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Abstract machine that models computation
In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two
Interactive_proof_system
Proving validity without revealing other data
functions. One way this was done was with multi-prover interactive proof systems (see interactive proof system), which have multiple independent provers instead
Zero-knowledge_proof
Interactive theorem prover software
mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration
Proof_assistant
Cryptographic primitive
Non-interactive zero-knowledge proofs are cryptographic primitives, where information between a prover and a verifier can be authenticated by the prover
Non-interactive zero-knowledge proof
Non-interactive_zero-knowledge_proof
Topics referred to by the same term
Interactive proof can refer to: The abstract concept of an Interactive proof system Interactive theorem proving software This disambiguation page lists
Interactive_proof
Class of interactive proof
In cryptography, a proof of knowledge is an interactive proof in which the prover succeeds in 'convincing' a verifier that the prover knows something.
Proof_of_knowledge
Complexity class from interactive proofs
of an interactive proof system was first introduced by Shafi Goldwasser, Silvio Micali, and Charles Rackoff in 1985. An interactive proof system consists
IP_(complexity)
Placeholder characters
genders are alternated: Alice, Bob, Carol, Dave, Eve, etc. For interactive proof systems there are other characters: The names Alice and Bob are often
Alice_and_Bob
Establishment of a theorem using inference from the axioms
sequence. Formal proofs often are constructed with the help of computers in interactive theorem proving (e.g., through the use of proof checker and automated
Formal_proof
Complexity class
Quantum Interactive Proof) is the quantum computing analogue of the classical complexity class IP, which is the set of problems solvable by an interactive proof
QIP_(complexity)
American cryptographer (born 1974)
Jens Groth, "Efficient Non-interactive Proof Systems for Bilinear Groups," introduced a new type of non-interactive proof system offering improved efficiency
Amit_Sahai
Interactive proof system in computational complexity theory
an Arthur–Merlin protocol, introduced by Babai (1985), is an interactive proof system in which the verifier's coin tosses are constrained to be public
Arthur–Merlin_protocol
Set of problems in computational complexity theory
other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships
Complexity_class
Topics referred to by the same term
a material's porous structure MIP, an interactive proof system complexity class; see Interactive proof system Minimum ionizing particle, in particle
MIP
System that regulates the formation of blocks on a blockchain
Back formally described a proof of work system called Hashcash as a protection against email spam in 1997. The term "proof of work" was first coined and
Proof_of_work
Abstract machine used to study decision problems
box (i.e., as a random oracle). Black box group Turing reduction Interactive proof system Matroid oracle Demand oracle Padding oracle attack van Melkebeek
Oracle_machine
Concept in computational complexity theory
provers gives it great power. See interactive proof system#MIP for more details. Another interactive proof system characterizing NEXPTIME is a certain
NEXPTIME
3/4. A quantum interactive proof with two competing provers is a generalization of the single prover quantum interactive proof system. It can be modelled
Quantum_refereed_game
Proof assistant
Coq) is an interactive theorem prover first released in 1989. It allows the expression of mathematical assertions, mechanical checking of proofs of these
Rocq
Class of computational complexity
languages recognizable by a particular interactive proof system, the one defining the class IP. In this system, there is an all-powerful prover trying
PSPACE
protocol is a cryptographic protocol for the construction of interactive proof systems, used widely in zero-knowledge protocols. The sum-check protocol
Sum-check_protocol
American computer scientist
1980s on the invention of interactive proof systems, where a proof could be developed by answering a series of interactive questions between different
Charles_Rackoff
Israeli computer scientist
well known for his work on interactive proof systems. His two most-cited papers are Raz (1998) on multi-prover interactive proofs and Raz & Safra (1997) on
Ran_Raz
American computer scientist (born 1963)
scientist known for major results in computational complexity and interactive proof systems. Since 2019, he has been at the Illinois Institute of Technology
