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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
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
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.
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Notion in combinatorial game theory
places it in the important complexity class PSPACE; with more work, it can be shown to be PSPACE-complete. Due to the large size of game complexities
Game_complexity
Abstract strategy board game
rectangular grid was unavailable. In 1981, Stefan Reisch showed that Hex is PSPACE-complete. In 2002, the first explicit winning strategy (a reduction-type strategy)
Hex_(board_game)
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
Calculations of the game complexity of go
Without ko, Go is PSPACE-hard. This is proved by reducing True Quantified Boolean Formula, which is known to be PSPACE-complete, to generalized geography
Go_and_mathematics
model Existential theory of the reals § Complete problems Karp's 21 NP-complete problems List of PSPACE-complete problems Reduction (complexity) Grigoriev
List_of_NP-complete_problems
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 first
Reversi
Computational Formula that can be measured in terms of True or False
\exists y\ \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
True quantified Boolean formula
True_quantified_Boolean_formula
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 is
Context-sensitive_grammar
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)
in mathematics List of undecidable problems List of NP-complete problems List of PSPACE-complete problems List of problems in loop theory and quasigroup
Lists_of_problems
Abstract strategy board game
S2CID 207577292. Stefan Reisch (1980). "Gobang ist PSPACE-vollständig (Gomoku is PSPACE-complete)". Acta Informatica. 13: 59–66. doi:10.1007/bf00288536
Gomoku
Puzzle video game
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
Complexity class
Wiley & Sons, ISBN 0-471-90413-9. More precisely, this language is PSPACE-complete; see, for example, Wegener, Ingo (2005), Complexity Theory: Exploring
NP-hardness
can be higher; in particular, testing reachability for Sokoban is PSPACE-complete. Rotation distance in binary trees and related problems of flip distance
Reconfiguration
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 problem
computational complexity theory, generalized geography is a well-known PSPACE-complete problem. Geography is a children's game, where players take turns naming
Generalized_geography
Algorithm for linear programming
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 and
Simplex_algorithm
Game generalized so that it can be played on a board or grid of any size
for the first player in a given position is PSPACE-complete. Generalized hex and reversi are PSPACE-complete. For many generalized games which may last
Generalized_game
Finite-state machine
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
Abstract strategy game
Computer Othello refers to computer architecture encompassing computer hardware and computer software capable of playing the game of Othello. A version
Computer_Othello
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 generalized
Rush_Hour_(puzzle)
Algorithmic complexity class
often PSPACE-complete. The same is true of exponentially long games in which non-repetition is automatic. Another set of important EXPTIME-complete problems
EXPTIME
Two-player board game
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, into
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 asks
Intersection non-emptiness problem
Intersection_non-emptiness_problem
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 Knuth
Instant_Insanity
Board game
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 reduction
Havannah_(board_game)
Preprocessor – Primitive recursive function – Programming language – Prolog – PSPACE-complete – Pulse-code modulation (PCM) – Pushdown automaton – Python QuarkXPress
Index_of_computing_articles
Problem of determining if a Boolean formula could be made true
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
1990 video game
problem of determining whether an Atomix puzzle has a solution is PSPACE-complete. Some heuristic approaches have been considered. Several open source
Atomix_(video_game)
Mathematical game
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 vertex
Kayles
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)
1991 video game
whether it is possible to complete a level of Lemmings is NP-hard. Later, Giovanni Viglietta showed that the task is PSPACE-complete, even for levels where
Lemmings_(video_game)
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
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 (centralised
Distributed_computing
minimization is PSPACE-complete. No efficient (polynomial time) algorithms are known, and under the standard assumption that P ≠ PSPACE, none exist. The
NFA_minimization
such as the emptiness problem for non-erasing stack automata, are PSPACE-complete. The emptiness problem in machine learning and formal languages determines
Emptiness_problem
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
