Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.

There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory. In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area. Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available

Expressive power and complexity of a logic with quantifiers that count proportions of sets

We present a second-order logic of proportional quantifiers, SOLP, which is essentially a first-order language extended with quantifiers that act upon second-order variables of a given arity r and count the fraction of elements in a subset of r-tuples of a model that satisfy a formula. Our logic is capable of expressing proportional versions of different problems of complexity up to NP-hard as, for example, the problem of deciding if at least a fraction 1/n of the set of vertices of a graph form a clique; and fragments within our logic capture complexity classes as NL and P, with auxiliary ordering relation. When restricted to monadic second-order variables, our logic of proportional quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting quantifiers (restricted to almost orders), for it does not have the bounded number of degrees property. Moreover, we show that, in this almost-ordered setting, different fragments of this logic vary in their expressive power, and show the existence of an infinite hierarchy inside our monadic language. We extend our inexpressibility result of almost-ordered structure to a fragment of SOLP, which in the presence of full order captures P. To obtain all our inexpressibility results, we developed combinatorial games appropriate for these logics, whose application could go beyond the almost-ordered models and hence are interesting by themselves.Peer ReviewedPreprin

A logic for document spanners

Document spanners are a formal framework for information extraction that was introduced by Fagin, Kimelfeld, Reiss, and Vansummeren (PODS 2013, JACM 2015). One of the central models in this framework are core spanners, which formalize the query language AQL that is used in IBM’s SystemT. As shown by Freydenberger and Holldack (ICDT 2016, ToCS 2018), there is a connection between core spanners and ECreg, the existential theory of concatenation with regular constraints. The present paper further develops this connection by defining SpLog, a fragment of ECreg that has the same expressive power as core spanners. This equivalence extends beyond equivalence of expressive power, as we show the existence of polynomial time conversions between SpLog and core spanners. Consequences and applications include an alternative way of defining relations for spanners, a pumping lemma for core spanners, and insights into the relative succinctness of various classes of spanner representations and their connection to graph querying languages. We also briefly discuss the connection between SpLog with negation and core spanners with a difference operator

Algorithms for Near-Term and Noisy Quantum Devices

Quantum computing promises to revolutionise many fields, including chemical simulations and machine learning. At the present moment those promises have not been realised, due to the large resource requirements of fault tolerant quantum computers, not excepting the scientific and engineering challenges to building a fault tolerant quantum computer. Instead, we currently have access to quantum devices that are both limited in qubit number, and have noisy qubits. This thesis deals with the challenges that these devices present, by investigating applications in quantum simulation for molecules and solid state systems, quantum machine learning, and by presenting a detailed simulation of a real ion trap device. We firstly build on a previous algorithm for state discrimination using a quantum machine learning model, and we show how to adapt the algorithm to work on a noisy device. This algorithm outperforms the analytical best POVM if ran on a noisy device. We then discuss how to build a quantum perceptron - the building block of a quantum neural network. We also present an algorithm for simulating the Dynamical Mean Field Theory (DMFT) using a quantum device, for two sites. We also discuss some of the difficul- ties found in scaling up that system, and present an algorithm for building the DMFT ansatz using the quantum device. We also discuss modifications to the algorithm that make it more ‘device-aware’. Finally we present a pule-level simulation of the noise in an ion trap device, designed to match the specifications of a device at the National Physical Laboratory (NPL), which we can use to direct future experimental focus. Each of these sections is preceded by a review of the relevant literature

Circumscribing datalog: Expressive power and complexity

AbstractIn this paper we study a generalization of datalog, the language of function-free definite clauses. It is known that standard datalog semantics (i.e., least Herbrand model semantics) can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized. The extension proposed here, called datalogcirc, consists in considering the general form of circumscription, where some predicates are minimized, some predicates are fixed, and some vary. We study the complexity and the expressive power of the language thus obtained. We show that this language (and, actually, its non-recursive fragment) is capable of expressing all the queries in DB-co-NP and, as such, is much more powerful than standard datalog, whose expressive power is limited to a strict subset of PTIME queries. Both data and combined complexities of answering datalogcirc queries are studied. Data complexity is proved to be co-NP-complete. Combined complexity is shown to be in general hard for co-NE and complete for co-NE in the case of Herbrand bases containing k distinct constant symbols, where k is bounded

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)