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, named after Henry Gordon Rice and Norman Shapiro. It states that
Rice–Shapiro_theorem
Necklace splitting problem
mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions
Hobby–Rice_theorem
Problem in computer science
that would determine whether the original program halts. Rice's theorem generalizes the theorem that the halting problem is unsolvable. It states that for
Halting_problem
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Study of computable functions and Turing degrees
reducible to E, that is, can be mapped using a many-one reduction to E (see Rice's theorem for more detail). But, many of these index sets are even more complicated
Computability_theory
Theorem in category theory
theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem.
Lawvere's_fixed-point_theorem
Analysis of computer programs without executing them
of Church, Gödel and Turing in the 1930s (see: Halting problem and Rice's theorem). As with many undecidable questions, one can still attempt to give
Static_program_analysis
Classes of partial recursive functions
non-computable, aside from two trivial exceptions. This is stated in Rice's theorem: Let C {\displaystyle {\mathcal {C}}} be a class of partial computable
Index_set_(computability)
Theorem implying that no algorithm can optimally perform a task done by humans
science and mathematics, a full employment theorem is a term used, often humorously, to refer to a theorem which states that no algorithm can optimally
Full-employment_theorem
American mathematician (1920–2003)
Henry Gordon Rice (July 20, 1920 – April 14, 2003) was an American logician and mathematician best known as the author of Rice's theorem, which he proved
Henry_Gordon_Rice
British mathematician
inequivalent prefixes. In computability theory, the Rice–Myhill–Shapiro theorem, more commonly known as Rice's theorem, states that, for any nontrivial property
John_Myhill
Proof by Alan Turing
to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture
Turing's_proof
Yes-or-no question that cannot ever be solved by a computer
the second sense of the term. This result was later generalized by Rice's theorem. In 1973, Saharon Shelah showed the Whitehead problem in group theory
Undecidable_problem
Ability of a computing system to simulate Turing machines
loop Loop (computing) Machine that always halts Rice's theorem S m n theorem Structured program theorem Turing tarpit Virtualization Emulation (computing)
Turing_completeness
Proof in set theory
a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem. Diagonalization arguments
Cantor's_diagonal_argument
Process of writing a self-compiling compiler
variation of the proof that the halting problem is undecidable that uses Rice's Theorem. Due to security concerns regarding the Trusting Trust Attack (which
Bootstrapping_(compilers)
Computational problems no algorithm can solve
halting Turing machines with the same number of states and symbols). Rice's theorem states that for all nontrivial properties of partial functions, it is
List_of_undecidable_problems
Approach to static program analysis
information is in general not computable within finite time and memory (see Rice's theorem and the halting problem). Abstraction is used to allow for generalized
Abstract_interpretation
Theorem in political science
In political science and social choice, Black's median voter theorem says that if voters and candidates are distributed along a one-dimensional political
Median_voter_theorem
American mathematician (1934–2024)
Machinery. Hobby–Rice theorem Faculty profile Archived 2017-02-22 at the Wayback Machine, Purdue University, retrieved 2011-01-29. "John R. Rice: Biographical
John R. Rice (computer scientist)
John_R._Rice_(computer_scientist)
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
equation Quotient rule Ramsey's theorem Rao–Blackwell theorem Rice's theorem Rolle's theorem Splitting lemma squeeze theorem Sum rule in differentiation Sum
List_of_mathematical_proofs
Graph data structure
AST size or performance considerations. E-graphs are used in automated theorem proving. They are a crucial part of modern SMT solvers such as Z3 and CVC4
E-graph
Surname list
Henry Gordon Rice (1920–2003), U.S. logician known for Rice's theorem Henry Mower Rice (1816–1894), U.S. Senator from Minnesota Homer Rice (born 1927)
Rice_(surname)
Language for controlling a computer
Machine", The Perl Review. Papers 2 and 3 prove, using respectively Rice's theorem and direct reduction to the halting problem, that the parsing of Perl
Programming_language
Type of Turing machine
was shown to be, in general, undecidable in Turing's original paper. Rice's theorem shows that any non-trivial question about the output of a Turing machine
Universal_Turing_machine
Automated theorem proving ACL2 theorem prover E equational theorem prover Gandalf theorem prover HOL theorem prover Isabelle theorem prover LCF theorem prover
List of mathematical logic topics
List_of_mathematical_logic_topics
Concepts in theoretical computer science
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Safety and liveness properties
Safety_and_liveness_properties
Of a function, an additional effect besides returning a value
