### Abstract

We introduce a problem set-up we call the Iterated Matching Pennies (IMP) game and show that it is a powerful framework for the study of three problems: adversarial learnability, conventional (i.e. non-adversarial) learnability and approximability. Using it, we are able to derive the following theorems. (i) It is possible to learn by example all of σ_{1} 1 as well as some supersets; (ii) in adversarial learning (which we describe as a pursuit-evasion game), the pursuer has a winning strategy (in other words, σ_{1} can be learned adversarially, but 0 1 not); (iii) some languages in σ_{1} cannot be approximated by any language in σ_{1}. In particular, we show that some languages that are not computable are nevertheless learnable by example. We show corresponding results also for σ_{i} and 0 i for arbitrary i.

Original language | English |
---|---|

Pages (from-to) | 2171-2192 |

Number of pages | 22 |

Journal | Journal of Logic and Computation |

Volume | 27 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Oct 2017 |

### Keywords

- adversarial learning
- approximability
- approximation
- decidable
- elusive model paradox
- halting
- halting problem
- learnability
- matching pennies
- Nash equilibrium
- recursively enumerable
- red herring sequence
- Turing machine

### Cite this

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_{1}'

*Journal of Logic and Computation*, vol. 27, no. 7, pp. 2171-2192. https://doi.org/10.1093/logcom/exw031

**The IMP game : Learnability, approximability and adversarial learning beyond Σ _{1}.** / Brand, Michael; Dowe, David L.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - The IMP game

T2 - Learnability, approximability and adversarial learning beyond Σ1

AU - Brand, Michael

AU - Dowe, David L.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - We introduce a problem set-up we call the Iterated Matching Pennies (IMP) game and show that it is a powerful framework for the study of three problems: adversarial learnability, conventional (i.e. non-adversarial) learnability and approximability. Using it, we are able to derive the following theorems. (i) It is possible to learn by example all of σ1 1 as well as some supersets; (ii) in adversarial learning (which we describe as a pursuit-evasion game), the pursuer has a winning strategy (in other words, σ1 can be learned adversarially, but 0 1 not); (iii) some languages in σ1 cannot be approximated by any language in σ1. In particular, we show that some languages that are not computable are nevertheless learnable by example. We show corresponding results also for σi and 0 i for arbitrary i.

AB - We introduce a problem set-up we call the Iterated Matching Pennies (IMP) game and show that it is a powerful framework for the study of three problems: adversarial learnability, conventional (i.e. non-adversarial) learnability and approximability. Using it, we are able to derive the following theorems. (i) It is possible to learn by example all of σ1 1 as well as some supersets; (ii) in adversarial learning (which we describe as a pursuit-evasion game), the pursuer has a winning strategy (in other words, σ1 can be learned adversarially, but 0 1 not); (iii) some languages in σ1 cannot be approximated by any language in σ1. In particular, we show that some languages that are not computable are nevertheless learnable by example. We show corresponding results also for σi and 0 i for arbitrary i.

KW - adversarial learning

KW - approximability

KW - approximation

KW - decidable

KW - elusive model paradox

KW - halting

KW - halting problem

KW - learnability

KW - matching pennies

KW - Nash equilibrium

KW - recursively enumerable

KW - red herring sequence

KW - Turing machine

UR - http://www.scopus.com/inward/record.url?scp=85032645393&partnerID=8YFLogxK

U2 - 10.1093/logcom/exw031

DO - 10.1093/logcom/exw031

M3 - Article

VL - 27

SP - 2171

EP - 2192

JO - Journal of Logic and Computation

JF - Journal of Logic and Computation

SN - 0955-792X

IS - 7

ER -

_{1}. Journal of Logic and Computation. 2017 Oct 1;27(7):2171-2192. https://doi.org/10.1093/logcom/exw031