Lance_Fortnow
Computer science award
Micali, S.; Rackoff, C. (1989), "The knowledge complexity of interactive proof systems" (PDF), SIAM Journal on Computing, 18 (1): 186–208, CiteSeerX 10
Gödel_Prize
Complexity class used to classify decision problems
NP can be seen as a very simple type of interactive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic
NP_(complexity)
Danish computer scientist (born 1963)
characterizing complexity classes such as PSPACE and NEXPTIME in terms of interactive proof systems; this work became part of his 1991 Ph.D. thesis from the University
Carsten_Lund
Israeli American computer scientist (born 1959)
additional knowledge. They began by studying interactive proofs more broadly, where a proof is developed interactively by answering a series of questions about
Shafi_Goldwasser
Branch of mathematical logic
system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof
Proof_theory
Israeli computer scientist
computational complexity of pseudorandom number generators, and interactive proof systems, among other topics. Nisan won an ACM Distinguished Dissertation
Noam_Nisan
Italian-American computer scientist (born 1954)
invented interactive proofs in the 1980s, at the same time as László Babai and Shlomo Moran. In an interactive proof system, participants develop a proof by
Silvio_Micali
Israeli computer scientist
Charles Rackoff for their work on Arthur–Merlin protocols and interactive proof systems. Faculty profile, the Technion, retrieved 2010-12-02. Shlomo Moran
Shlomo_Moran
Computational Formula that can be measured in terms of True or False
result IP = PSPACE was shown (see interactive proof system), it was done by exhibiting an interactive proof system that could solve QBF by solving a particular
True quantified Boolean formula
True_quantified_Boolean_formula
Nonconstructive method for mathematical proofs
Mathematics portal Interactive proof system Las Vegas algorithm Incompressibility method Method of conditional probabilities Probabilistic proofs of non-probabilistic
Probabilistic_method
Cryptographic proof technique
with n-bit output. Interactive proof system Universal composability Bellare, Mihir, and Phillip Rogaway. "Code-based game-playing proofs and the security
Hybrid argument (cryptography)
Hybrid_argument_(cryptography)
integer determinant of a matrix IP Solvable in polynomial time by an interactive proof system L Solvable with logarithmic (small) space LOGCFL Logspace-reducible
List_of_complexity_classes
Hungarian-American mathematician and computer scientist
papers. His notable accomplishments include the introduction of interactive proof systems, the introduction of the term Las Vegas algorithm, and the introduction
László_Babai
Practice and study of secure communication techniques
electronic cash systems, signcryption systems, etc. Some more 'theoretical'[clarification needed] cryptosystems include interactive proof systems, (like zero-knowledge
Cryptography
Mathematical model of computation
computation is also critical for the definition of most classes of interactive proof systems, in which the verifier machine depends on randomness to avoid
Probabilistic_Turing_machine
optical recorders (Ryder hauled his around in an 11-ton truck). Ryder's system, which today involves recorders weighing ounces rather than tons, was first
List of University of California, Berkeley alumni
List_of_University_of_California,_Berkeley_alumni
algorithm Non-determinism Non-deterministic Turing machine Interactive computation Interactive proof system Probabilistic Turing Machine Approximation algorithm
List of computability and complexity topics
List_of_computability_and_complexity_topics
Greek mathematician and logician (born 1947)
on randomized complexity classes, Arthur–Merlin protocols, and interactive proof systems has been very influential in proving important theorems and is
Stathis_Zachos
Software for automated theorem proving
IsaPlanner is a proof planner for the interactive proof assistant, Isabelle, originally developed by Lucas Dixon. IsaPlanner 2: A Proof Planner in Isabelle
IsaPlanner
Proof assistant and programming language
Lean is a proof assistant and a functional programming language. It is based on the calculus of constructions with inductive types. It is a free and open-source
Lean_(proof_assistant)
Theorem in computational complexity theory
work on interactive proofs and probabilistically checkable proofs. The first theorem relating standard proofs and probabilistically checkable proofs is the
PCP_theorem
theorem prover Interactive proof system Mizar system QED project Rocq, formerly Coq Automated Mathematician Eurisko Begriffsschrift Systems of Logic Based
List of mathematical logic topics