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Side effect (computer science)
Side_effect_(computer_science)
Academic subfield of computer science
problem result. Another important step in computability theory was Rice's theorem, which states that for all non-trivial properties of partial functions
Theory_of_computation
Rules to verify computer program correctness
KeY-Hoare is a semi-automatic verification system built on top of the KeY theorem prover. It features a Hoare calculus for a simple while language. j-Algo
Hoare_logic
Validates computer program operations
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Typestate_analysis
Scott–Curry theorem applies equally to sets of terms in combinatory logic with weak equality. It has parallels to Rice's theorem in computability theorem, which
Scott–Curry_theorem
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Dependence_analysis
Every graph has evenly many odd vertices
Sperner's lemma and to fair subdivision of resources according to the Hobby–Rice theorem. Hein, James L. (2015), "Example 3: The Handshaking Problem", Discrete
Handshaking_lemma
Sequence of program instructions invokable by other software
callable has a side effect is difficult – indeed, undecidable by virtue of Rice's theorem. So, while this optimization is safe in a purely functional programming
Function (computer programming)
Function_(computer_programming)
Computer program for the Boolean satisfiability problem
optimizations to work efficiently. By a result known as the Cook–Levin theorem, Boolean satisfiability is an NP-complete problem in general. As a result
SAT_solver
Logical formalism using combinators instead of variables
M. An analogue of Rice's theorem for this toy model then says that every complete predicate is trivial. The proof of this theorem is rather simple. Proof
Combinatory_logic
Ability to solve a problem by an effective procedure
non-computable or undecidable. An extension of the halting problem is called Rice's theorem, which states that it is undecidable (in general) whether a given language
Computability
Limit of collective decision-making rules
nonempty set of ordinal numbers has a least element. See a section for Rice's theorem for the definition of a computable simple game. In particular, all finite
Nakamura_number
Computer science field
for developing drivers for Windows. Abstract interpretation Automated theorem proving Binary decision diagram Büchi automaton Computation tree logic
Model_checking
Software that provides access that hides details
non-trivial properties of computer programs are essentially undecidable (see Rice's theorem). As a consequence, automatic methods for deriving information on the
Abstraction (computer science)
Abstraction_(computer_science)
Overview of and topical guide to algorithms
BPP (complexity) BQP Undecidable problem Halting problem Rice's theorem No free lunch theorem List of algorithms List of artificial intelligence algorithms
Outline_of_algorithms
Breadth of ideas which can be represented in a formal language
regarding the set of strings they describe is undecidable, a fact known as Rice's Theorem. There are some results on conciseness as well; for instance, nondeterministic
Expressive power (computer science)
Expressive_power_(computer_science)
Process of analyzing computer program behavior
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Program_analysis
because several important results like the Kleene's recursion theorem and Rice's theorem, which were originally proven for the Gödel-numbered set of computable
Complete_numbering
Fundamental problem in computer science
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Path_explosion
Topics referred to by the same term
Frederick Triebel Henry Gordon Rice (1920–2003), American logician and mathematician, author of Rice's theorem Henry Rice Guild, (1928-2019), American lawyer
Henry_Rice
Mathematical problem
t {\displaystyle (k-1)t} cuts. This is a generalization of the Hobby–Rice theorem, and it is used to get an exact division of a cake. Each problem can
Necklace_splitting_problem
Complexity class
program given a finite input finishes running or will run forever. By Rice's theorem, deciding membership of a in any nontrivial subset of the set of partial
RE_(complexity)
Set of software engineering methods
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Program_slicing
Difference between two descriptions of an object by different linguistic representations
which is proven by Rice's theorem. The general expression of limitations for rule based deduction by Gödel's incompleteness theorem indicates that the
Semantic_gap
One is to ensure that all locations are written before they are read. Rice's theorem establishes that this problem cannot be solved in general for all programs;
Definite_assignment_analysis
Concept in computer science
Proofs have been done using embeddings of Separation Logic into interactive theorem provers such as Rocq (previously known as Coq) and HOL (proof assistant)
Separation_logic
Interpreted programming language first released in 1987
from the original on September 3, 2013. Retrieved September 16, 2013. "Rice's Theorem". The Perl Review. 4 (3): 23–29. Summer 2008. and "Perl is Undecidable"
Perl
American mathematician (1932–2021)
