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Class of computational complexity
{\mathsf {P{\overset {?}{=}}PSPACE}}} More unsolved problems in computer science In computational complexity theory, PSPACE is the set of all decision
PSPACE
Type of decision problem in computer science
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input
PSPACE-complete
Complexity class from interactive proofs
problems solvable by an interactive proof system. It is equal to the class PSPACE. The result was established in a series of papers: the first by Lund, Karloff
IP_(complexity)
Set of problems in computational complexity theory
complexity classes relate to each other in the following way: L⊆NL⊆P⊆NP⊆PSPACE⊆EXPTIME⊆NEXPTIME⊆EXPSPACE Where ⊆ denotes the subset relation. However,
Complexity_class
Theoretical computer scientist
interactive proofs, and the quantum analogue of the celebrated result IP = PSPACE: QIP = PSPACE. This was preceded by a series of results, showing QIP can be constrained
John Watrous (computer scientist)
John_Watrous_(computer_scientist)
Complexity class
is contained in PSPACE, which also proves that QIP = IP = PSPACE, since PSPACE is easily shown to be in QIP using the result IP = PSPACE. Watrous, John
QIP_(complexity)
Quantum Merlin Arthur
in PSPACE. It is unknown if any of these inclusions is unconditionally strict, as it is not even known whether P is strictly contained in PSPACE or P
QMA
Abstract strategy board game
1145/321978.321989. S2CID 8845949. Stefan Reisch (1981). "Hex ist PSPACE-vollständig (Hex is PSPACE-complete)". Acta Informatica. 15 (2): 167–191. doi:10.1007/bf00288964
Hex_(board_game)
Branch of mathematical logic
Second-order logic with a transitive closure operator (commutative or not) yields PSPACE, the problems solvable in polynomial space. Second-order logic with a least
Descriptive_complexity_theory
Here are some of the more commonly known problems that are PSPACE-complete when expressed as decision problems. This list is in no way comprehensive. Generalized
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Inherent difficulty of computational problems
PSPACE {\displaystyle {\textsf {P}}\subseteq {\textsf {NP}}\subseteq {\textsf {PP}}\subseteq {\textsf {PSPACE}}} , but it is possible that P = PSPACE
Computational complexity theory
Computational_complexity_theory
Algorithmic complexity class
basic time and space complexity classes in the following way: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE. Furthermore, by the time hierarchy theorem
EXPTIME
Notion in combinatorial game theory
need not store game states; however many games of interest are known to be PSPACE-hard, and it follows that their space complexity will be lower-bounded by
Game_complexity
Abstract machine that models computation
exponential time, a very large class. NEXPTIME contains PSPACE, and is believed to strictly contain PSPACE. Adding a constant number of additional provers beyond
Interactive_proof_system
Class of problems solvable in polynomial time
than PSPACE, the class of problems decidable in polynomial space. PSPACE is equivalent to NPSPACE by Savitch's theorem. Again, whether P = PSPACE is an
P_(complexity)
mathematics List of undecidable problems List of NP-complete problems List of PSPACE-complete problems List of problems in loop theory and quasigroup theory
Lists_of_problems
Computer science concept
the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing
Polynomial_hierarchy
Theoretical model of computation
solution among the exponentially many branches. Probabilistic Turing machine PSPACE Garey, Michael R.; David S. Johnson (1979). Computers and Intractability:
Nondeterministic Turing machine
Nondeterministic_Turing_machine
Computational Formula that can be measured in terms of True or False
\exists z\ ((x\lor z)\land y)} QBF is the canonical complete problem for PSPACE, the class of problems solvable by a deterministic or nondeterministic Turing
True quantified Boolean formula
True_quantified_Boolean_formula
Concept in the philosophy of mathematics
capture mathematics associated with various complexity classes like P and PSPACE. Buss's work can be considered the continuation of Edward Nelson's work
Ultrafinitism
Complexity class used to classify decision problems
ignoring the proof and solving it. NP is contained in PSPACE—to show this, it suffices to construct a PSPACE machine that loops over all proof strings and feeds
NP_(complexity)
Written or spoken word game
is in EXPSPACE, and is PSPACE-hard. It's proved to be PSPACE-hard by reducing Generalized Geography, a problem known to be PSPACE-hard, to a game of Ghost
Ghost_(game)