List_of_mathematical_logic_topics
Type of computational problem
complexity classes of P, NP, PH, etc, in circuit complexity, and in interactive proof systems. Let R be a search problem, formalised as a binary relation R
Counting_problem_(complexity)
Theoretical computer scientist
[quant-ph]. Watrous, John (2003). "PSPACE has constant-round quantum interactive proof systems". Theor. Comput. Sci. 292 (3). Essex, UK: Elsevier Science Publishers
John Watrous (computer scientist)
John_Watrous_(computer_scientist)
Set of problems solved by small circuits
{P}}} , even PSPACE = MA. Proof: Consider a language L from PSPACE. It is known that there exists an interactive proof system for L, where actions of the
P/poly
Subfield of automated reasoning and mathematical logic
reduced to a proof checker, with the user providing the proof in a formal way, or significant proof tasks can be performed automatically. Interactive provers
Automated_theorem_proving
Inherent difficulty of computational problems
decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems. For the complexity
Computational complexity theory
Computational_complexity_theory
Cryptographic technique
heuristic, or Fiat–Shamir transformation, is a technique for taking an interactive proof of knowledge and creating a digital signature based on it. This way
Fiat–Shamir_heuristic
American mathematician
As a student of Umesh Vazirani at Berkeley, Mahadev discovered interactive proof systems that could demonstrate with high certainty, to an observer using
Urmila_Mahadev
logic); Judea Pearl (artificial intelligence); Shafi Goldwasser (Interactive proof system); Asher Peres (Quantum information); Adi Shamir (RSA, Differential
Culture_of_Israel
2026 action-adventure video game
adaptations. IO Interactive announced the development of a James Bond game in November 2020. The game entered full production at IO Interactive following the
007_First_Light
Mathematical proof at least partially generated by computer
new proofs for known theorems.[citation needed] Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which
Computer-assisted_proof
Higher-order logic (HOL) automated theorem prover
trustworthiness of proofs without requiring, yet supporting, explicit proof objects. Isabelle is available inside a flexible system framework allowing
Isabelle_(proof_assistant)
Designed to be proof against misuse or error
2025-11-20 Wasserman, Anthony I. (1973-06-04). "The design of 'idiot-proof' interactive programs". Proceedings of the June 4-8, 1973, national computer conference
Idiot-proof
(TPMs), interactive proofs, probabilistically checkable proofs, efficient arguments, and Micali's CS proofs. These verifications are either interactive which
Verifiable_computing
Problem a computer might be able to solve
complexity, including hardness of approximation, property testing, and interactive proof systems. Lateral computing, alternative approaches to solving problems
Computational_problem
American theoretical computer scientist (born 1954)
complexity theory, specifically efficient error correcting codes, interactive proof systems, randomness, quantum computation, and establishing the inherent
Michael_Sipser
Proof assistant
At the interactive level, the system implements a small step execution of structured tactics allowing a much better management of the proof development
Matita
Realization of a certain method or idea in order to demonstrate its feasibility
A proof of concept (POC or PoC), also known as proof of principle, is an inchoate realization of a certain idea or method in order to demonstrate its feasibility
Proof_of_concept
Quantum Merlin Arthur
S2CID 6869749. Watrous, John (2003). "PSPACE has constant-round quantum interactive proof systems". Theoretical Computer Science. 292 (3): 575–588. doi:10
QMA
focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and
Focused_proof
Subfield of computer science and logic
the less automated but more pragmatic subfield of interactive theorem proving) and automated proof checking (viewed as guaranteed correct reasoning under
Automated_reasoning
Planetary system consisting of the Sun and objects orbiting it
The Solar System is an isolated single-star planetary system (not part of a larger star system) within the Milky Way Galaxy. The system formed about
Solar_System
science and discrete mathematics, especially for his work on interactive proof systems; Gödel Prize winner Walter Lewis Baily, Jr. – known for work in
List of University of Chicago faculty
List_of_University_of_Chicago_faculty
Cryptographic protocol for two-party computation
Micali, S; Rackoff, C (1985-12-01). "The knowledge complexity of interactive proof-systems". Proceedings of the seventeenth annual ACM symposium on Theory