2021) was an American mathematician, who was the co-author of the Rice–Shapiro theorem. Shapiro obtained a BS in mathematics at University of Illinois in
Norman_Shapiro
function must also be in the set). In this context, this notion can extend Rice's theorem, stating that: Let A {\displaystyle A} be a subset such that A ≠ ∅
Saturated_set
Game where groups of players may enforce cooperative behaviour
doi:10.1016/S1574-0110(02)80012-1. ISBN 9780444829146. See a section for Rice's theorem for the definition of a computable simple game. In particular, all finite
Cooperative_game_theory
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Hyperproperty
Theory and paradigm of statistics
Bayesian statistical methods use Bayes' theorem to compute and update probabilities after obtaining new data. Bayes' theorem describes the conditional probability
Bayesian_statistics
the House of Representatives (1998–2002). John R. Rice, 89, American mathematician (Hobby–Rice theorem) and computer scientist, founder of ACM Transactions
Deaths_in_January_2024
Theorem in category theory
In category theory in mathematics, Mac Lane's coherence theorem, after Saunders Mac Lane, states that in any monoidal category, every well-formed diagram
Mac_Lane's_coherence_theorem
Correctness Hyperproperties Invariants Path explosion Polyvariance Rice's theorem Runtime verification Safety and liveness Undefined behavior Semantics
Polyvariance
French mathematician (1789–1857)
physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field of complex
Augustin-Louis_Cauchy
Mathematical property of sets
being total is famously not a decidable property of functions. Indeed, Rice's theorem on index sets, most domains of indices are, in fact, not computable
Subcountability
Branch of mathematics
curves. These two branches are related to each other by the fundamental theorem of calculus. Calculus uses convergence of infinite sequences and infinite
Calculus
more secure version of the Anshel–Anshel–Goldfeld protocol. A famous Rice's theorem states that if F is a subset of the set of partial computable functions
Generic-case_complexity
Numbers whose differences are not squares
Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets
Square-difference-free_set
Theorem In probability theory and statistics
In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the
Campbell's theorem (probability)
Campbell's_theorem_(probability)
Type of fair division
This is a direct corollary of the Hobby–Rice theorem. It can also be proved using the Borsuk-Ulam theorem: Every partition of an interval using n {\displaystyle
Consensus_splitting
Mathematical integral
the gamma function which cancels with the gamma from Ramanujan's Master Theorem. A closely related integral frequently occurs in the discussion of Riesz
Nørlund–Rice_integral
Economic model for international trade
Stolper–Samuelson theorem). The Magnification effect on production quantity-shifts induced by endowment changes (via the Rybczynski theorem) predicts a larger
Heckscher–Ohlin_model
English mathematician, mathematical physicist (born 1931)
Prize in Physics with Stephen Hawking for the Penrose–Hawking singularity theorems, and the 2020 Nobel Prize in Physics "for the discovery that black hole
Roger_Penrose
Branch of discrete mathematics
none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory. The types of questions
Combinatorics
American mathematician (born 1941)
particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words:
Dennis_Sullivan
History Faculty at the University of Oxford 25 October 2012 Fermat's Last Theorem Marcus du Sautoy, Professor of Mathematics & Simonyi Professor for the
List of In Our Time programmes
List_of_In_Our_Time_programmes
Process of repeating items in a self-similar way
this is a theorem guaranteeing that recursively defined functions exist. Given a set X, an element a of X and a function f: X → X, the theorem states that
Recursion
In probability theory, the central limit theorem states conditions under which the average of a sufficiently large number of independent random variables
Central limit theorem for directional statistics
Central_limit_theorem_for_directional_statistics
Set of the values of a function
Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group
Image_(mathematics)
Austrian–American mathematician
regions; as well as topology. In graph theory, he is credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory
Karl_Menger
Prime number of the form 2^n – 1
because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne
Mersenne_prime
Graphical representation of the distribution of numerical data
that Terrell and Scott were at Rice University when the proposed it suggests that this is also the origin of the Rice rule. The Freedman–Diaconis rule
Histogram
Standard that diagrams must satisfy up to isomorphism
coherence theorem. Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one". A coherence theorem theorem for
Coherency_(homotopy_theory)
topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem Alternating factorial Antichain Beta function Bhargava factorial Binomial
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Field of mathematics and science based on non-linear systems and initial conditions
the system into two open sets. An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit
