# Two-Variable Logic with Two Order Relations

@article{Schwentick2012TwoVariableLW, title={Two-Variable Logic with Two Order Relations}, author={Thomas Schwentick and Thomas Zeume}, journal={Log. Methods Comput. Sci.}, year={2012}, volume={8} }

It is shown that the finite satisfiability problem for two-variable logic over structures with one total preorder relation, its induced successor relation, one linear order relation and some further unary relations is EXPSPACE-complete. Actually, EXPSPACE-completeness already holds for structures that do not include the induced successor relation. As a special case, the EXPSPACE upper bound applies to two-variable logic over structures with two linear orders. A further consequence is that…

## 45 Citations

### Order-Invariance of Two-Variable Logic is Decidable

- Mathematics2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2016

It is shown that order-invariance of two-variable first-logic is decidable in the finite and the finite satisfiability problem for ESO2 on structures with one linear order and its induced successor relation is shown to be decidability in non-deterministic exponential time.

### Decidability Issues for Two-Variable Logics with Several Linear Orders

- MathematicsCSL
- 2011

It is shown that the satisfiability and the finite satisfiability problems for two-variable logic, FO2, over the class of structures with three linear orders, are undecidable and that GF2 with an arbitrary number of linear orders which can be used only as guards becomes decidable if except linear orders only unary relations are allowed.

### Decidability of weak logics with deterministic transitive closure

- MathematicsCSL-LICS
- 2014

This paper considers the extension of the two-variable fragment of first-order logic by the deterministic transitive closure of a single binary relation, and proves that the satisfiability and finite satisfiability problems for the obtained logic are decidable and ExpSpace-complete.

### Two-Variable Logic on 2-Dimensional Structures

- Mathematics, Computer ScienceCSL
- 2013

It is obtained that finite satisfiability of FO^2 is decidable on structures with two linear order successors and a linear order corresponding to one of the successors on structures where the <=_p-equivalence classes are of bounded size.

### Two-Variable Separation Logic and Its Inner Circle

- PhilosophyACM Trans. Comput. Log.
- 2015

This work shows that first-order separation logic with a unique record field restricted to two quantified variables and no program variables is undecidable, and establishes insightful and concrete relationships between two-variable separation logic and propositional interval temporal logic (PITL), data logics, and modal logics.

### A Decidable Two-Way Logic on Data Words

- Mathematics, Computer Science2011 IEEE 26th Annual Symposium on Logic in Computer Science
- 2011

This work reduces the satisfiability problem for a logic on data words to a derivation problem for an infinite transition system, and shows how to abstract this problem into a reachability problem of a finite transition system.

### Two-Variable Logic over Countable Linear Orderings

- MathematicsMFCS
- 2016

It is proved that the satisfiability problems for two-variable logic over arbitrary, countable, and scattered linear orderings are NEXPTIME-complete.

### Satisfiability for two-variable logic with two successor relations on finite linear orders

- MathematicsArXiv
- 2012

We study the finitary satisfiability problem for first order logic with two variables and two binary relations, corresponding to the induced successor relations of two finite linear orders. We show…

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

- Computer Science, MathematicsLICS
- 2020

This work proves 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.

### Finite satisfiability for two‐variable, first‐order logic with one transitive relation is decidable

- MathematicsMath. Log. Q.
- 2018

We consider two‐variable, first‐order logic in which a single distinguished predicate is required to be interpreted as a transitive relation. We show that the finite satisfiability problem for this…

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