Computer memory needed by an algorithm
use O ( f ( n ) ) {\displaystyle O(f(n))} space. The complexity classes PSPACE and NPSPACE allow f {\displaystyle f} to be any polynomial, analogously
Space_complexity
Computational problem
computational complexity theory, generalized geography is a well-known PSPACE-complete problem. Geography is a children's game, where players take turns
Generalized_geography
Calculations of the game complexity of Go
complexity. Without ko, Go is PSPACE-hard. This is proved by reducing True Quantified Boolean Formula, which is known to be PSPACE-complete, to generalized
Go_and_mathematics
Strategy board game
the drawing rule in standard Checkers), then the problem is in PSPACE, thus it is PSPACE-complete. However, without this bound, Checkers is EXPTIME-complete
Checkers
Game generalized so that it can be played on a board or grid of any size
win for the first player in a given position is PSPACE-complete. Generalized hex and reversi are PSPACE-complete. For many generalized games which may
Generalized_game
Single-player game played with mahjong tiles
removing all tiles is PSPACE-complete, and the game is NP-complete if looking below tiles is allowed. It has been proven that it is PSPACE-hard to approximate
Mahjong_solitaire
Computational complexity class of problems
PP\subseteq PSPACE\subseteq EXP}}} As the problem of P = ? P S P A C E {\displaystyle {\mathsf {P}}\ {\stackrel {?}{=}}\ {\mathsf {PSPACE}}} has
BQP
Unsolved problem in computer science
prove that IP = PSPACE. However, in 2008, Scott Aaronson and Avi Wigderson showed that the main technical tool used in the IP = PSPACE proof, known as
P_versus_NP_problem
Complexity class
polynomial space, but not in non-deterministic polynomial time (unless NP = PSPACE). NP-hard problems do not have to be elements of the complexity class NP
NP-hardness
Concept in computer science
are strict subsets, since we don't even know if P is a strict subset of PSPACE. BPP is contained in the second level of the polynomial hierarchy and therefore
BPP_(complexity)
Sliding block puzzle
solution is PSPACE-complete. This is proved by reducing a graph game called nondeterministic constraint logic, which is known to be PSPACE-complete, to
Rush_Hour_(puzzle)
Randomized polynomial time class of computational complexity theory
probabilistic complexity classes (ZPP, co-RP, BPP, BQP, PP), which generalise P within PSPACE. It is unknown if any of these containments are strict.
RP_(complexity)
Class of problems in computer science
are uniform (generated by a polynomial-time algorithm). PP is included in PSPACE. This can be easily shown by exhibiting a polynomial-space algorithm for
PP_(complexity)
Abstract strategy board game
3233/ICG-2001-24104. S2CID 207577292. Stefan Reisch (1980). "Gobang ist PSPACE-vollständig (Gomoku is PSPACE-complete)". Acta Informatica. 13: 59–66. doi:10.1007/bf00288536
Gomoku
Complexity class
is symmetrical. co-NP is a subset of PH, which itself is a subset of PSPACE. An example of a problem that is known to belong to both NP and co-NP (but
Co-NP
Strategy board game
determining if the first player has a winning move in a given position is PSPACE-complete. The World Othello Championship (WOC), which started in 1977, was
Reversi
Computer hardware technology that uses quantum mechanics
P, NP, and PSPACE is not known. However, it is known that P ⊆ B Q P ⊆ P S P A C E {\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}} ; that is
Quantum_computing
complexity can be higher; in particular, testing reachability for Sokoban is PSPACE-complete. Rotation distance in binary trees and related problems of flip
Reconfiguration
Two-player board game
configuration) is PSPACE-complete. This can be proved in two ways. The first way is by reducing a generalized Hex position, which is known to be PSPACE-complete
Game_of_the_Amazons
intersection problem or the non-emptiness of intersection problem, is a PSPACE-complete decision problem from the field of automata theory. The problem
Intersection non-emptiness problem
Intersection_non-emptiness_problem
Method for solving one problem using another
computational problem that is known to be NP-hard and in PSPACE, but is not known to be complete for NP, PSPACE, or any language in the polynomial hierarchy. ∃
Polynomial-time_reduction
Topics referred to by the same term
Italy, vehicle registration code AP, an alternative characterization of PSPACE In computational complexity theory Application Processor, usually means
AP
Concept in computer science
other probabilistic complexity classes (RP, co-RP, BPP, BQP, PP), which generalise P within PSPACE. It is unknown if any of these containments are strict.