Chaos_theory
Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Austrian mathematician (1899–1982)
integral, as it is now called, for vector-valued functions. Bochner's theorem on Fourier transforms appeared in a 1932 book. His techniques came into
Salomon_Bochner
Random process independent of past history
Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains. In 1912 Henri Poincaré studied Markov chains on finite
Markov_chain
Ancient Chinese mathematics text
also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of
The Nine Chapters on the Mathematical Art
The_Nine_Chapters_on_the_Mathematical_Art
Topics referred to by the same term
theory Myhill graph Myhill isomorphism theorem Myhill–Nerode theorem Myhill's property Rice-Myhill-Shapiro theorem This disambiguation page lists articles
Myhill
Process forming a path from many random steps
approximation theorem. The convergence of a random walk toward the Wiener process is controlled by the central limit theorem, and by Donsker's theorem. For a
Random_walk
Unsolved problem in computer science
resolution of Fermat's Last Theorem also shows that very simple questions may be settled only by very deep theories. — Moshe Y. Vardi, Rice University Being attached
P_versus_NP_problem
Geometric model of the physical space
Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem. Later in the
Three-dimensional_space
President of the United States in 1881
aptitude for mathematics extended to his own proof of the Pythagorean theorem, published in 1876, and his advocacy of using statistics to inform government
James_A._Garfield
Statistical method
analytical form or an asymptotic theory (e.g., an applicable central limit theorem) to help estimate the distribution of the statistics of interest. This
Bootstrapping_(statistics)
Inverse of a finite difference
complex analysis (related to Carlson's theorem, the Phragmén–Lindelöf principle, and the Paley–Wiener theorem) which states that a non-constant periodic
Indefinite_sum
Number of subsets of a given size
coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥
Binomial_coefficient
American computer scientist
Alan Robinson's major contribution is to the foundations of automated theorem proving. His unification algorithm eliminated one source of combinatorial
John_Alan_Robinson
American annual computer science prize
way"; the citation in particular mentions his paper "The Complexity of Theorem Proving Procedures," which is credited with founding the theory of NP-completeness
Turing_Award
RICES THEOREM
RICES THEOREM
Boy/Male
Arabic, Muslim, Sindhi
Riches; Happiness
Boy/Male
Tamil
King
Boy/Male
Welsh American Anglo Saxon
Ardent.
Boy/Male
Muslim/Islamic
Riches happiness
Surname or Lastname
English
English : patronymic from Rich 2.
Biblical
riches
Boy/Male
Biblical
Riches.
Girl/Female
Indian, Telugu
Sun Rises
Boy/Male
Hindu, Indian
Rice
Surname or Lastname
English and German
English and German : patronymic from a short form of Richard.English : topographic name for someone who lived where rushes grew, Middle English rexe, rixe (Old English rix).
Girl/Female
Arabic, Muslim
Wealth; Riches
Surname or Lastname
English
English : variant spelling of Ryles.
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Rice
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Telugu
Holy Rice; Sacred Rice of Worship
Boy/Male
Hindu, Indian
Rice
Surname or Lastname
English
English : unexplained.
Girl/Female
Hindu, Indian
Rice
Boy/Male
Hindu
King
Girl/Female
Muslim/Islamic
Wealth riches
Boy/Male
Hindu, Indian
Lord Shiva
RICES THEOREM
RICES THEOREM
Boy/Male
Arabic, Muslim
Arrow
Surname or Lastname
Dutch, German, Danish, and Jewish (Ashkenazic)
Dutch, German, Danish, and Jewish (Ashkenazic) : occupational name for a baker of bread, or brick and tiles, from backen ‘to bake’.English : occupational name for a maker or user of mattocks or pickaxes, from an agent derivative of Old English becca ‘mattock’.This name is recorded in Beverwijck in New Netherland in the mid 17th century, but it was also brought independently to North America by many other bearers.
Girl/Female
African, Arabic
Reward
Girl/Female
Greek
meaning gift. Famous bearer: In Greek mythology, Doris was the daughter of Oceanus and mother of...
Boy/Male
Hindu
To see
Girl/Female
American, British, English, Gaelic
Boomerang; Female Version of Kyle
Girl/Female
Tamil
Girl/Female
Hindu
True
Boy/Male
Bengali, Gujarati, Hindu, Indian, Tamil, Traditional
A King
Male
Egyptian
, a high-priest of Mentu.
RICES THEOREM
RICES THEOREM
RICES THEOREM
RICES THEOREM
RICES THEOREM
n.
Riches; wealth; the god of riches; riches, personified.
n.
Wealth; riches; affluence.
n.
One who rides on a velocipede.
n.
A genus of grasses including the rice plant; rice.
n.
Wealth; riches. See the Note under Riches.
n.
One who rides a bicycle.
n.
One who, or that which, rides.
a.
Full of rimes, fissures, or chinks.
n.
A woman who rides on horseback.
superl.
Abounding in riches; affluent; fortunate.
n.
One who rides out on horseback.
v. t.
To bury with funeral rites.
n.
Boiled rice; rice gruel.
v. t.
Money; riches; wealth.
n.
A genus of grasses including Indian rice. See Indian rice, under Rice.
v. t.
To purify with sacred rites.
a.
Pertaining to all the Slavic races.
n.
One who rives or splits.