---
_id: '9402'
abstract:
- lang: eng
text: Direct and indirect reciprocity are key mechanisms for the evolution of cooperation.
Direct reciprocity means that individuals use their own experience to decide whether
to cooperate with another person. Indirect reciprocity means that they also consider
the experiences of others. Although these two mechanisms are intertwined, they
are typically studied in isolation. Here, we introduce a mathematical framework
that allows us to explore both kinds of reciprocity simultaneously. We show that
the well-known ‘generous tit-for-tat’ strategy of direct reciprocity has a natural
analogue in indirect reciprocity, which we call ‘generous scoring’. Using an equilibrium
analysis, we characterize under which conditions either of the two strategies
can maintain cooperation. With simulations, we additionally explore which kind
of reciprocity evolves when members of a population engage in social learning
to adapt to their environment. Our results draw unexpected connections between
direct and indirect reciprocity while highlighting important differences regarding
their evolvability.
acknowledgement: 'This work was supported by the European Research Council CoG 863818
(ForM-SMArt) (to K.C.), the European Research Council Start Grant 279307: Graph
Games (to K.C.), and the European Research Council Starting Grant 850529: E-DIRECT
(to C.H.). The funders had no role in study design, data collection and analysis,
decision to publish or preparation of the manuscript.'
article_processing_charge: No
article_type: original
author:
- first_name: Laura
full_name: Schmid, Laura
id: 38B437DE-F248-11E8-B48F-1D18A9856A87
last_name: Schmid
orcid: 0000-0002-6978-7329
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Christian
full_name: Hilbe, Christian
id: 2FDF8F3C-F248-11E8-B48F-1D18A9856A87
last_name: Hilbe
orcid: 0000-0001-5116-955X
- first_name: Martin A.
full_name: Nowak, Martin A.
last_name: Nowak
citation:
ama: Schmid L, Chatterjee K, Hilbe C, Nowak MA. A unified framework of direct and
indirect reciprocity. Nature Human Behaviour. 2021;5(10):1292–1302. doi:10.1038/s41562-021-01114-8
apa: Schmid, L., Chatterjee, K., Hilbe, C., & Nowak, M. A. (2021). A unified
framework of direct and indirect reciprocity. Nature Human Behaviour. Springer
Nature. https://doi.org/10.1038/s41562-021-01114-8
chicago: Schmid, Laura, Krishnendu Chatterjee, Christian Hilbe, and Martin A. Nowak.
“A Unified Framework of Direct and Indirect Reciprocity.” Nature Human Behaviour.
Springer Nature, 2021. https://doi.org/10.1038/s41562-021-01114-8.
ieee: L. Schmid, K. Chatterjee, C. Hilbe, and M. A. Nowak, “A unified framework
of direct and indirect reciprocity,” Nature Human Behaviour, vol. 5, no.
10. Springer Nature, pp. 1292–1302, 2021.
ista: Schmid L, Chatterjee K, Hilbe C, Nowak MA. 2021. A unified framework of direct
and indirect reciprocity. Nature Human Behaviour. 5(10), 1292–1302.
mla: Schmid, Laura, et al. “A Unified Framework of Direct and Indirect Reciprocity.”
Nature Human Behaviour, vol. 5, no. 10, Springer Nature, 2021, pp. 1292–1302,
doi:10.1038/s41562-021-01114-8.
short: L. Schmid, K. Chatterjee, C. Hilbe, M.A. Nowak, Nature Human Behaviour 5
(2021) 1292–1302.