problem (the recognition of true quantified Boolean formulas) that is PSPACE-complete. Analogously, dependency quantified boolean formulas encode computation
Cook–Levin_theorem
Two-player strategy board game from Hawaii
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 nondeterministic
Kōnane
Strategy board game
Checkers), then the problem is in PSPACE, thus it is PSPACE-complete. However, without this bound, Checkers is EXPTIME-complete. However, other problems have
Checkers
Formal language that can be expressed using a regular expression
universality problem is NP-complete already for a singleton alphabet. For larger alphabets, that problem is PSPACE-complete. If regular expressions are
Regular_language
Algorithm that begins on possibly incomplete inputs
between the online and offline algorithms' performance. This problem is PSPACE-complete. There are many formal problems that offer more than one online algorithm
Online_algorithm
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 simplest
Nondeterministic constraint logic
Nondeterministic_constraint_logic
Impossible task in computing
{\displaystyle {\rm {{Sat}([\exists ^{n}\forall \exists ]_{=})}}} are PSPACE-complete (Section 5.4.3). Börger et al. (2001) describes the level of computational
Entscheidungsproblem
Connection board game in the 3M bookshelf game series
been proven to be PSPACE-complete for determining the game value, via a reduction from Hex. TwixT has also been shown to be NP-complete regarding whether
TwixT
Where all data references are valid
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 for
Referential_integrity
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 generally
Hypercomputation
the game represented by Φ {\displaystyle \Phi } . FORMULA-GAME is PSPACE-complete because it is exactly the same decision problem as True quantified
Formula_game
Logical formulation of graph properties
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 on
Logic_of_graphs
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
Type of finite-state machine in automata theory
initial state and check if some final state can be reached. It is PSPACE-complete to test, given an NFA, whether it is universal, i.e., if there is a
Nondeterministic finite automaton
Nondeterministic_finite_automaton
Modal temporal logic with modalities referring to time
formula are PSPACE-complete problems. LTL synthesis and the problem of verification of games against an LTL winning condition is 2EXPTIME-complete. Automata-theoretic
Linear_temporal_logic
building up arithmetic functions. PSPACE Solvable with polynomial space. PSPACE-complete The hardest problems in PSPACE. PTAS Polynomial-time approximation
List_of_complexity_classes
Automated planner
is PSPACE-complete. Various restrictions can be enforced in order to decide if a plan exists in polynomial time or at least make it an NP-complete problem
Stanford Research Institute Problem Solver
Stanford_Research_Institute_Problem_Solver
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
Game and demonstration of logic gates
sufficiently large. This follows because the game is P-complete by the circuit value problem and PSPACE-complete if an exponential number of marbles is allowed
Turing_Tumble
between the online and offline algorithms' performance. This problem is PSPACE-complete. There are many formal problems that offer more than one online algorithm
Online_optimization
Concept in computer science
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
Mathematical model of computer
unreasonable amounts of computational power, enabling it to solve PSPACE-complete problems in polynomial time. When analyzing algorithms for the real
Real_RAM
Computational graph problem
paper analysed the complexity of the problem and reported it to be PSPACE-complete. It was also shown that finding an optimal path in the case where each
Canadian_traveller_problem
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
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
Pencil and paper map-coloring game
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)
negative answer means "don't know". The decision problem for SCT is PSPACE-complete; however, there exists an algorithm that computes an approximation
Size-change termination principle
Size-change_termination_principle
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*
Game whose outcome can be correctly predicted
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 player
Solved_game
Language defined by context-sensitive grammar
or by an arbitrary deterministic context-sensitive grammar, is a PSPACE-complete problem. List of parser generators for context-sensitive languages
Context-sensitive_language
Branch of logic
whether a given sentence has probability tending to zero or to one is PSPACE-complete. A similar analysis has been performed for more expressive logics than
Finite_model_theory
Infinite graph containing all countable graphs
can be done more quickly than exponential time, as the problem is PSPACE-complete. The Rado graph is ultrahomogeneous, and thus is the Fraïssé limit
Rado_graph
Question in theoretical computer science
finite-state automata, equivalence is decidable, and the problem is PSPACE-complete. Further, in the case of deterministic pushdown automata, equivalence
