Topic
Undecidable problem
About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.
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TL;DR: In this article, it was shown that the spectral gap problem is undecidable and that the existence or absence of a spectral gap is independent of the axioms of mathematics, which implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless.
Abstract: The spectral gap--the energy difference between the ground state and first excited state of a system--is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.
215 citations
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Abstract: We show that the equality problem for rational series with multiplicities in the tropical semiring is undecidable.
212 citations
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TL;DR: A variant of the alternating-time temporal logic (ATL) where each agent has a given memory is studied and it is shown that it is an interesting compromise, rather realistic but with a reasonable complexity.
211 citations
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06 Jan 2007TL;DR: It is proved that conservative extensions are 2ExpTime-complete in ALCQI, but undecidable in A LCQIO, and it is shown that ifconservative extensions are defined model-theoretically rather than in terms of the consequence relation, they are undec formidable already in ALP.
Abstract: The notion of a conservative extension plays a central role in ontology design and integration: it can be used to formalize ontology refinements, safe mergings of two ontologies, and independent modules inside an ontology. Regarding reasoning support, the most basic task is to decide whether one ontology is a conservative extension of another. It has recently been proved that this problem is decidable and 2ExpTime-complete if ontologies are formulated in the basic description logic ALC. We consider more expressive description logics and begin to map out the boundary between logics for which conservativity is decidable and those for which it is not. We prove that conservative extensions are 2ExpTime-complete in ALCQI, but undecidable in ALCQIO. We also show that if conservative extensions are defined model-theoretically rather than in terms of the consequence relation, they are undecidable already in ALC.
208 citations
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30 Jun 2009TL;DR: This thesis develops the idea of game as computation to a greater degree than has been done previously, and presents a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games.
Abstract: There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games.
In the first half of this thesis. I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis.
In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Konane, Cross Purposes, TipOver, and others. Some of these have been well-known open problems for some time. For other games, including Minesweeper, the Warehouseman's Problem, Sokoban, and Rush Hour, I either strengthen existing results, or provide new, simpler hardness proofs than the original proofs. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
205 citations