Some Turing-Complete Extensions of First-Order Logic

We introduce a natural Turing-complete extension of first-order logic FO. The extension adds two novel features to FO. The first one of these is the capacity to add new points to models and new tuples to relations. The second one is the possibility of recursive looping when a formula is evaluated using a semantic game. We first define a game-theoretic semantics for the logic and then prove that the expressive power of the logic corresponds in a canonical way to the recognition capacity of Turing machines. Finally, we show how to incorporate generalized quantifiers into the logic and argue for a highly natural connection between oracles and generalized quantifiers.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Counting and enumerating in first-order team logics

Descriptive complexity theory is the study of the expressibility of computational problems in certain logics. Most of the results in this field use (fragments or extensions of) first-order logic or second-order logic to describe decision complexity classes. For example the complexity class NP can be characterized as the set of problems that are describable by a dependence logic formula, in short NP = FO(=(...)). Dependence logic is a certain team logic, where a team logic is an extension of first-order logic by some new atoms, with new semantics, called team semantics. Compared to decision complexity where one is interested in the existence of a solution to an input instance, in counting complexity one is interested in the number of solutions and in enumeration complexity one wants to compute all solutions. Counting complexity has been less studied in terms of descriptive complexity than decision complexity, whereas there are no results for enumeration complexity in this field. The latter is because the concept of hardness in the enumeration setting was first introduced rather recently. In this thesis, we characterize counting and enumeration complexity classes with team logics and compare the results to the corresponding results for decision complexity classes. To study the framework of hard enumeration a bit more, we investigate further team logic based enumeration problems. In the counting setting we characterize the classes #P and #•NP as #P = #FOT and #•NP = #FO(⊥). Furthermore, we establish two team logic based classes #FO(⊆) and #FO(=(...)) which seem to have no direct counterpart in classical counting complexity, but contain problems that are complete under Turing reductions for #P and #•NP, respectively. To show the latter we identify a new #•NP-complete problem with respect to Turing reductions. We show that in the enumeration setting the classes behave very similarly to the corresponding classes in the decision setting. We translate the results P = FO(⊆) and NP = FO(=(...)) to the enumeration setting which results in DelP = DelFO(⊆) and DelNP = DelFO(=(...)). Furthermore, we identify several DelP and DelNP-complete problems which yield additional characterisations of DelP and DelNP. For one of the investigated problems we were only able to show Del+NP membership (and DelNP-hardness), a precise classification remains open

Complexity of Prioritized Default Logics

In default reasoning, usually not all possible ways of resolving conflicts between default rules are acceptable. Criteria expressing acceptable ways of resolving the conflicts may be hardwired in the inference mechanism, for example specificity in inheritance reasoning can be handled this way, or they may be given abstractly as an ordering on the default rules. In this article we investigate formalizations of the latter approach in Reiter's default logic. Our goal is to analyze and compare the computational properties of three such formalizations in terms of their computational complexity: the prioritized default logics of Baader and Hollunder, and Brewka, and a prioritized default logic that is based on lexicographic comparison. The analysis locates the propositional variants of these logics on the second and third levels of the polynomial hierarchy, and identifies the boundary between tractable and intractable inference for restricted classes of prioritized default theories

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

Quantified CTL: Expressiveness and Complexity

While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy)