First-order Definition Research Articles

Computing difference abstractions of linear equation systems

Abstract interpretation was proposed for predicting changes of reaction networks with partial kinetic information in systems biology. This requires to compute the set of difference abstractions of a system of linear equations under nonlinear constraints. We present the first practical algorithm that can compute the difference abstractions of linear equation systems exactly. We also present a new heuristics based on minimal support consequences for overapproximating the set of difference abstractions. Our algorithms rely on elementary modes, first-order definitions, and finite domain constraint programming. We implemented our algorithms and applied them to change prediction in systems biology. It turns out experimentally that the new heuristics is often exact in practice, while outperforming the exact algorithm.This journal article extends on a paper published at the 17th International Conference on Computational Methods in Systems Biology (CMSB'2019) [1].

Logical complexity of induced subgraph isomorphism for certain families of graphs

We investigate the problem of the most efficient first-order definition of the property of containing an induced subgraph isomorphic to a given pattern graph, which is closely related to the time complexity of the decision problem for this property. We derive a series of new bounds for the minimum quantifier depth of a formula defining this property for pattern graphs on five vertices, as well as for disjoint unions of isomorphic complete multipartite graphs. Moreover, we prove that for any there exists a graph on vertices and a first-order formula of quantifier depth at most that defines the property of containing an induced subgraph isomorphic to this graph. Bibliography: 12 titles.

Universally defining finitely generated subrings of global fields

It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in number fields and rings of S -integers in global function fields of odd characteristic. In this article a proof is presented which is uniform in all global fields, including the characteristic two case, where the result is entirely novel. Furthermore, the proposed method results in universal formulae requiring significantly fewer quantifiers than the formulae that can be derived through the previous approaches.

First-order definitions of subgraph isomorphism through the adjacency and order relations

We study first-order definitions of graph properties over the vocabulary consisting of the adjacency and order relations. We compare logical complexities of subgraph isomorphism in terms of the minimum quantifier depth in two settings: with and without the order relation. We prove that, for pattern-trees, it is at least (roughly) two times smaller in the former case. We find the minimum quantifier depths of <-sentences defining subgraph isomorphism for all pattern graphs with at most 4 vertices.

Irreducibility of polynomials over global fields is diophantine

Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have a root in $K$. This strengthens a result by Colliot-Thélène and Van Geel [Compositio Math. 151 (2015), 1965–1980] stating that the set of non-$n$th powers in a number field $K$ is diophantine. We also deduce a diophantine criterion for a polynomial over $K$ of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of $\mathbb{Z}$ in $\mathbb{Q}$.

On first-order definitions of subgraph isomorphism properties

Let φ(F) be the property of containing (as a subgraph) an isomorphic copy of a graph F. It is easy to show that this property cannot be defined in a first-order language by a sentence with a quantifier depth (or variable width) strictly less than the number of vertices in F. Nevertheless, such a definition exists in some classes of graphs. Three classes of graphs are considered: connected graphs with a large number of vertices, graphs with large treewidth, and graphs with high connectivity.

The Complexity of Phylogeny Constraint Satisfaction Problems

We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains, for example, the rooted triple consistency problem, forbidden subtree problems, the quartet consistency problem, and many other problems studied in the bioinformatics literature. The studied problems can be described as constraint satisfaction problems , where the constraints have a first-order definition over the rooted triple relation. We show that every such phylogeny problem can be solved in polynomial time or is NP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithm of Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithm repeatedly solves linear equation systems to construct a solution in polynomial time. We then show that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Our classification establishes a dichotomy for a large class of infinite structures that we believe is of independent interest in universal algebra, model theory, and topology. The proof of our main result combines results and techniques from various research areas: a recent classification of the model-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb’s Ramsey theorem for rooted trees, and universal algebra.

On Kalimullin pairs

We study Kalimullin pairs, a definable class (of pairs) of enumeration degrees that has been used to give first-order definitions of other important classes and relations, including the enumeration jump and the total enumeration degrees. We show that the global definition of Kalimullin pairs is also valid in the local structure of the enumeration degrees, giving a simpler local definition than was previously known. We prove that the typical enumeration degree is not half of a nontrivial Kalimullin pair, both in the sense of category and measure. Using genericity, we show that not all members of nontrivial Kalimullin pairs are half of a maximal Kalimullin pair. Finally, we construct such a set that has no maximal Kalimullin partner, making it qualitatively different from half of a maximal Kalimullin pair.

Permutations on the Random Permutation

The random permutation is the Fraïssé limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure. Equivalently, we classify all structures which have a first-order definition in the random permutation.

Constraining fluid migration in low density data area: successful use of integrated modelling in the Southern Perth Basin

Constraining trap integrity or CO2 containment potential in areas with low density of data is challenging. We show here how the integration of hard data and modelling improved our understanding of the subsurface tectonic and stratigraphic architecture and ensured an initial definition of hydraulic behaviour of faults and fractures in the onshore Southern Perth Basin. A first-order geomodel was built based on interpretation of low-density 2D-seismic data and constrained by well, geophysical, and outcrop data. Forward stratigraphic models were deformed and implemented in the geomodel to predict the distribution of the phyllosilicate content. Membrane fault-seal prediction was performed based on the Shale Gouge Ratio algorithm to estimate the lateral migration potential. Characterisation of the stress state of fault planes was used to define the likelihood of reactivation and to predict vertical migration and caprock bypass. The elastic dislocation method was used to model the density and mode of failure of subsidiary sub-seismic faults that can impact on the caprock integrity. This was also used to constrain and validate the structural model. The integrated modelling significantly decreased uncertainties on the subsurface architecture and ensured the first-order definition of the CO2 migration and containment potential for the Wonnerup reservoir in the SW Hub. The maximum CO2 column heights supported by faults and potential leak points were estimated and migration scenarios and risks were defined.