Garbled_circuit
Proof assistant program
system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written
Mizar_system
Concept in computer science
"Mechanical Verification of a Constructive Proof for FLP", in Blanchette, Jasmin Christian; Merz, Stephan (eds.), Interactive Theorem Proving, Lecture Notes in
Consensus_(computer_science)
Proof checkable by a randomized algorithm
computational complexity theory, a probabilistically checkable proof (PCP) is a type of proof that can be checked by a randomized algorithm using a bounded
Probabilistically checkable proof
Probabilistically_checkable_proof
Formal language and associated computer program
language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. Several databases of proved theorems have
Metamath
American computer scientist (born 1946)
the constraints. This result even led to the breakthroughs of interactive proof systems Karloff-Nisan and Shamir, including the result IP = PSPACE. In
Richard_Lipton
Interactive theorem proving systems
denotes a family of interactive theorem proving systems using similar (higher-order) logics and implementation strategies. Systems in this family follow
HOL_(proof_assistant)
Type system used in computer programming and mathematics
factored by him into the soundness proof. Before formulating the proof obligations, a deviation between the rules systems D and S and the algorithms presented
Hindley–Milner_type_system
Process that helps an equipment operator avoid mistakes
of the Toyota Production System. Poka-yoke was originally baka-yoke, but as this means "fool-proofing" (or "idiot-proofing") the name was changed to
Poka-yoke
Functional programming language
PhD Thesis. Chalmers University of Technology, 2007. [1] "Agda: An Interactive Proof Editor". Archived from the original on 8 October 2011. Retrieved 20
Agda_(programming_language)
American live-streaming platform
Curse, from Twitch Interactive for an undisclosed amount. Curse was dissolved and its assets were moved under Twitch Interactive. Twitch's new headquarters
Twitch_(service)
Operating system for mobile devices
CNET. CBS Interactive. Archived from the original on March 11, 2017. Retrieved March 11, 2017. "Global smartphone sales by operating system 2009–2016
Android_(operating_system)
cryptologist, won the Godel Prize for his work on interactive proof systems and zero-knowledge proofs Geoffrey Hinton (professor of computer science, 1987–98
List of University of Toronto faculty
List_of_University_of_Toronto_faculty
platform that supports Qt) Proofs: No Extra: Editable Live Geometry is a free CodePlex project that lets you create interactive ruler and compass constructions
List of interactive geometry software
List_of_interactive_geometry_software
Methodology of programming
formal proof system, will generally be required for full verification of non-trivial programs. In this case a high degree of automation of proofs is also
Invariant-based_programming
Type of consensus algorithm
Proof of space (PoS) is a type of consensus algorithm achieved by demonstrating one's legitimate interest in a service (such as sending an email) by allocating
Proof_of_space
Micali, S; Rackoff, C (1985-12-01). "The knowledge complexity of interactive proof-systems". Proceedings of the seventeenth annual ACM symposium on Theory
Secure_two-party_computation
Theorem concerning quantum field theory ground states
applies to any eigenstate of the Hamiltonian. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the
Gell-Mann_and_Low_theorem
1970s automated theorem prover
provides similar trustworthiness to systems that generate explicit proof certificates but without the need to store proof objects in memory. The Theorem data
Logic for Computable Functions
Logic_for_Computable_Functions
Elementary cellular automaton
but they do not feature as prominently in the universality proof. The cyclic tag system machinery has three main components: A data string which is stationary;
Rule_110
Style of formal logical argumentation
calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead
Sequent_calculus
Proving or disproving the correctness of certain intended algorithms
imply conformance of the system to its specification, and discharging these obligations using either proof assistants (interactive theorem provers) (such
Formal_verification
System for dealing with email spam
Hashcash is a proof-of-work system used to limit email spam and denial-of-service attacks. Hashcash was proposed in 1997 by Adam Back and described more
Hashcash
French computer scientist and mathematician
a 1985 collaboration with Huet titled "Constructions: A Higher Order Proof System for Mechanizing Mathematics". Coquand and Huet published another joint
