# Two-Variable First Order Logic with Counting Quantifiers: Complexity Results

@inproceedings{Lodaya2017TwoVariableFO, title={Two-Variable First Order Logic with Counting Quantifiers: Complexity Results}, author={Kamal Lodaya and A. V. Sreejith}, booktitle={International Conference on Developments in Language Theory}, year={2017} }

Etessami et al. [5] showed that satisfiability of two-variable first order logic \(\mathrm {FO}^2\)[<] on word models is Nexptime-complete. We extend this upper bound to the slightly stronger logic \(\mathrm {FO}^2\)[\(<,succ ,\equiv \)], which allows checking whether a word position is congruent to r modulo q, for some divisor q and remainder r. If we allow the more powerful modulo counting quantifiers of Straubing, Therien et al. [22] (we call this two-variable fragment FOmod \(^2\)[\(<,succ…

## 2 Citations

### Modulo Counting on Words and Trees

- Computer Science, MathematicsFSTTCS
- 2017

A small-model property of this logic is proved, which gives a technique for deciding the satisfiability problem for the two-variable fragment of the first-order logic extended with modulo counting quantifiers and interpreted over finite words or trees.

### “Most of” leads to undecidability: Failure of adding frequencies to LTL

- Computer ScienceFoSSaCS
- 2021

The main goal of this work is to study the effect of adding weak forms of percentage constraints to fragments of LTL, and sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple.

## References

SHOWING 1-10 OF 26 REFERENCES

### Undecidability results on two-variable logics

- MathematicsArch. Math. Log.
- 1997

It is shown that going beyond L2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, such as transitive closure or monadic fixed-point operations.

### Structure Theorem and Strict Alternation Hierarchy for FO2 on Words

- Computer ScienceCSL
- 2006

This work proves precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words, and proves, among other results, that there is a strict hierarchy of alternating quantifiers for both languages.

### Arithmetic, first-order logic, and counting quantifiers

- Mathematics, PhilosophyTOCL
- 2005

This article shows that Presburger arithmetic is closed under unary counting quantifiers and addition, and obtains an easy proof of Ruhl's result that reachability in finite graphs is not expressible in first-order logic with unARY counting quantifier and addition.

### First-order logic with two variables and unary temporal logic

- Mathematics, Computer ScienceProceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science
- 1997

It is proved that FO/sup 2/ can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "s sometime in the past", a logic the authors denote by unary-TL.

### Complexity of two-variable logic with counting

- MathematicsProceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science
- 1997

It is proved that the problem of satisfiability of sentences of C/sub 1//sup 2/ is NEXPTIME-complete, which easily implies that the satisfiability problem for C/sup2/ is in non-deterministic, doubly exponential time.

### The Descriptive Complexity Approach to LOGCFL

- Computer ScienceJ. Comput. Syst. Sci.
- 1998

It is proven that FO with unary groupoidal quantifiers is strictly more expressive with predicates for plus and times than without, and it is proved that first-order logic with the “majority of pairs” quantifier is strictlyMore expressive than first- order with majority of individuals.

### LTL Can Be More Succinct

- Computer ScienceATVA
- 2010

This work considers a restricted logic which allows only the modulo counting of length from the beginning of the word, and shows that in some cases (such as the modular and symmetric group modalities and for threshold counting) the authors can use numeric constants in binary notation, and still maintain the Pspace upper bound.

### 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.

### Logic Meets Algebra: the Case of Regular Languages

- Computer ScienceLog. Methods Comput. Sci.
- 2007

This work surveys the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata and shows that many of the best known results share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid.

### Restricted set-theoretical definitions in arithmetic

- Philosophy, Mathematics
- 1958

It follows that the only sets of natural numbers which are definable are the finite sets and their complements, and the set of even numbers is not definable.