**First-Order logic and its Infinitary Quantifier Extensions over Countable Words**
with Bharat Adsul and Saptarshi Sarkar,

FCT 2021
(

pdf,

full,

abstract)

** Abstract **:
We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order
properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as
well as boolean closure of existential fragment of FO via a strengthening of Simon’s theorem about piecewise testable
languages. We propose a new extension of FO which admits infinitary quantifiers to reason about the inherent infinitary
properties of countable words. We provide a very natural and hierarchical block-product based characterization of the
new extension. We also explicate its role in view of other natural and classical logical systems such as WMSO and
FO[cut] - an extension of FO where quantification over Dedekind cuts is allowed. We also rule out the possibility of a
finite-basis for a block-product based characterization of these logical systems. Finally, we report simple but novel
algebraic characterizations of one variable fragments of the hierarchies of the new proposed extension of FO.

**Coupling threshold theory and satellite image derived channel width to
estimate the formative discharge of Himalayan Foreland rivers.
**
with K. Gaurav , F. Métivier , R. Sinha , A. Kumar , and S. K. Tandon,

Earth Science Dynamics 2021
(

pdf,

abstract)

** Abstract **:
We propose an innovative methodology to estimate the formative discharge of alluvial rivers from remote sensing images.
This procedure involves automatic extraction of the width of a channel from Landsat Thematic Mapper,
Landsat 8, and Sentinel-1 satellite images. We translate the channel width extracted from satellite images to
discharge by using a width- discharge regime curve established previously by us for the Himalayan Rivers.
This regime curve is based on the threshold theory, a simple physical force balance that explains the
first-order geometry of alluvial channels. Using this procedure, we estimate the discharge of six major
rivers of the Himalayan Foreland: the Brahmaputra, Chenab, Ganga, Indus, Kosi, and Teesta rivers.
Except highly regulated rivers (Indus and Chenab), our estimates of the discharge from satellite images
can be compared with the mean annual discharge obtained from historical records of gauging stations.
We have shown that this procedure applies both to braided and single-thread rivers over a large territory.
Further our methodology to estimate discharge from remote sensing images does not rely on continuous ground calibration.

**Pooled testing**
with Somenath Biswas
(

pdf,

abstract)

** Abstract **:
We look at one-time pooled testing and multiple pooled testing.

**Notes on Parity games**
(

pdf,

abstract)

** Abstract **:
The notes were made during a lecture series on Parity games. We used notes from Mikołaj Bojańczyk's
page for the lecture series.

**Undecidability of MSO+“ultimately periodic”**, with
Mikołaj Bojańczyk, Laure Daviaud, Bruno Guillon, and Vincent Penelle,

LMCS 2020
(

pdf,

abstract)

** Abstract **:
We prove that MSO on omega-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e., sets X such that for some positive integer p, ultimately either both or none of positions x and x+p belong to X. We obtain it as a corollary of the undecidability of MSO on omega-words extended with the second-order predicate U_{1}(X) which says that the distance between consecutive positions in a set X ⊆ N is unbounded. This is achieved by showing that adding U_{1} to mso gives a logic with the same expressive power as MSO+U, a logic on omega-words with undecidable satisfiability.

**Block Product for finite monoids with generalized associativity**, with
Saptarshi Sarkar, and Bharat Adsul
(

pdf,

abstract)

** Abstract **:
We give the definition of Block Product for finite moniods with generalized associativity.

**Block products for algebras over countable words and applications to logic**, with
Saptarshi Sarkar, and Bharat Adsul,

LICS 2019.
(

pdf,

abstract)

** Abstract **:
We propose a seamless integration of the block product operation to the recently developed algebraic framework for regular languages of countable words. A simple but subtle accompanying block product principle has been established. Building on this, we generalize the well-known algebraic char- acterizations of first-order logic (resp. first-order logic with two variables) in terms of strongly (resp. weakly) iterated block products. We use this to arrive at a complete analogue of Schu ̈tzenberger-McNaughton-Papert theorem for countable words. We also explicate the role of block products for linear temporal logic by formulating a novel algebraic characterization of a natural fragment.

**Lecture notes: Logic for computer scientists**
(

pdf,

abstract)

** Abstract **:
Lecture notes prepared for the Logic course (CS228)

**Two-variable first order logic with counting quantifiers: Complexity Results**,
with

Kamal Lodaya,

DLT 2017.
(

pdf,

abstract)

** Abstract **:
Etessami,
Vardi and Wilke showed that satisfiability of two-variable first order
logic FO^{2}[<] on word models is Nexptime-complete.
We extend this upper bound to the slightly stronger logic
FO^{2}[<,successor,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 and Thomas (we call this two-variable fragment
FOMOD^{2}[<,successor]), satisfiability becomes
Expspace-complete. A
more general counting quantifier, FOCOUNT^{2}[<,successor],
makes the logic undecidable.