date_created: 2021-05-18T16:56:57Z
date_published: 2021-05-13T00:00:00Z
date_updated: 2024-03-27T23:30:44Z
day: '13'
ddc:
- '000'
department:
- _id: KrCh
- _id: GradSch
doi: 10.1038/s41562-021-01114-8
ec_funded: 1
external_id:
isi:
- '000650304000002'
pmid:
- '33986519'
file:
- access_level: open_access
checksum: 34f55e173f90dc1dab731063458ac780
content_type: application/pdf
creator: dernst
date_created: 2023-11-07T08:27:23Z
date_updated: 2023-11-07T08:27:23Z
file_id: '14496'
file_name: 2021_NatureHumanBehaviour_Schmid_accepted.pdf
file_size: 5232761
relation: main_file
success: 1
file_date_updated: 2023-11-07T08:27:23Z
has_accepted_license: '1'
intvolume: ' 5'
isi: 1
issue: '10'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Submitted Version
page: 1292–1302
pmid: 1
project:
- _id: 0599E47C-7A3F-11EA-A408-12923DDC885E
call_identifier: H2020
grant_number: '863818'
name: 'Formal Methods for Stochastic Models: Algorithms and Applications'
- _id: 2581B60A-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '279307'
name: 'Quantitative Graph Games: Theory and Applications'
publication: Nature Human Behaviour
publication_identifier:
eissn:
- 2397-3374
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
link:
- description: News on IST Homepage
relation: press_release
url: https://ist.ac.at/en/news/the-emergence-of-cooperation/
record:
- id: '10293'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: A unified framework of direct and indirect reciprocity
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 5
year: '2021'
...
---
_id: '7346'
abstract:
- lang: eng
text: 'The Price of Anarchy (PoA) is a well-established game-theoretic concept to
shed light on coordination issues arising in open distributed systems. Leaving
agents to selfishly optimize comes with the risk of ending up in sub-optimal states
(in terms of performance and/or costs), compared to a centralized system design.
However, the PoA relies on strong assumptions about agents'' rationality (e.g.,
resources and information) and interactions, whereas in many distributed systems
agents interact locally with bounded resources. They do so repeatedly over time
(in contrast to "one-shot games"), and their strategies may evolve. Using a more
realistic evolutionary game model, this paper introduces a realized evolutionary
Price of Anarchy (ePoA). The ePoA allows an exploration of equilibrium selection
in dynamic distributed systems with multiple equilibria, based on local interactions
of simple memoryless agents. Considering a fundamental game related to virus propagation
on networks, we present analytical bounds on the ePoA in basic network topologies
and for different strategy update dynamics. In particular, deriving stationary
distributions of the stochastic evolutionary process, we find that the Nash equilibria
are not always the most abundant states, and that different processes can feature
significant off-equilibrium behavior, leading to a significantly higher ePoA compared
to the PoA studied traditionally in the literature. '
alternative_title:
- LIPIcs
article_number: '21'
article_processing_charge: No
author:
- first_name: Laura
full_name: Schmid, Laura
id: 38B437DE-F248-11E8-B48F-1D18A9856A87
last_name: Schmid
orcid: 0000-0002-6978-7329
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Stefan
full_name: Schmid, Stefan
last_name: Schmid
citation:
ama: 'Schmid L, Chatterjee K, Schmid S. The evolutionary price of anarchy: Locally
bounded agents in a dynamic virus game. In: Proceedings of the 23rd International
Conference on Principles of Distributed Systems. Vol 153. Schloss Dagstuhl
- Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.OPODIS.2019.21'
apa: 'Schmid, L., Chatterjee, K., & Schmid, S. (2020). The evolutionary price
of anarchy: Locally bounded agents in a dynamic virus game. In Proceedings
of the 23rd International Conference on Principles of Distributed Systems
(Vol. 153). Neuchâtel, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.OPODIS.2019.21'
chicago: 'Schmid, Laura, Krishnendu Chatterjee, and Stefan Schmid. “The Evolutionary
Price of Anarchy: Locally Bounded Agents in a Dynamic Virus Game.” In Proceedings
of the 23rd International Conference on Principles of Distributed Systems,
Vol. 153. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.OPODIS.2019.21.'
ieee: 'L. Schmid, K. Chatterjee, and S. Schmid, “The evolutionary price of anarchy:
Locally bounded agents in a dynamic virus game,” in Proceedings of the 23rd
International Conference on Principles of Distributed Systems, Neuchâtel,
Switzerland, 2020, vol. 153.'
ista: 'Schmid L, Chatterjee K, Schmid S. 2020. The evolutionary price of anarchy:
Locally bounded agents in a dynamic virus game. Proceedings of the 23rd International
Conference on Principles of Distributed Systems. OPODIS: International Conference
on Principles of Distributed Systems, LIPIcs, vol. 153, 21.'