Equivalence_problem
Theory in computer science
satisfiability are tower complete. the structure semantics. We label states. QCTL* = QCTL = MSO over graphs. Model checking is PSPACE-complete but satisfiability
Computation_tree_logic
PSPACE-complete PSPACE-complete ∪,∩,+ PSPACE-complete NP-complete ∪,+ NP-complete NP-complete ∩,+ C=L-complete L-complete + C=L-complete L-complete ∪
Integer_circuit
Theoretical model of computation
with unlimited precision are unreasonably powerful (able to solve PSPACE-complete problems in polynomial time). The transdichotomous model makes a minimal
Transdichotomous_model
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 chromatic
Graph_coloring_game
Nash equilibrium of a bimatrix game algorithm
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
Both deterministic and nondeterministic machines can solve more problems given more space
is this corollary along with the fact that TQBF ∉ NL since TQBF is PSPACE-complete. This could also be proven using the non-deterministic space hierarchy
Space_hierarchy_theorem
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)
Category of positional games
Maker-Breaker game called an Avoider-Enforcer game. Maker-Breaker game is PSPACE-complete even if the size of each set is restricted to 5. The first result was
Maker-Breaker_game
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
Extension of propositional modal logic
EXPTIME-complete. Like for linear temporal logic, the model checking, satisfiability and validity problems of linear modal μ-calculus are PSPACE-complete. Actually
Modal_μ-calculus
examples.[H88] A simpler method, showing also that the problem is PSPACE-complete, was provided in 2005 by Kirsten. Earlier, in 1979, Hashiguchi had
Kosaburo_Hashiguchi
Economical computational problem
is PLS-complete. Their reduction also implies that computing an ε-approximate equilibrium reachable from a given initial state is PSPACE-complete. They
Nash_equilibrium_computation
equivalence problem (do two input AFAs recognize the same language) are PSPACE-complete for AFAs. Chandra, Ashok K.; Kozen, Dexter C.; Stockmeyer, Larry J
Alternating_finite_automaton
Type of database query
queries and are thus at least as hard (in fact, relational algebra is PSPACE-complete with respect to combined complexity and is therefore even harder under
Conjunctive_query
Type of computational algorithm
case for showing that the true quantified Boolean formula problem is PSPACE-complete. This is because the need for memory in such reduction constructions
Log-space_reduction
for simply typed lambda calculus the type inhabitation problem is PSPACE-complete. For other calculi, like System F, the problem is even undecidable
Type_inhabitation
DLOGCFL ∪,∩,−,+ PSPACE-complete PSPACE-complete ∪,∩,+ PSPACE-complete NP-complete ∪,+ NP-complete NP-complete ∩,+ C=L-complete in L + C=L-complete in L ∪,∩,−
Circuits over sets of natural numbers
Circuits_over_sets_of_natural_numbers
Class of games in game theory
that finding a Nash equilibrium reachable from a specified state is PSPACE-complete. For every problem in the complexity class PLS (essentially, every
Congestion_game
Mathematical model
and checking whether it accepts the empty language. This problem is PSPACE-complete. The universality problem of non-deterministic timed automaton is undecidable
Timed_automaton
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
Unsolved problem in computer science
That is, any NP problem can be transformed into any NP-complete problem. Informally, an NP-complete problem is an NP problem that is at least as "tough"
P_versus_NP_problem
American computer scientist (born 1946)
that the type inhabitation problem in simply typed lambda calculus is PSPACE-complete, lower bounds on simply typed lambda calculus, logical relations, and
Richard_Statman
problem is solvable in polynomial time and its satisfiability problem is PSPACE-complete. Muddy children puzzle formalized with PAL: Here are some of the statements
Dynamic_epistemic_logic
2009 book by Robert Hearn and Erik Demaine
computationally difficult: sudoku is NP-complete, Rush Hour and reversi are PSPACE-complete, and chess is EXPTIME-complete. Beyond proving new results along
Games, Puzzles, and Computation
Games,_Puzzles,_and_Computation
Algebraic structure used in logic
1979, who showed it was PSPACE-complete and hence at least as hard as deciding equations of Boolean algebra (shown coNP-complete in 1971 by Stephen Cook)
Heyting_algebra
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
axioms of GLP. GLP is complete with respect to the class of all GLP-spaces. The problem of being a theorem of GLP is PSPACE-complete. So is the same problem
Japaridze's_polymodal_logic
PSPACE COMPLETE
PSPACE COMPLETE
Surname or Lastname
English
English : from a vernacular short form of the Latin personal name Paschalis (see Pascal, Italian Pasquale).nickname for a mild-mannered and peaceable person, from Middle English pace, pece ‘peace’, ‘concord’, ‘amity’ (via Anglo-Norman French from Latin pax, genitive pacis).Italian : from the medieval personal name Pace, used for both men and women, from the word pace ‘peace’ (see 1).
