A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields

On the strictness of the quantifier structure hierarchy in first-order logic

We study a natural hierarchy in first-order logic, namely the quantifier structure hierarchy, which gives a systematic classification of first-order formulas based on structural quantifier resource. We define a variant of Ehrenfeucht-Fraisse games that characterizes quantifier classes and use it to prove that this hierarchy is strict over finite structures, using strategy compositions. Moreover, we prove that this hierarchy is strict even over ordered finite structures, which is interesting in the context of descriptive complexity.Comment: 38 pages, 8 figure

Data Integration on the (Semantic) Web with Rules and Rich Unification

For the last decade a multitude of new data formats for the World Wide Web have been developed, and a huge amount of heterogeneous semi-structured data is flourishing online. With the ever increasing number of documents on the Web, rules have been identified as the means of choice for reasoning about this data, transforming and integrating it. Query languages such as SPARQL and rule languages such as Xcerpt use compound queries that are matched or unified with semi-structured data. This notion of unification is different from the one that is known from logic programming engines in that it (i) provides constructs that allow queries to be incomplete in several ways (ii) in that variables may have different types, (iii) in that it results in sets of substitutions for the variables in the query instead of a single substitution and (iv) in that subsumption between queries is much harder to decide than in logic programming. This thesis abstracts from Xcerpt query term simulation, SPARQL graph pattern matching and XPath XML document matching, and shows that all of them can be considered as a form of rich unification. Given a set of mappings between substitution sets of different languages, this abstraction opens up the possibility for format-versatile querying, i.e. combination of queries in different formats, or transformation of one format into another format within a single rule. To show the superiority of this approach, this thesis introduces an extension of Xcerpt called Xcrdf, and describes use-cases for the combined querying and integration of RDF and XML data. With XML being the predominant Web format, and RDF the predominant Semantic Web format, Xcrdf extends Xcerpt by a set of RDF query terms and construct terms, including query primitives for RDF containers collections and reifications. Moreover, Xcrdf includes an RDF path query language called RPL that is more expressive than previously proposed polynomial-time RDF path query languages, but can still be evaluated in polynomial time combined complexity. Besides the introduction of this framework for data integration based on rich unification, this thesis extends the theoretical knowledge about Xcerpt in several ways: We show that Xcerpt simulation unification is decidable, and give complexity bounds for subsumption in several fragments of Xcerpt query terms. The proof is based on a set of subsumption monotone query term transformations, and is only feasible because of the injectivity requirement on subterms of Xcerpt queries. The proof gives rise to an algorithm for deciding Xcerpt query term simulation. Moreover, we give a semantics to locally and weakly stratified Xcerpt programs, but this semantics is applicable not only to Xcerpt, but to any rule language with rich unification, including multi-rule SPARQL programs. Finally, we show how Xcerpt grouping stratification can be reduced to Xcerpt negation stratification, thereby also introducing the notion of local grouping stratification and weak grouping stratification

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.