Baire categories on small complexity classes and meager–comeager laws

We introduce two resource-bounded Baire category notions on small complexity classes such as P, QUASIPOLY, SUBEXP and PSPACE and on probabilistic classes such as BPP, which differ on how the corresponding finite extension strategies are computed. We give an alternative characterization of small sets via resource-bounded Banach-Mazur games. As an application of the first notion, we show that for almost every language A (i.e. all except a meager class) computable in subexponential time, PA = BPPA. We also show that almost all languages in PSPACE do not have small nonuniform complexity. We then switch to the second Baire category notion (called locally-computable), and show that the class SPARSE is meager in P. We show that in contrast to the resource-bounded measure case, meager–comeager laws can be obtained for many standard complexity classes, relative to locally-computable Baire category on BPP and PSPACE. Another topic where locally-computable Baire categories differ from resource-bounded measure is regarding weak-completeness: we show that there is no weak-completeness notion in P based on locally-computable Baire categories, i.e. every P-weakly-complete set is complete for P. We also prove that the class of complete sets for P under Turing-logspace reductions is meager in P, if P is not equal to DSPACE (log n), and that the same holds unconditionally for QUASIPOLY. Finally we observe that locally-computable Baire categories are incomparable with all existing resource-bounded measure notions on small complexity classes, which might explain why those two settings seem to differ so fundamentally

Martingale families and dimension in P

AbstractWe introduce a new measure notion on small complexity classes (called F-measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0’s and 1’s. On larger complexity classes (E and above), F-measure is equivalent to Lutz resource-bounded measure. As applications to F-measure, we answer a question raised in [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818] by improving their result to: for almost every language A decidable in subexponential time, PA=BPPA. We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F-measure to previous notions and prove that martingale families are strictly stronger than Γ-measure [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [J.H. Lutz, Dimension in complexity classes, in: Proceedings of the 15th Annual IEEE Conference on Computational Complexity, 2000, pp. 158–169] on P, which meets the intuition behind Lutz’s notion. We show that P-dimension lies between finite-state dimension and dimension on E. We prove an analogue of a Theorem of Eggleston in P, i.e. the class of languages whose characteristic sequence contains 1’s with frequency α, has dimension the Shannon entropy of α in P

Finite-State Genericity : on the Diagonalization Strength of Finite Automata

Algorithmische Generizit¨atskonzepte spielen eine wichtige Rolle in der Berechenbarkeitsund Komplexit¨atstheorie. Diese Begriffe stehen in engem Zusammenhang mit grundlegenden Diagonalisierungstechniken, und sie wurden zur Erzielung starker Trennungen von Komplexit¨atsklassen verwendet. Da f¨ur jedes Generizit¨atskonzept die zugeh¨origen generischen Mengen eine co-magere Klasse bilden, ist die Analyse generischer Mengen ein wichtiges Hifsmittel f¨ur eine quantitative Analyse struktureller Ph¨anomene. Typischerweise werden Generizit¨atskonzepte mit Hilfe von Erweiterungsfunktionen definiert, wobei die St¨arke eines Konzepts von der Komplexit¨at der zugelassenen Erwiterungsfunktionen abh¨angt. Hierbei erweisen sich die sog. schwachen Generizit¨atskonzepte, bei denen nur totale Erweiterungsfunktionen ber¨ucksichtigt werden, meist als wesentlich schw¨acher als die vergleichbaren allgemeinen Konzepte, bei denen auch partielle Funktionen zugelassen sind. Weiter sind die sog. beschr¨ankten Generizit¨atskonzepte – basierend auf Erweiterungen konstanter L¨ange – besonders interessant, da hier die Klassen der zugeh¨origen generischen Mengen nicht nur co-mager sind sondern zus¨atzlich Maß 1 haben. Generische Mengen diesen Typs sind daher typisch sowohl im topologischen wie im maßtheoretischen Sinn. In dieser Dissertation initiieren wir die Untersuchung von Generizit¨at im Bereich der Theorie der Formalen Sprachen: Wir f¨uhren finite-state-Generizit¨atskonzepte ein und verwenden diese, um die Diagonalisierungsst¨arke endlicher Automaten zu erforschen. Wir konzentrieren uns hierbei auf die beschr¨ankte finite-state-Generizit¨at und Spezialf ¨alle hiervon, die wir durch die Beschr¨ankung auf totale Erweiterungsfunktionen bzw. auf Erweiterungen konstanter L¨ange erhalten. Wir geben eine rein kombinatorische Charakterisierung der beschr¨ankt finite-state-generischen Mengen: Diese sind gerade die Mengen, deren charakteristische Folge saturiert ist, d.h. jedes Bin¨arwort als Teilwort enth¨alt. Mit Hilfe dieser Charakterisierung bestimmen wir die Komplexit¨at der beschr¨ankt finitestate- generischen Mengen und zeigen, dass solch eine generische Menge nicht regul¨ar sein kann es aber kontext-freie Sprachen mit dieser Generizit¨atseigenschaft gibt. Die von uns betrachteten unbeschr¨ankten finite-state-Generizit¨atskonzepte basieren auf Moore-Funktionen und auf Verallgemeinerungen dieser Funktionen. Auch hier vergleichen wir die St¨arke der verschiedenen korrespondierenden Generizit¨atskonzepte und er¨ortern die Frage, inwieweit diese Konzepte m¨achtiger als die beschr¨ankte finite-state-Generizit ¨at sind. Unsere Untersuchungen der finite-state-Generizit¨at beruhen zum Teil auf neuen Ergebnissen ¨uber Bi-Immunit¨at in der Chomsky-Hierarchie, einer neuen Chomsky-Hierarchie f¨ur unendliche Folgen und einer gr¨undlichen Untersuchung der saturierten Folgen. Diese Ergebnisse – die von unabh¨angigem Interesse sind – werden im ersten Teil der Dissertation vorgestellt. Sie k¨onnen unabh¨angig von dem Hauptteil der Arbeit gelesen werden

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.