Thierry_Coquand
Term in logic and deductive reasoning
completeness proof applies to all classical models, not some special proper subclass of intended ones. Philosophy portal Soundness (interactive proof) Type soundness
Soundness
Branch of logic
(\varphi \to \psi )} . Proof systems in propositional logic can be broadly classified into semantic proof systems and syntactic proof systems, according to the
Propositional_logic
Concept in computer science
specifications and proofs of a program component mention only the portion of memory used by the component, and not the entire global state of the system. Applications
Separation_logic
Software system
powerful version, called PC-Nqthm (Proof-checker Nqthm) was developed by Matt Kaufmann. This gave the proof tools that the system uses automatically to the user
Nqthm
Cryptographic protocol
[citation needed] A common use of a zero-knowledge password proof is in authentication systems where one party wants to prove its identity to a second party
Zero-knowledge_password_proof
to be embedded in the same type system. A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typed
Logical_framework
The Super Nintendo Entertainment System has a library of 1,749 official releases, of which 717 were released in North America plus 4 championship cartridges
List of Super Nintendo Entertainment System games
List_of_Super_Nintendo_Entertainment_System_games
Free and open-source tool for making interactive fiction in the form of web pages
Twine is a free open-source tool created by Chris Klimas for making interactive fiction and hypertext fiction in the form of web pages. It is available
Twine_(software)
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
Boy/Male
Muslim
Proof
Boy/Male
Muslim/Islamic
Proof
Surname or Lastname
English
English : variant of Rolfe.German : from Ruffo, a short form of a personal name formed with hrÅd ‘renown’, ‘victory’.Probably an Americanized spelling of German Ruf and Ruff.
Girl/Female
Muslim
Proof
Boy/Male
Arabic, Muslim
The Proof
Boy/Male
Arabic
Proof; Evidence
Boy/Male
Arabic, Muslim
Evidence; Proof
Girl/Female
Indian
Witness; Proof
Boy/Male
Muslim
Evidence. Proof.
Boy/Male
Muslim/Islamic
Proof
Boy/Male
Indian
Proof
Boy/Male
Indian
Argument, Reasoning, Proof
Girl/Female
Muslim
Guide, Proof
Boy/Male
Indian
Proof
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Good Looking; Interactive; Brightness
Boy/Male
Muslim
Proof
Boy/Male
Arabic, French, German, Gujarati, Hindu, Indian, Malaysian, Muslim, Turkish
Proof; Evidence
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
Boy/Male
Hindu, Indian, Tamil
Writer
Boy/Male
Muslim
Supporter, Helper, One who helps
Boy/Male
Tamil
Jalendu | ஜாலேஂதà¯Â
Moon in the water
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Ancient Name of a State
Boy/Male
Bengali, Hindu, Indian, Kannada, Tamil, Telugu
Miracle; Good King; Good Partner; Kind
Girl/Female
Arabic, Muslim
Helper
Male
Hebrew
(רֶבַע) Variant spelling of Hebrew Reba, REVA means "a fourth part."
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Red Lotus; Bright
Boy/Male
Gujarati, Hindu, Indian, Kannada
Earth
Female
Hungarian
Feminine form of Hungarian Krisztián, KRISZTINA means "believer" or "follower of Christ."
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
n.
Demonstration; proof.
n.
Proof.
n.
Mutual or reciprocal action or influence; as, the interaction of the heart and lungs on each other.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
n.
That which resembles, or corresponds to, the covering or the ceiling of a house; as, the roof of a cavern; the roof of the mouth.
v. t.
To cover with a roof.
n.
Proof.
a.
Used in proving or testing; as, a proof load, or proof charge.
n.
The cover of any building, including the roofing (see Roofing) and all the materials and construction necessary to carry and maintain the same upon the walls or other uprights. In the case of a building with vaulted ceilings protected by an outer roof, some writers call the vault the roof, and the outer protection the roof mask. It is better, however, to consider the vault as the ceiling only, in cases where it has farther covering.
a.
Proof against proofs; obstinate in the wrong.
v. t.
To arm with proof armor; to arm securely; as, to proof-arm herself.
n.
Proof; trial.
n.
Proof.
a.
Having the power to prohibit; as, an interdictive sentence.
a.
Firm or successful in resisting; as, proof against harm; waterproof; bombproof.
n.
A trial impression, as from type, taken for correction or examination; -- called also proof sheet.
v. t.
Armor of excellent or tried quality, and deemed impenetrable; properly, armor of proof.
n.
Proof.