Succinctness in subsystems of the spatial mu-calculus

In this paper we systematically explore questions of succinctness in modal logics employed in spatial reasoning. We show that the closure operator, despite being less expressive, is exponentially more succinct than the limit-point operator, and that the $\mu$-calculus is exponentially more succinct than the equally-expressive tangled limit operator. These results hold for any class of spaces containing at least one crowded metric space or containing all spaces based on ordinals below $\omega^\omega$, with the usual limit operator. We also show that these results continue to hold even if we enrich the less succinct language with the universal modality

Logics of knowledge and action: critical analysis and challenges

International audienceWe overview the most prominent logics of knowledge and action that were proposed and studied in the multiagent systems literature. We classify them according to these two dimensions, knowledge and action, and moreover introduce a distinction between individual knowledge and group knowledge, and between a nonstrategic an a strategic interpretation of action operators. For each of the logics in our classification we highlight problematic properties. They indicate weaknesses in the design of these logics and call into question their suitability to represent knowledge and reason about it. This leads to a list of research challenges

Formula size games for modal logic and μ\mu-calculus

We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\mathrm{FO}$ and (basic) modal logic $\mathrm{ML}$. We also present a version of the game for the modal $\mu$-calculus $\mathrm{L}_\mu$ and show that $\mathrm{FO}$ is also non-elementarily more succinct than $\mathrm{L}_\mu$.Comment: This is a preprint of an article published in Journal of Logic and Computation Published by Oxford University Press. arXiv admin note: substantial text overlap with arXiv:1604.0722

Succinctness and Formula Size Games

Tämä väitöskirja tutkii erilaisten logiikoiden tiiviyttä kaavan pituuspelien avulla. Logiikan tiiviys viittaa ominaisuuksien ilmaisemiseen tarvittavien kaavojen kokoon. Kaavan pituuspelit ovat hyväksi todettu menetelmä tiiviystulosten todistamiseen. Väitöskirjan kontribuutio on kaksiosainen. Ensinnäkin väitöskirjassa määritellään kaavan pituuspeli useille logiikoille ja tarjotaan näin uusia menetelmiä tulevaan tutkimukseen. Toiseksi näitä pelejä ja muita menetelmiä käytetään tiiviystulosten todistamiseen tutkituille logiikoille. Tarkemmin sanottuna väitöskirjassa määritellään uudet parametrisoidut kaavan pituuspelit perusmodaalilogiikalle, modaaliselle μ-kalkyylille, tiimilauselogiikalle ja yleistetyille säännöllisille lausekkeille. Yleistettyjen säännöllisten lausekkeiden pelistä esitellään myös variantit, jotka vastaavat säännöllisiä lausekkeita ja uusia “RE over star-free” -lausekkeita, joissa tähtiä ei esiinny komplementtien sisällä. Pelejä käytetään useiden tiiviystulosten todistamiseen. Predikaattilogiikan näytetään olevan epäelementaarisesti tiiviimpi kuin perusmodaalilogiikka ja modaalinen μ-kalkyyli. Tiimilauselogiikassa tutkitaan systemaattisesti yleisten riippuvuuksia ilmaisevien atomien määrittelemisen tiiviyttä. Klassinen epäelementaarinen tiiviysero predikaattilogiikan ja säännöllisten lausekkeiden välillä osoitetaan uudelleen yksinkertaisemmalla tavalla ja saadaan tähtien lukumäärälle “RE over star-free” -lausekkeissa hierarkia ilmaisuvoiman suhteen. Monissa yllämainituista tuloksista hyödynnetään eksplisiittisiä kaavoja peliargumenttien lisäksi. Tällaisia kaavoja ja tyyppien laskemista hyödyntäen saadaan epäelementaarisia ala- ja ylärajoja yksittäisten sanojen määrittelemisen tiiviydelle predikaattilogiikassa ja monadisessa toisen kertaluvun logiikassa.This thesis studies the succinctness of various logics using formula size games. The succinctness of a logic refers to the size of formulas required to express properties. Formula size games are some of the most successful methods of proof for results on succinctness. The contribution of the thesis is twofold. Firstly, we define formula size games for several logics, providing methods for future research. Secondly, we use these games and other methods to prove results on the succinctness of the studied logics. More precisely, we develop new parameterized formula size games for basic modal logic, modal μ-calculus, propositional team logic and generalized regular expressions. For the generalized regular expression game we introduce variants that correspond to regular expressions and the newly defined RE over star-free expressions, where stars do not occur inside complements. We use the games to prove a number of succinctness results. We show that first-order logic is non-elementarily more succinct than both basic modal logic and modal μ-calculus. We conduct a systematic study of the succinctness of defining common atoms of dependency in propositional team logic. We reprove a classic non-elementary succinctness gap between first-order logic and regular expressions in a much simpler way and establish a hierarchy of expressive power for the number of stars in RE over star-free expressions. Many of the above results utilize explicit formulas in addition to game arguments. We use such formulas and a type counting argument to obtain non-elementary lower and upper bounds for the succinctness of defining single words in first-order logic and monadic second-order logic

Succinctness of Epistemic Languages

Tim French, Wiebe van der Hoek, Petar Iliev and Barteld Kooi. Succinctness of Epistemic Languages. In: T. Walsh (editor). Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (IJCAI-11), pp. 881-886, AAAI Press, Menlo Park.