mla: 'Schmid, Laura, et al. “The Evolutionary Price of Anarchy: Locally Bounded
Agents in a Dynamic Virus Game.” Proceedings of the 23rd International Conference
on Principles of Distributed Systems, vol. 153, 21, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2020, doi:10.4230/LIPIcs.OPODIS.2019.21.'
short: L. Schmid, K. Chatterjee, S. Schmid, in:, Proceedings of the 23rd International
Conference on Principles of Distributed Systems, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2020.
conference:
end_date: 2019-12-19
location: Neuchâtel, Switzerland
name: 'OPODIS: International Conference on Principles of Distributed Systems'
start_date: 2019-12-17
date_created: 2020-01-21T16:00:26Z
date_published: 2020-02-10T00:00:00Z
date_updated: 2023-02-23T13:05:49Z
day: '10'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.4230/LIPIcs.OPODIS.2019.21
external_id:
arxiv:
- '1906.00110'
file:
- access_level: open_access
checksum: 9a91916ac2c21ab42458fcda39ef0b8d
content_type: application/pdf
creator: dernst
date_created: 2020-03-23T09:14:06Z
date_updated: 2020-07-14T12:47:56Z
file_id: '7608'
file_name: 2019_LIPIcS_Schmid.pdf
file_size: 630752
relation: main_file
file_date_updated: 2020-07-14T12:47:56Z
has_accepted_license: '1'
intvolume: ' 153'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Preprint
project:
- _id: 25832EC2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S 11407_N23
name: Rigorous Systems Engineering
publication: Proceedings of the 23rd International Conference on Principles of Distributed
Systems
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'The evolutionary price of anarchy: Locally bounded agents in a dynamic virus
game'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 153
year: '2020'
...
---
_id: '8600'
abstract:
- lang: eng
text: 'A vector addition system with states (VASS) consists of a finite set of states
and counters. A transition changes the current state to the next state, and every
counter is either incremented, or decremented, or left unchanged. A state and
value for each counter is a configuration; and a computation is an infinite sequence
of configurations with transitions between successive configurations. A probabilistic
VASS consists of a VASS along with a probability distribution over the transitions
for each state. Qualitative properties such as state and configuration reachability
have been widely studied for VASS. In this work we consider multi-dimensional
long-run average objectives for VASS and probabilistic VASS. For a counter, the
cost of a configuration is the value of the counter; and the long-run average
value of a computation for the counter is the long-run average of the costs of
the configurations in the computation. The multi-dimensional long-run average
problem given a VASS and a threshold value for each counter, asks whether there
is a computation such that for each counter the long-run average value for the
counter does not exceed the respective threshold. For probabilistic VASS, instead
of the existence of a computation, we consider whether the expected long-run average
value for each counter does not exceed the respective threshold. Our main results
are as follows: we show that the multi-dimensional long-run average problem (a)
is NP-complete for integer-valued VASS; (b) is undecidable for natural-valued
VASS (i.e., nonnegative counters); and (c) can be solved in polynomial time for
probabilistic integer-valued VASS, and probabilistic natural-valued VASS when
all computations are non-terminating.'
alternative_title:
- LIPIcs
article_number: '23'
article_processing_charge: No
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Thomas A
full_name: Henzinger, Thomas A
id: 40876CD8-F248-11E8-B48F-1D18A9856A87
last_name: Henzinger
orcid: 0000-0002-2985-7724
- first_name: Jan
full_name: Otop, Jan
id: 2FC5DA74-F248-11E8-B48F-1D18A9856A87
last_name: Otop
citation:
ama: 'Chatterjee K, Henzinger TA, Otop J. Multi-dimensional long-run average problems
for vector addition systems with states. In: 31st International Conference
on Concurrency Theory. Vol 171. Schloss Dagstuhl - Leibniz-Zentrum für Informatik;
2020. doi:10.4230/LIPIcs.CONCUR.2020.23'
apa: 'Chatterjee, K., Henzinger, T. A., & Otop, J. (2020). Multi-dimensional
long-run average problems for vector addition systems with states. In 31st
International Conference on Concurrency Theory (Vol. 171). Virtual: Schloss
Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.CONCUR.2020.23'
chicago: Chatterjee, Krishnendu, Thomas A Henzinger, and Jan Otop. “Multi-Dimensional
Long-Run Average Problems for Vector Addition Systems with States.” In 31st
International Conference on Concurrency Theory, Vol. 171. Schloss Dagstuhl
- Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.CONCUR.2020.23.
