Two-Variable Logic with Two Order Relations

Schwentick, Thomas; Zeume, Thomas

doi:10.1007/978-3-642-15205-4_38

Cited by 25 publications

(47 citation statements)

References 12 publications

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“…The above rough sketch can be used to obtain a new proof of decidability of finite satisfiability problem of FO 2 with two linear orders [SZ12]. This automata-based approach generalizes well to various other results in this vein, by considering various domains D or various shapes of the input structures.…”

Section: Introductionmentioning

confidence: 82%

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

Toruńczyk

Zeume

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable. As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable.

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Section: Introductionmentioning

confidence: 82%

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

Toruńczyk

Zeume

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…The connection of intervals to the two-dimensional plane becomes clear when one interprets an interval [a, b] as point (a, b). In [SZ10] decidability results for FO 2 on two-dimensional structures have been transferred to those two logics.…”

Section: Transfer Of Techniquesmentioning

confidence: 99%

“…Here, the difficulty lies in finding a good notion of row type. For details we refer the reader to [SZ10,SZ12].…”

Section: Indexmentioning

confidence: 99%

Small Dynamic Complexity Classes

Zeume

2017

Lecture Notes in Computer Science

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Everything changes and nothing stands still. -Heraclitus (quoted by Platon in Cratylus) PrefaceOn the first few days of my PhD studies in the summer of 2009, my adviser Thomas Schwentick introduced me to dynamic complexity. Very recently Wouter Gelade, Marcel Marquardt and Thomas had obtained a very nice characterization of regular languages in terms of dynamic complexity, and also a lower bound for the dynamic complexity of the alternating reachability problem. Many interesting problems in this area seemed to wait for a solution; and so I started to try to prove a lower bound for reachability. I was not successful. After two (at the end frustrating) months, I abandoned this project.In the following two and a half years I almost forgot about dynamic complexity. Decidability issues for the two-variable fragment of first-order logic had turned out to be a much more accessible and fruitful field. Thomas and I had obtained several results, and a PhD in this field seemed not to be too far away. This was the moment when Thomas asked whether I would be interested in applying for funds from the DFG. If successful, such funding could relieve me from my teaching obligations.We decided to have a second, deeper look into dynamic complexity, and to apply for funds for doing an extensive study of the power of logics in dynamic settings. At that time, the decision to spend more time on dynamic complexity was not easy for me. I was in the third year of my PhD and already had results and further ideas for two-variable logics; and it was not clear whether an application for funding would be successful. On the other hand, now I had more experience, which might turn out to be helpful to attack the very same problems that I had tried to solve at the beginning of my PhD. I do not regret the decision.With the thesis at hand I want to document the progress in dynamic complexity that we have made in the last two and a half years. The focus of this thesis is on small dynamic descriptive complexity classes, in particular on lower bound methods for them. A short summary of results on decidability issues for two-variable logic is presented at the end of the thesis.

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“…Stronger complexity‐theoretic upper bounds are available when the distinguished predicates are required to be interpreted as linear orders: the satisfiability and finite satisfiability problems for

L^{2}

together with one linear order are both NExpTime ‐complete ; the finite satisfiability problem for

L^{2}

together with two linear orders is in 2 ‐NExpTime (falling to ExpSpace when all non‐navigational predicates are unary ); with three linear orders, satisfiability and finite satisfiability are both undecidable . Also somewhat related to

{scriptL}^{2} 1 {PO}^{normalu}

is the propositional modal logic known as navigational XPATH , which features a signature of proposition letters interpreted over vertices of some finite, ordered tree, together with modal operators giving access to vertices standing in the relations of daughter and next‐sister , as well as their transitive closures.…”

Section: Introductionmentioning

confidence: 99%