Journal of the ACM

A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

Communication Complexity: Basics

World Scientific Publishing Company

We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose.

/pdf/a-survey-on-small-fragments-of-first-order-logic-over-finite-20ykpq5fal.pdf

A survey on small fragments of first-order logic over finite words

We study several classical decision problems on finite automata under the (Strong) Exponential Time Hypothesis. We focus on three types of problems: universality, equivalence, and emptiness of intersection. All these problems are known to be CoNP-hard for nondeterministic finite automata, even when restricted to unary input alphabets. A different type of problems on finite automata relates to aperiodicity and to synchronizing words. We also consider finite automata that work on commutative alphabets and those working on two-dimensional words.

https://www.mdpi.com/1999-4893/10/1/24/pdf

Problems on Finite Automata and the Exponential Time Hypothesis

We characterize the languages in TC0 = L(Maj[<,Bit]) and L(Maj[<]) as inverse morphic images of certain groups. Necessarily these are infinite, since nonregular sets are concerned. To limit the power of these infinite algebraic objects, we equip them with a finite type set and introduce the notion of a finitely typed (infinite) monoid. Following this approach we investigate type-respecting mappings and construct a new type of block product that more adequately deals with infinite monoids. We exhibit two classes of finitely typed groups which exactly characterize TC0 and L(Maj[<]) via inverse morphisms.

Characterizing TC 0 in Terms of Infinite Groups

We introduce two variants of computation tree logic CTL based on team semantics: an asynchronous one and a synchronous one. For both variants we investigate the computational complexity of the satisfiability as well as the model checking problem. The satisfiability problem is shown to be EXPTIME-complete. Here it does not matter which of the two semantics are considered. For model checking we prove a PSPACE-completeess for the synchronous case, and show P-completeness for the asynchronous case. Furthermore we prove several interesting fundamental properties of both semantics.

A Team Based Variant of CTL

We develop team semantics for Linear Temporal Logic (LTL) to express hyperproperties, which have recently been identified as a key concept in the verification of information flow properties. Conceptually, we consider an asynchronous and a synchronous variant of team semantics. We study basic properties of this new logic and classify the computational complexity of its satisfiability, path, and model checking problem. Further, we examine how extensions of these basic logics react on adding other atomic operators. Finally, we compare its expressivity to the one of HyperLTL, another recently introduced logic for hyperproperties. Our results show that LTL under team semantics is a viable alternative to HyperLTL, which complements the expressivity of HyperLTL and has partially better algorithmic properties.

Team semantics for the specification and verification of hyperproperties

We study an extension of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, ‘the letter a appears between positions x and y’. This is, in a sense, the simplest property that is not expressible using only two variables.We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We also give an effective necessary condition, in terms of the syntactic monoid of a regular language, for a property to be expressible in this logic. We show that this condition is also sufficient for words over a two-letter alphabet. This algebraic analysis allows us us to prove, among other things, that our new logic has strictly less expressive power than full first-order logic FO[<].

Andreas Krebs

Papers

Problems on Finite Automata and the Exponential Time Hypothesis

Characterizing TC 0 in Terms of Infinite Groups

A Team Based Variant of CTL

Team semantics for the specification and verification of hyperproperties

Two-variable Logic with a Between Relation