ZPP_(complexity)
Combinatorial reconfiguration problem
proven to be PSPACE-complete. These hardness results form the basis for proofs that various games and puzzles are PSPACE-hard or PSPACE-complete. In the
Nondeterministic constraint logic
Nondeterministic_constraint_logic
Board game
board. During this competition the pie rule is used. Solving Havannah is PSPACE-complete with respect to the size of the input graph. The proof is by a
Havannah_(board_game)
Form of logic that allows quantification over predicates
formulas. PH is the set of languages definable by second-order formulas. PSPACE is the set of languages definable by second-order formulas with an added
Second-order_logic
Algorithm for linear programming
computing its output is PSPACE-complete. In 2015, this was strengthened to show that computing the output of Dantzig's pivot rule is PSPACE-complete. Analyzing
Simplex_algorithm
Both deterministic and nondeterministic machines can solve more problems given more space
required, but runs for infinite time. The above proof holds for the case of PSPACE, but some changes need to be made for the case of NPSPACE. The crucial point
Space_hierarchy_theorem
System with multiple networked computers
non-deterministic) finite-state machines can reach a deadlock. This problem is PSPACE-complete, i.e., it is decidable, but not likely that there is an efficient
Distributed_computing
Problem in computer science
cannot easily be proven, since this would prove that P is different from PSPACE. Simon's problem considers access to a function f : { 0 , 1 } n → { 0 ,
Simon's_problem
List of unsolved computational problems
NC = P problem NP = co-NP problem P = BPP problem P = PSPACE problem L = NL problem PH = PSPACE problem L = P problem L = RL problem Unique games conjecture
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Set of problems solved by small circuits
furthermore, NP ⊆ P/poly implies AM = MA If PSPACE ⊆ P/poly then P S P A C E = Σ 2 P ∩ Π 2 P {\displaystyle {\mathsf {PSPACE}}=\Sigma _{2}^{\mathsf {P}}\cap \Pi
P/poly
equivalence problem (do two input AFAs recognize the same language) are PSPACE-complete for AFAs. Chandra, Ashok K.; Kozen, Dexter C.; Stockmeyer, Larry
Alternating_finite_automaton
Computational complexity of quantum algorithms
classes relate to classical complexity classes such as P, NP, BPP, and PSPACE. One of the reasons quantum complexity theory is studied are the implications
Quantum_complexity_theory
Two-player strategy board game from Hawaii
player eventually cannot perform a capture. Bob Hearn proved that Kōnane is PSPACE-complete with respect to the dimensions of the board, by a reduction from
Kōnane
Economical computational problem
that the problem of finding a PNE reachable from a given input state is PSPACE-complete. The class of ordinal potential games is even larger than the class
Nash_equilibrium_computation
Simulation of a dynamical system of particles
poly(n) is in PSPACE. On the other hand, if the question is whether the body eventually reaches the destination ball, the problem is PSPACE-hard. These
N-body_simulation
Model of computation
ISBN 978-3-540-64310-4. Yang, Ke (2001). "Integer Circuit Evaluation Is PSPACE-Complete". Journal of Computer and System Sciences. 63 (2, September 2001):
Circuit_(computer_science)
Class of mathematical games
interesting open problem". Only in 2020 it was proved that the game is PSPACE-Complete. Acyclic coloring. Every graph G {\displaystyle G} with acyclic
Graph_coloring_game
LOGCFL +,× P-hard, in co-NP L-hard, in LOGCFL ∪,∩,−,+ PSPACE-complete PSPACE-complete ∪,∩,+ PSPACE-complete NP-complete ∪,+ NP-complete NP-complete ∩,+
Integer_circuit
Game and demonstration of logic gates
follows because the game is P-complete by the circuit value problem and PSPACE-complete if an exponential number of marbles is allowed. The device has
Turing_Tumble
Finite-state machine
solved efficiently also for NFAs. The non-universality problem for NFAs is PSPACE complete since there are small NFAs with shortest rejecting word in exponential
Deterministic finite automaton
