## Abstract

Traditional formal methods are based on a Boolean satisfaction notion: a reactive system satisfies, or not, a given specification. We generalize formal methods to also address the quality of systems. As an adequate specification formalism we introduce the linear temporal logic LTL[F]. The satisfaction value of an LTL[F] formula is a number between 0 and 1, describing the quality of the satisfaction. The logic generalizes traditional LTL by augmenting it with a (parameterized) set F of arbitrary functions over the interval [0,1]. For example, F may contain the maximum or minimum between the satisfaction values of subformulas, their product, and their average. The classical decision problems in formal methods, such as satisfiability, model checking, and synthesis, are generalized to search and optimization problems in the quantitative setting. For example, model checking asks for the quality in which a specification is satisfied, and synthesis returns a system satisfying the specification with the highest quality. Reasoning about quality gives rise to other natural questions, like the distance between specifications. We formalize these basic questions and study them for LTL[F]. By extending the automata-theoretic approach for LTL to a setting that takes quality into an account, we are able to solve the above problems and show that reasoning about LTL[F] has roughly the same complexity as reasoning about traditional LTL.

Original language | American English |
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Title of host publication | Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Proceedings |

Pages | 15-27 |

Number of pages | 13 |

Edition | PART 2 |

DOIs | |

State | Published - 2013 |

Event | 40th International Colloquium on Automata, Languages, and Programming, ICALP 2013 - Riga, Latvia Duration: 8 Jul 2013 → 12 Jul 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 7966 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 40th International Colloquium on Automata, Languages, and Programming, ICALP 2013 |
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Country/Territory | Latvia |

City | Riga |

Period | 8/07/13 → 12/07/13 |

### Bibliographical note

Funding Information:This work was supported in part by the Austrian Science Fund NFN RiSE (Rigorous Systems Engineering), by the ERC Advanced Grant QUAREM (Quantitative Reactive Modeling), and the ERC Grant QUALITY. The full version is available at the authors’ URLs.