Surname or Lastname
English
English : from the Old Norse personal name Spakr.Respelling of Jewish, Ukrainian, and Belorussian Shpak, a nickname from Ukrainian and Belorussian shpak ‘starling’. In the case of Jewish bearers, it is generally an ornamental name.
Boy/Male
Hindu
Space
Boy/Male
Hindu
Space
Girl/Female
Latin English
Tranquil.
Surname or Lastname
English
English : metonymic occupational name for a spicer (see Spicer).
Boy/Male
Australian, English
A Calm Person
Surname or Lastname
English
English : variant of Speake.
Boy/Male
British, Christian, English, Italian
Form of Pascal; Passover
Male
English
English surname transferred to forename use, derived from the French personal name Pascal, PACE means "Passover; Easter."
Boy/Male
Hindu, Indian
Space; Outer Space; Sky
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Girl/Female
Indian, Telugu
Space
Boy/Male
Hindu
Space
Girl/Female
American, Australian, British, Christian, English, German, Latin
Tranquility; Peaceful
Boy/Male
Australian, French, Hebrew, Polish
Holy Place; Myrrh; Fragrant Oil; Peace
Surname or Lastname
English
English : variant of Pace, found mainly in Yorkshire but also in Orkney.
Surname or Lastname
English and Irish
English and Irish : variant of Stacey.
Female
English
English name derived from the vocabulary word, PEACE means "peace."Â
Surname or Lastname
English or Scottish
English or Scottish : unexplained.
PSPACE COMPLETE
PSPACE COMPLETE
Boy/Male
Christian, Hindu, Indian
Good Man; Victorious God
Boy/Male
English
ModernJaron 'cry of rejoicing.
Boy/Male
Indian, Sanskrit
A Cluster of Blossoms
Girl/Female
Indian
Leader
Surname or Lastname
English
English : from the Old English personal name TÄta, possibly a short form of various compound names with the obscure first element tÄt, or else a nursery formation. This surname is common and widespread in Britain; the chief area of concentration is northeastern England, followed by northern Ireland.
Boy/Male
Indian
Fame; Gift of God
Boy/Male
Tamil
Meditation
Boy/Male
English
Boisterous. Western nickname.
Girl/Female
Muslim
Merciful, Companionate, Kind
Boy/Male
Hindu, Indian, Marathi
Name of Lord Vishnu
PSPACE COMPLETE
PSPACE COMPLETE
PSPACE COMPLETE
PSPACE COMPLETE
PSPACE COMPLETE
v. t.
Scanty; not abundant or plentiful; as, a spare diet.
n.
To set; to fix; to repose; as, to place confidence in a friend.
v. t.
To dig with a spade; to pare off the sward of, as land, with a spade.
n.
Ordinal relation; position in the order of proceeding; as, he said in the first place.
n.
Any portion of space regarded as measured off or distinct from all other space, or appropriated to some definite object or use; position; ground; site; spot; rarely, unbounded space.
n.
Space; place; room.
n.
A broad way in a city; an open space; an area; a court or short part of a street open only at one end.
n.
A quantity or portion of extension; distance from one thing to another; an interval between any two or more objects; as, the space between two stars or two hills; the sound was heard for the space of a mile.
n.
Place, having more or less extension; room.
n.
To arrange or adjust the spaces in or between; as, to space words, lines, or letters.
v. t.
Held in reserve, to be used in an emergency; as, a spare anchor; a spare bed or room.
v. t.
To pace again; to walk over again in a contrary direction.
imp. & p. p.
of Space
n.
One of that suit of cards each of which bears one or more figures resembling a spade.
n.
To put out at interest; to invest; to loan; as, to place money in a bank.
v. t.
To season with spice, or as with spice; to mix aromatic or pungent substances with; to flavor; to season; as, to spice wine; to spice one's words with wit.
adv.
With a quick pace; quick; fast; speedily.
n.
Space.
n.
To assign a place to; to put in a particular spot or place, or in a certain relative position; to direct to a particular place; to fix; to settle; to locate; as, to place a book on a shelf; to place balls in tennis.
n.
Vacated or relinquished space; room; stead (the departure or removal of another being or thing being implied).