**Two-variable logic over
countable linear orderings**, with

Amaldev
Manuel,

MFCS 2016.
(

pdf,

full,

abstract)

** Abstract **:
We
study the class of
languages of finitely-labelled countable linear orderings definable in
two-variable first-order logic. We give a number of characterisations,
in particular an algebraic one in terms of circle monoids, using
equations. This generalises the corresponding characterisation, namely
variety DA, over finite words to the countable case. A corollary is
that the membership in this class is decidable: for instance given an
MSO formula it is possible to check if there is an equivalent
two-variable logic formula
over countable linear orderings. In addition, we prove that the
satisfiability problems for two-variable logic over arbitrary,
countable, and scattered linear orderings are
Nexptime-complete.

**Limited Set quantifiers
over
Countable Linear Orderings**, with

Thomas
Colcombet,

ICALP 2015.
(

pdf,

full,

abstract)

** Abstract **:
In
this paper, we study several sublogics of monadic second-order logic
over countable linear orderings, such as first-order logic, first-order
logic on cuts, weak monadic second-order logic, weak monadic
second-order logic with cuts, as well as fragments of monadic
second-order logic in which sets have to be well ordered or scattered.
We give decidable algebraic characterizations of all these logics and
compare their respective expressive power.

**Counting quantifiers
and
linear arithmetic on word models**, with

Kamal Lodaya,

Asian
Logic Conference (ALC), 2014.
(

pdf)

**On lower bounds for
multiplicative circuits and linear circuits in
noncommutative domains**, with

V.
Arvind and

S. Raja,
CSR, 2014.
(

pdf,

abstract)

** Abstract **:
In
this paper we show some lower bounds for the size of multiplicative
circuits computing multi-output functions in some noncommutative
domains like monoids and finite groups. We also introduce and study a
generalization of linear circuits in which the goal is to compute **MY**
where **Y** is a vector of indeterminates and **M** is a matrix
whose
entries come from noncommutative rings. We show some lower
bounds in this setting as well.

**Non-definability of
Languages by Generalized First-order Formulas
over (N, +)**, with

Andreas
Krebs,

LICS 2012.
(

pdf,

full,

abstract)

** Abstract **:
We
consider first-order logic with monoidal quantifiers. We show that all
languages with a neutral letter, definable using the addition numerical
predicate are also definable with the order predicate as the only
numerical predicate. Since we prove this result for arbitrary subsets
of the monoidal quantifiers, the following holds in the presence of a
neutral letter: For aperiodic monoids, we get the result of Libkin that
FO[<, +] collapses to FO[<]; For solvable monoids, we get the
result of Roy and Straubing that FO+MOD[<, +] collapses to
FO+MOD[<]; For cyclic groups, we answer an open question of Roy and
Straubing, proving that MOD[<, +] collapses to MOD[<].
All these results can be viewed as collapse results for the uniformity
of constant depth circuits, and in this sense as a separation result
for very uniform circuit classes. For example we separate
FO[<,+]-uniform CC^{0} from FO[<,+]-uniform ACC^{0}.

**Expressive Completeness**
for
LTL With Modulo Counting and Group
Quantifiers,

Electronic
Notes in Theoretical Comp Sci
(ENTCS),
2011.
(

pdf,

abstract)

** Abstract **:
Kamp
showed that linear temporal logic is expressively complete for
first order logic over words. We give a Gabbay style proof to show
that linear temporal logic extended with modulo counting and group
quantifiers (introduced by Baziramwabo,McKenzie,Th\'erien) is
expressively complete for first order logic with modulo counting
(introduced by Straubing, Th\'erien, Thomas) and
group quantifiers (introduced by Barrington, Immerman,
Straubing).

**LTL can be more succinct**,
with

Kamal Lodaya,

ATVA 2010.
(

pdf,

abstract)

** Abstract **:
It
is well known that modelchecking and satisfiability of Linear Temporal
Logic (LTL) are Pspace-complete. Wolper showed that
with grammar operators, this result can be extended to increase the
expressiveness of the logic to all regular languages. Other ways of
extending the expressiveness of LTL using modular and group modalities
have been explored by Baziramwabo, McKenzie and Th\'erien, which are
expressively complete for regular languages recognized by solvable
monoids and for all regular languages, respectively. In all the papers
mentioned, the numeric constants used in the modalities are in unary
notation. We show that in some cases (such as the modular and symmetric
group modalities) we can use numeric constants in binary notation,
and still maintain the Pspace upper bound. Adding modulo counting to
LTL[F] (with just the unary future modality) already makes the logic
Pspace-hard. We also consider a restricted logic which allows only
the modulo counting of length from the beginning of the word. Its
satisfiability is Sigmathree-complete.