ieee: K. Chatterjee, T. A. Henzinger, and J. Otop, “Multi-dimensional long-run average
problems for vector addition systems with states,” in 31st International Conference
on Concurrency Theory, Virtual, 2020, vol. 171.
ista: 'Chatterjee K, Henzinger TA, Otop J. 2020. Multi-dimensional long-run average
problems for vector addition systems with states. 31st International Conference
on Concurrency Theory. CONCUR: Conference on Concurrency Theory, LIPIcs, vol.
171, 23.'
mla: Chatterjee, Krishnendu, et al. “Multi-Dimensional Long-Run Average Problems
for Vector Addition Systems with States.” 31st International Conference on
Concurrency Theory, vol. 171, 23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2020, doi:10.4230/LIPIcs.CONCUR.2020.23.
short: K. Chatterjee, T.A. Henzinger, J. Otop, in:, 31st International Conference
on Concurrency Theory, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
conference:
end_date: 2020-09-04
location: Virtual
name: 'CONCUR: Conference on Concurrency Theory'
start_date: 2020-09-01
date_created: 2020-10-04T22:01:36Z
date_published: 2020-08-06T00:00:00Z
date_updated: 2021-01-12T08:20:15Z
day: '06'
ddc:
- '000'
department:
- _id: KrCh
- _id: ToHe
doi: 10.4230/LIPIcs.CONCUR.2020.23
external_id:
arxiv:
- '2007.08917'
file:
- access_level: open_access
checksum: 5039752f644c4b72b9361d21a5e31baf
content_type: application/pdf
creator: dernst
date_created: 2020-10-05T14:04:25Z
date_updated: 2020-10-05T14:04:25Z
file_id: '8610'
file_name: 2020_LIPIcsCONCUR_Chatterjee.pdf
file_size: 601231
relation: main_file
success: 1
file_date_updated: 2020-10-05T14:04:25Z
has_accepted_license: '1'
intvolume: ' 171'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/3.0/
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: 25832EC2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S 11407_N23
name: Rigorous Systems Engineering
- _id: 25F2ACDE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S11402-N23
name: Rigorous Systems Engineering
- _id: 25F42A32-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z211
name: The Wittgenstein Prize
publication: 31st International Conference on Concurrency Theory
publication_identifier:
isbn:
- '9783959771603'
issn:
- '18688969'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Multi-dimensional long-run average problems for vector addition systems with
states
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode
name: Creative Commons Attribution 3.0 Unported (CC BY 3.0)
short: CC BY (3.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 171
year: '2020'
...
---
_id: '8533'
abstract:
- lang: eng
text: Game of Life is a simple and elegant model to study dynamical system over
networks. The model consists of a graph where every vertex has one of two types,
namely, dead or alive. A configuration is a mapping of the vertices to the types.
An update rule describes how the type of a vertex is updated given the types of
its neighbors. In every round, all vertices are updated synchronously, which leads
to a configuration update. While in general, Game of Life allows a broad range
of update rules, we focus on two simple families of update rules, namely, underpopulation
and overpopulation, that model several interesting dynamics studied in the literature.
In both settings, a dead vertex requires at least a desired number of live neighbors
to become alive. For underpopulation (resp., overpopulation), a live vertex requires
at least (resp. at most) a desired number of live neighbors to remain alive. We
study the basic computation problems, e.g., configuration reachability, for these
two families of rules. For underpopulation rules, we show that these problems
can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.
acknowledgement: "Krishnendu Chatterjee: The research was partially supported by the
Vienna Science and\r\nTechnology Fund (WWTF) Project ICT15-003.\r\nIsmaël Jecker:
This project has received funding from the European Union’s Horizon 2020 research\r\nand
innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411."