Deterministic_finite_automaton
Problem of determining if a Boolean formula could be made true
Boolean formula problem (QBF), which can be shown to be PSPACE-complete. It is widely believed that PSPACE-complete problems are strictly harder than any problem
Boolean satisfiability problem
Boolean_satisfiability_problem
Pencil and paper map-coloring game
the outcome in Snort is PSPACE-complete on general graphs. This is proven by reducing partizan node Kayles, which is PSPACE-complete, to a game of Snort
Col_(game)
Set of decision problems
of as the hardest problems in EXPSPACE. EXPSPACE is a strict superset of PSPACE, NP, and P. It contains EXPTIME and is believed to strictly contain it,
EXPSPACE
Mathematical game
wins), Schaefer proved in 1978 that deciding the outcome of these games is PSPACE-complete (the same holds for the partisan versions, in which, for every
Kayles
Logical formulation of recursion
structures, a property is expressible in FO(PFP,X) if and only if it lies in PSPACE. Since the iterated predicates involved in calculating the partial fixed
Fixed-point_logic
Given more time, a Turing machine can solve more problems
questions of computational complexity theory: whether P and NP, NP and PSPACE, PSPACE and EXPTIME, or EXPTIME and NEXPTIME are equal or not. The gap of approximately
Time_hierarchy_theorem
Israeli cryptographer (born 1952)
2-satisfiability and showing the equivalence of the complexity classes PSPACE and IP. 2002 ACM Turing Award, together with Rivest and Adleman, in recognition
Adi_Shamir
{\displaystyle \mu (\varphi )=0} . Moreover this problem has been shown to be PSPACE-complete. The following logics have the zero-one law: First-order logic
Zero–one_law_(logic)
Logical formulation of graph properties
sentence has probability tending to zero or to one is high: the problem is PSPACE-complete. If a first-order graph property has probability tending to one
Logic_of_graphs
Puzzle video game
any given Sokoban puzzle is solvable is a problem known to be NP-hard and PSPACE-complete. In artificial intelligence research, Sokoban serves as an experimental
Sokoban
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
a problem (the recognition of true quantified Boolean formulas) that is PSPACE-complete. Analogously, dependency quantified boolean formulas encode computation
Cook–Levin_theorem
Type of formal grammar
context-sensitive grammar G, is PSPACE-complete. Moreover, there are context-sensitive grammars whose languages are PSPACE-complete. In other words, there
Context-sensitive_grammar
Where all data references are valid
axiomatized by inference rules and can be decided by a PSPACE algorithm. The problem can be shown to be PSPACE-complete by reduction from the acceptance problem
Referential_integrity
Cryptographic model of a random function
later shown to be false, as the two acceptable complexity classes IP and PSPACE were shown to be equal despite IPA ⊊ PSPACEA for a random oracle A with
Random_oracle
Two-person board game
center, determining whether the current player has a winning strategy is PSPACE-hard. Schmittberger, R. Wayne (1992). New Rules for Classic Games. John
Phutball
Field of artificial intelligence around Go computer programs
Wolfe in their book Mathematical Go. Go endgames have been proven to be PSPACE-hard if the absolute best move must be calculated on an arbitrary mostly
Computer_Go
Quantified formulas with real-number variables
semialgebraic set is non-empty. This decision problem is NP-hard and lies in PSPACE, giving it significantly lower complexity than Alfred Tarski's quantifier
Existential theory of the reals
Existential_theory_of_the_reals
System of resource-aware logic
multiplicatives and additives (i.e., exponential-free). MALL entailment is PSPACE-complete. Multiplicative-exponential linear logic (MELL): only multiplicatives
Linear_logic
Relation between deterministic and nondeterministic space complexity
the Turing machine. Some important corollaries of the theorem include: PSPACE = NPSPACE That is, the languages that can be recognized by deterministic
