90 research outputs found
Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees
We consider a specific class of tree structures that can represent basic
structures in linguistics and computer science such as XML documents, parse
trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We
present axiomatizations of the monadic second-order logic (MSO), monadic
transitive closure logic (FO(TC1)) and monadic least fixed-point logic
(FO(LFP1)) theories of this class of structures. These logics can express
important properties such as reachability. Using model-theoretic techniques, we
show by a uniform argument that these axiomatizations are complete, i.e., each
formula that is valid on all finite trees is provable using our axioms. As a
backdrop to our positive results, on arbitrary structures, the logics that we
study are known to be non-recursively axiomatizable
Queries with Guarded Negation (full version)
A well-established and fundamental insight in database theory is that
negation (also known as complementation) tends to make queries difficult to
process and difficult to reason about. Many basic problems are decidable and
admit practical algorithms in the case of unions of conjunctive queries, but
become difficult or even undecidable when queries are allowed to contain
negation. Inspired by recent results in finite model theory, we consider a
restricted form of negation, guarded negation. We introduce a fragment of SQL,
called GN-SQL, as well as a fragment of Datalog with stratified negation,
called GN-Datalog, that allow only guarded negation, and we show that these
query languages are computationally well behaved, in terms of testing query
containment, query evaluation, open-world query answering, and boundedness.
GN-SQL and GN-Datalog subsume a number of well known query languages and
constraint languages, such as unions of conjunctive queries, monadic Datalog,
and frontier-guarded tgds. In addition, an analysis of standard benchmark
workloads shows that most usage of negation in SQL in practice is guarded
negation
The partition semantics of questions, syntactically
Groenendijk and Stokhof (1984, 1996; Groenendijk 1999) provide a logically
attractive theory of the semantics of natural language questions, commonly
referred to as the partition theory. Two central notions in this theory are
entailment between questions and answerhood. For example, the question "Who is
going to the party?" entails the question "Is John going to the party?", and
"John is going to the party" counts as an answer to both. Groenendijk and
Stokhof define these two notions in terms of partitions of a set of possible
worlds.
We provide a syntactic characterization of entailment between questions and
answerhood . We show that answers are, in some sense, exactly those formulas
that are built up from instances of the question. This result lets us compare
the partition theory with other approaches to interrogation -- both linguistic
analyses, such as Hamblin's and Karttunen's semantics, and computational
systems, such as Prolog. Our comparison separates a notion of answerhood into
three aspects: equivalence (when two questions or answers are interchangeable),
atomic answers (what instances of a question count as answers), and compound
answers (how answers compose).Comment: 14 page
The Product Homomorphism Problem and Applications
The product homomorphism problem (PHP) takes as input a finite collection of structures A_1, ..., A_n and a structure B, and asks if there is a homomorphism from the direct product between A_1, A_2, ..., and A_n, to B. We pinpoint the computational complexity of this problem. Our motivation stems from the fact that PHP naturally arises in different areas of database theory. In particular, it is equivalent to the problem of determining whether a relation is definable by a conjunctive query, and the existence of a schema mapping that fits a given collection of positive and negative data examples. We apply our results to obtain complexity bounds for these problems
Craig Interpolation for Decidable First-Order Fragments
We show that the guarded-negation fragment (GNFO) is, in a precise sense, the
smallest extension of the guarded fragment (GFO) with Craig interpolation. In
contrast, we show that the smallest extension of the two-variable fragment
(FO2), and of the forward fragment (FF) with Craig interpolation, is full
first-order logic. Similarly, we also show that all extensions of FO2 and of
the fluted fragment (FL) with Craig interpolation are undecidable.Comment: Submitted for FoSSaCS 2024. arXiv admin note: substantial text
overlap with arXiv:2304.0808
Characterising Modal Formulas with Examples
We study the existence of finite characterisations for modal formulas. A
finite characterisation of a modal formula is a finite collection of
positive and negative examples that distinguishes from every other,
non-equivalent modal formula, where an example is a finite pointed Kripke
structure. This definition can be restricted to specific frame classes and to
fragments of the modal language: a modal fragment admits finite
characterisations with respect to a frame class if every formula
has a finite characterisation with respect to consting of
examples that are based on frames in . Finite characterisations are useful
for illustration, interactive specification, and debugging of formal
specifications, and their existence is a precondition for exact learnability
with membership queries. We show that the full modal language admits finite
characterisations with respect to a frame class only when the modal logic
of is locally tabular. We then study which modal fragments, freely
generated by some set of connectives, admit finite characterisations. Our main
result is that the positive modal language without the truth-constants
and admits finite characterisations w.r.t. the class of all frames. This
result is essentially optimal: finite characterizability fails when the
language is extended with the truth constant or with all but very
limited forms of negation.Comment: Expanded version of material from Raoul Koudijs's MSc thesis (2022
Conjunctive Queries: Unique Characterizations and Exact Learnability
We answer the question of which conjunctive queries are uniquely characterized by polynomially many positive and negative examples, and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts
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