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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
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
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
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
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)
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
Israeli computer scientist
computational complexity of pseudorandom number generators, and interactive proof systems, among other topics. Nisan won an ACM Distinguished Dissertation
Noam_Nisan
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
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)
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
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
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
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
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
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
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
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
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)
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
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
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)
logic); Judea Pearl (artificial intelligence); Shafi Goldwasser (Interactive proof system); Asher Peres (Quantum information); Adi Shamir (RSA, Differential
Culture_of_Israel
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
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
Designed to be proof against misuse or error
(link) 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
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
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
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
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
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
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
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
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
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)
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
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
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
(TPMs), interactive proofs, probabilistically checkable proofs, efficient arguments, and Micali's CS proofs. These verifications are either interactive which
Verifiable_computing
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
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
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
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)
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
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)
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
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
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
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
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)
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
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
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
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
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
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
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)
Style of formal logical argumentation
of a Hilbert-style system is that complete formal proofs tend to get extremely long. Concrete arguments about proofs in such a system almost always appeal
Sequent_calculus
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
Australian computer scientist
Previously he has worked on formal specification, interactive proof, automated verification and proof description languages. He holds a Ph.D. from the
Don_Syme
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
British cryptographer and cypherpunk (born 1970)
Candidate Buying Bitcoin No Matter What". Bloomberg. Retrieved 3 June 2020. "Proof of Work - An interview with Adam Back (Blockstream)". YouTube. 7 August
Adam_Back
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
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
Entertainment System with special coprocessors. This standardized selection of chips was available to licensed developers, to increase system performance
List of Super NES enhancement chips
List_of_Super_NES_enhancement_chips
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
Type of educational intervention
transferring knowledge and may be used as a part of a learning process. More interactive and specific than a book or a lecture, a tutorial seeks to teach by example
Tutorial
Fault in a computer system that presents different symptoms to different observers
blockchain with proof of work allowing the system to overcome Byzantine failures and reach a coherent global view of the system's state. Some proof of stake
Byzantine_fault
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)
Cryptocurrency aimed at privacy
web" compared to Monero. Legality of bitcoin by country Non-interactive zero-knowledge proof "Releases - zcash/zcash". Retrieved 21 March 2026 – via GitHub
Zcash
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
Boy/Male
Indian
Proof
Boy/Male
Arabic
Proof; Evidence
Boy/Male
Arabic, Muslim
Evidence; Proof
Girl/Female
Indian
Witness; Proof
Boy/Male
Muslim/Islamic
Proof
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Indian
Argument, Reasoning, Proof
Boy/Male
Indian
Proof
Boy/Male
Arabic, French, German, Gujarati, Hindu, Indian, Malaysian, Muslim, Turkish
Proof; Evidence
Girl/Female
Muslim
Guide, 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.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Good Looking; Interactive; Brightness
Boy/Male
Muslim
Evidence. Proof.
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Muslim/Islamic
Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
Boy/Male
Muslim
Proof
Boy/Male
Arabic, Muslim
The Proof
Girl/Female
Muslim
Proof
Boy/Male
Muslim
Proof
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
Girl/Female
Afghan, Arabic, Indian, Muslim, Parsi
Spell or Bewitchment; Charm; Spell
Surname or Lastname
English, North German, Dutch, Frisian, and Danish
English, North German, Dutch, Frisian, and Danish : from a Germanic personal name, Boio or Bogo, of uncertain origin. It may represent a variant of Bothe, with the regular Low German loss of the dental between vowels, but a cognate name appears to have existed in Old English (see Boyce), where this feature does not occur. Boje is still in use as a personal name in Friesland.Dutch : nickname from Middle Dutch boy(e) ‘boy’, ‘lad’.
Girl/Female
American, Australian, German, Latin
Unlucky; Army Counselor
Girl/Female
Gujarati, Hindu, Indian, Kannada, Marathi, Sindhi, Tamil
Illuminated
Boy/Male
Indian, Punjabi, Sikh
God
Female
Hindi/Indian
(अवनà¥à¤¤à¥€) Hindi name AVANTI means "modest."
Boy/Male
Tamil
God of mountain attributed to Lord Shiva
Boy/Male
Hindu
Reviver of lakshmanas life
Boy/Male
Indian, Punjabi, Sikh
Light of Peace
Boy/Male
African, Australian, Swahili
Residence
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
INTERACTIVE PROOF-SYSTEM
a.
Used in proving or testing; as, a proof load, or proof charge.
a.
Firm or successful in resisting; as, proof against harm; waterproof; bombproof.
n.
Mutual or reciprocal action or influence; as, the interaction of the heart and lungs on each other.
v. t.
To cover with a roof.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
n.
Proof.
a.
Proof against proofs; obstinate in the wrong.
n.
Proof.
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.
v. t.
Armor of excellent or tried quality, and deemed impenetrable; properly, armor of proof.
n.
Proof.
n.
Proof; trial.
n.
A trial impression, as from type, taken for correction or examination; -- called also proof sheet.
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 arm with proof armor; to arm securely; as, to proof-arm herself.
n.
Proof.
n.
Demonstration; proof.
a.
Having the power to prohibit; as, an interdictive sentence.