Savitch's_theorem
Formal language that can be expressed using a regular expression
already for a singleton alphabet. For larger alphabets, that problem is PSPACE-complete. If regular expressions are extended to allow also a squaring operator
Regular_language
Deciding the winner of an arbitrary finite poset game is PSPACE-complete. This means that unless P=PSPACE, computing the Grundy value of an arbitrary poset game
Poset_game
Block puzzle with four colored cubes
proved that this game is PSPACE-complete, which illustrates the observation that NP-complete puzzles tend to lead to PSPACE-complete games. Devil's Dice
Instant_Insanity
Branching-time logic that is a superset of LTL and CTL
model checking in CTL* is not worse than that of LTL: they both lie in PSPACE. The language of well-formed CTL* formulae is generated by the following
CTL*
Rendering method
reflective objects represented by a system of rational linear inequalities is PSPACE-hard. For any dimension equal to or greater than 2, ray tracing with a finite
Ray_tracing_(graphics)
Concept in computational complexity
C2P = PPPP C3P = PPPPPP ... The counting hierarchy is contained within PSPACE. By Toda's theorem, the polynomial hierarchy PH is entirely contained in
Counting_hierarchy
Question in theoretical computer science
of finite-state automata, equivalence is decidable, and the problem is PSPACE-complete. Further, in the case of deterministic pushdown automata, equivalence
Equivalence_problem
Models of computation
inside the black hole. Access to a CTC may allow the rapid solution to PSPACE-complete problems, a complexity class which, while Turing-decidable, is
Hypercomputation
1991 video game
Lemmings is NP-hard. Later, Giovanni Viglietta showed that the task is PSPACE-complete, even for levels where there is only one lemming to save. In 2010
Lemmings_(video_game)
Concept in computer science
logic parameterized over the sorts of locations and data can be shown to be PSPACE-complete. An algorithm for solving this fragment in DPLL(T)-based SMT solvers
Separation_logic
Transformation of one computational problem to another
classes P, NP and PSPACE are closed under (many-one, "Karp") polynomial-time reductions. The complexity classes L, NL, P, NP and PSPACE are closed under
Reduction_(complexity)
Game whose outcome can be correctly predicted
solving Hex on an N×N board is unlikely as the problem has been shown to be PSPACE-complete.[citation needed] If Hex is played on an N×(N + 1) board then the
Solved_game
Nash equilibrium of a bimatrix game algorithm
pure strategies in the game. Subsequently, it has been shown that it is PSPACE-complete to find any of the solutions that can be obtained with the Lemke–Howson
Lemke–Howson_algorithm
Modal temporal logic with modalities referring to time
AG(EF(p)). Model checking and satisfiability against an LTL formula are PSPACE-complete problems. LTL synthesis and the problem of verification of games
Linear_temporal_logic
American mathematician
S2CID 206798056. 1988(over 740 citations) Regenar, James (April 1988). "A faster PSPACE algorithm for deciding the existential theory of the reals" (PDF). Technical
James_Renegar
World line of a particle in spacetime which returns to its starting point
implies also equivalence of quantum and classical computation (both in PSPACE). If Lloyd's prescription holds, quantum computations would be PP-complete
Closed_timelike_curve
Type of Turing reduction
under some type of many-one reducibility, including P, NP, L, NL, co-NP, PSPACE, EXP, and many others. It is known for example that the first four listed
Many-one_reduction
PSPACE
PSPACE
PSPACE
PSPACE
Boy/Male
Muslim
Brightness, Whiteness, Drought
Girl/Female
Gujarati, Hindu, Indian
Mother of Lord Hanuman
Boy/Male
Hindu, Indian
Golden
Girl/Female
German
Prospers in Battle
Girl/Female
Muslim/Islamic
Gift
Girl/Female
Hindu, Indian
Attraction; Grace; Beauty
Surname or Lastname
English
English : variant of Franey.
Boy/Male
Tamil
Possessed with fire a burning torch
Boy/Male
Muslim/Islamic
Generous
Boy/Male
Tamil
Rain
PSPACE
PSPACE
PSPACE
PSPACE
PSPACE