alternative_title:
- LIPIcs
article_number: 22:1-22:13
article_processing_charge: No
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Rasmus
full_name: Ibsen-Jensen, Rasmus
id: 3B699956-F248-11E8-B48F-1D18A9856A87
last_name: Ibsen-Jensen
orcid: 0000-0003-4783-0389
- first_name: Ismael R
full_name: Jecker, Ismael R
id: 85D7C63E-7D5D-11E9-9C0F-98C4E5697425
last_name: Jecker
- first_name: Jakub
full_name: Svoboda, Jakub
id: 130759D2-D7DD-11E9-87D2-DE0DE6697425
last_name: Svoboda
citation:
ama: 'Chatterjee K, Ibsen-Jensen R, Jecker IR, Svoboda J. Simplified game of life:
Algorithms and complexity. In: 45th International Symposium on Mathematical
Foundations of Computer Science. Vol 170. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2020. doi:10.4230/LIPIcs.MFCS.2020.22'
apa: 'Chatterjee, K., Ibsen-Jensen, R., Jecker, I. R., & Svoboda, J. (2020).
Simplified game of life: Algorithms and complexity. In 45th International Symposium
on Mathematical Foundations of Computer Science (Vol. 170). Prague, Czech
Republic: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2020.22'
chicago: 'Chatterjee, Krishnendu, Rasmus Ibsen-Jensen, Ismael R Jecker, and Jakub
Svoboda. “Simplified Game of Life: Algorithms and Complexity.” In 45th International
Symposium on Mathematical Foundations of Computer Science, Vol. 170. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.MFCS.2020.22.'
ieee: 'K. Chatterjee, R. Ibsen-Jensen, I. R. Jecker, and J. Svoboda, “Simplified
game of life: Algorithms and complexity,” in 45th International Symposium on
Mathematical Foundations of Computer Science, Prague, Czech Republic, 2020,
vol. 170.'
ista: 'Chatterjee K, Ibsen-Jensen R, Jecker IR, Svoboda J. 2020. Simplified game
of life: Algorithms and complexity. 45th International Symposium on Mathematical
Foundations of Computer Science. MFCS: Symposium on Mathematical Foundations of
Computer Science, LIPIcs, vol. 170, 22:1-22:13.'
mla: 'Chatterjee, Krishnendu, et al. “Simplified Game of Life: Algorithms and Complexity.”
45th International Symposium on Mathematical Foundations of Computer Science,
vol. 170, 22:1-22:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020,
doi:10.4230/LIPIcs.MFCS.2020.22.'
short: K. Chatterjee, R. Ibsen-Jensen, I.R. Jecker, J. Svoboda, in:, 45th International
Symposium on Mathematical Foundations of Computer Science, Schloss Dagstuhl -
Leibniz-Zentrum für Informatik, 2020.
conference:
end_date: 2020-08-28
location: Prague, Czech Republic
name: 'MFCS: Symposium on Mathematical Foundations of Computer Science'
start_date: 2020-08-24
date_created: 2020-09-20T22:01:36Z
date_published: 2020-08-18T00:00:00Z
date_updated: 2021-01-12T08:19:55Z
day: '18'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.4230/LIPIcs.MFCS.2020.22
ec_funded: 1
external_id:
arxiv:
- '2007.02894'
file:
- access_level: open_access
checksum: bbd7c4f55d45f2ff2a0a4ef0e10a77b1
content_type: application/pdf
creator: dernst
date_created: 2020-09-21T13:57:34Z
date_updated: 2020-09-21T13:57:34Z
file_id: '8550'
file_name: 2020_LIPIcs_Chatterjee.pdf
file_size: 491374
relation: main_file
success: 1
file_date_updated: 2020-09-21T13:57:34Z
has_accepted_license: '1'
intvolume: ' 170'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: 25892FC0-B435-11E9-9278-68D0E5697425
grant_number: ICT15-003
name: Efficient Algorithms for Computer Aided Verification
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: 45th International Symposium on Mathematical Foundations of Computer
Science
publication_identifier:
isbn:
- '9783959771597'
issn:
- '18688969'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Simplified game of life: Algorithms and complexity'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode
name: Creative Commons Attribution 3.0 Unported (CC BY 3.0)
short: CC BY (3.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 170
year: '2020'
...
---
_id: '8534'
abstract:
- lang: eng
text: A regular language L of finite words is composite if there are regular languages
L₁,L₂,…,L_t such that L = ⋂_{i = 1}^t L_i and the index (number of states in a
minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise,
L is prime. Primality of regular languages was introduced and studied in [O. Kupferman
and J. Mosheiff, 2015], where the complexity of deciding the primality of the
language of a given DFA was left open, with a doubly-exponential gap between the
upper and lower bounds. We study primality for unary regular languages, namely
regular languages with a singleton alphabet. A unary language corresponds to a
subset of ℕ, making the study of unary prime languages closer to that of primality
in number theory. We show that the setting of languages is richer. In particular,
while every composite number is the product of two smaller numbers, the number
t of languages necessary to decompose a composite unary language induces a strict
hierarchy. In addition, a primality witness for a unary language L, namely a word
that is not in L but is in all products of languages that contain L and have an
index smaller than L’s, may be of exponential length. Still, we are able to characterize
compositionality by structural properties of a DFA for L, leading to a LogSpace
algorithm for primality checking of unary DFAs.
acknowledgement: "Ismaël Jecker: This project has received funding from the European
Union’s Horizon\r\n2020 research and innovation programme under the Marie Skłodowska-Curie
Grant Agreement No.\r\n754411. Nicolas Mazzocchi: PhD fellowship FRIA from the F.R.S.-FNRS."
alternative_title:
- LIPIcs
article_number: 51:1-51:12
article_processing_charge: No
author:
- first_name: Ismael R
full_name: Jecker, Ismael R
id: 85D7C63E-7D5D-11E9-9C0F-98C4E5697425
last_name: Jecker
- first_name: Orna
full_name: Kupferman, Orna
last_name: Kupferman
- first_name: Nicolas
full_name: Mazzocchi, Nicolas
last_name: Mazzocchi
citation:
ama: 'Jecker IR, Kupferman O, Mazzocchi N. Unary prime languages. In: 45th International
Symposium on Mathematical Foundations of Computer Science. Vol 170. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.MFCS.2020.51'
apa: 'Jecker, I. R., Kupferman, O., & Mazzocchi, N. (2020). Unary prime languages.
In 45th International Symposium on Mathematical Foundations of Computer Science
(Vol. 170). Prague, Czech Republic: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.MFCS.2020.51'
chicago: Jecker, Ismael R, Orna Kupferman, and Nicolas Mazzocchi. “Unary Prime Languages.”
In 45th International Symposium on Mathematical Foundations of Computer Science,
Vol. 170. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.MFCS.2020.51.
ieee: I. R. Jecker, O. Kupferman, and N. Mazzocchi, “Unary prime languages,” in
45th International Symposium on Mathematical Foundations of Computer Science,
Prague, Czech Republic, 2020, vol. 170.
ista: 'Jecker IR, Kupferman O, Mazzocchi N. 2020. Unary prime languages. 45th International
Symposium on Mathematical Foundations of Computer Science. MFCS: Symposium on
Mathematical Foundations of Computer Science, LIPIcs, vol. 170, 51:1-51:12.'
mla: Jecker, Ismael R., et al. “Unary Prime Languages.” 45th International Symposium
on Mathematical Foundations of Computer Science, vol. 170, 51:1-51:12, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:10.4230/LIPIcs.MFCS.2020.51.
short: I.R. Jecker, O. Kupferman, N. Mazzocchi, in:, 45th International Symposium
on Mathematical Foundations of Computer Science, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2020.
conference:
end_date: 2020-08-28
location: Prague, Czech Republic
name: 'MFCS: Symposium on Mathematical Foundations of Computer Science'
start_date: 2020-08-24
date_created: 2020-09-20T22:01:36Z
date_published: 2020-08-18T00:00:00Z
date_updated: 2021-01-12T08:19:56Z
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doi: 10.4230/LIPIcs.MFCS.2020.51
ec_funded: 1
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publication: 45th International Symposium on Mathematical Foundations of Computer
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title: Unary prime languages
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