Certain query answering in partially consistent databases

A database is called uncertain if two or more tuples of the same relation are allowed to agree on their primary key. Intuitively, such tuples act as alternatives for each other. A repair (or possible world) of such uncertain database is obtained by selecting a maximal number of tuples without ever selecting two tuples of the same relation that agree on their primary key. For a Boolean queryq, the problem CERTAINTY(q) takes as input an uncertain databasedband asks whetherqevaluates to true on every repair ofdb.In recent years, the complexity of CERTAINTY(q) has been studied under different restrictions onq. These complexity studies have assumed no restrictions on the uncertain databases that are input to CERTAINTY(q). In practice, however, it may be known that these input databases are partially consistent, in the sense that they satisfy some dependencies (e.g., functional dependencies). In this article, we introduce the problem CERTAINTY(q) in the presence of a set Σ of dependencies. The problem CERTAINTY(q, Σ) takes as input an uncertain databasedbthat satisfiesΣ, and asks whether every repair ofdbsatisfiesq.We focus on the complexity of CERTAINTY(q, Σ) whenqis an acyclic conjunctive query without self-join, and Σ is a set of functional dependencies and join dependencies, the latter of a particular form. We provide an algorithm that, givenqand Σ, decides whether CERTAINTY(q, Σ) is first-order expressible. Moreover, we show how to effectively construct a first-order definition of CERTAINTY(q, Σ) if it exists.

Decidability of Definability

AbstractFor a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ0, ϕ1, …, ϕn that define relations R0, R1, …, Rn over Γ. The question is whether the relation R0 is primitive positive definable from R1, …, Rn, i.e., definable by a first-order formula that uses only relation symbols for R1, …, Rn, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden).We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

ELLIPTIC CURVE POINTS AND DIOPHANTINE MODELS OF ℤ IN LARGE SUBRINGS OF NUMBER FIELDS

Let K be a number field such that there exists an elliptic curve E of rank one over K. For a set [Formula: see text] of primes of K, let [Formula: see text]. Let P ∈ E(K) be a generator of E(K) modulo the torsion subgroup. Let (xn(P), yn(P)) be the affine coordinates of [n]P with respect to a fixed Weierstrass equation of E. We show that there exists a set [Formula: see text] of primes of K of natural density one such that in [Formula: see text] multiplication of indices (with respect to some fixed multiple of P) is existentially definable and therefore these indices can be used to construct a Diophantine model of ℤ. We also show that ℤ is definable over [Formula: see text] using just one universal quantifier. Both the construction of a Diophantine model using the indices and the first-order definition of ℤ can be lifted to the integral closure of [Formula: see text] in any infinite extension K∞of K as long as E(K∞) is finitely generated and of rank one.

Certain conjunctive query answering in first-order logic

Primary key violations provide a natural means for modeling uncertainty in the relational data model. A repair (or possible world) of a database is then obtained by selecting a maximal number of tuples without ever selecting two distinct tuples that have the same primary key value. For a Boolean query q , the problem CERTAINTY( q ) takes as input a database db and asks whether q evaluates to true on every repair of db . We are interested in determining queries q for which CERTAINTY( q ) is first-order expressible (and hence in the low complexity class AC °). For queries q in the class of conjunctive queries without self-join, we provide a necessary syntactic condition for first-order expressibility of CERTAINTY( q ). For acyclic queries (in the sense of Beeri et al. [1983]), this necessary condition is also a sufficient condition. So we obtain a decision procedure for first-order expressibility of CERTAINTY( q ) when q is acyclic and without self-join. We also show that if CERTAINTY( q ) is first-order expressible, its first-order definition, commonly called certain first-order rewriting, can be constructed in a rather straightforward way.

Horn versus full first-order: Complexity dichotomies in algebraic constraint satisfaction

We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain basic algebraic structures Δ, we prove definability theorems of the following form: for every first-order expansion Γ of Δ, either Γ has a quantifier-free Horn definition in Δ, or there is an element d of Γ such that all non-empty relations in Γ contain a tuple of the form (d,…,d), or all relations with a first-order definition in Δ have a primitive positive definition in Γ. The results imply that several families of constraint satisfaction problems exhibit a complexity dichotomy: the problems are either polynomial-time solvable or NP-hard depending on the choice of the allowed relations. As concrete examples, we investigate fundamental algebraic constraint satisfaction problems. The first class consists of all relational structures with a first-order definition in (ℚ; +) that contain the relation {(x, y, z) ∈ ℚ3 | x + y = z}. The second class is the affine variant of the first class. In both cases, we obtain full dichotomies by utilizing our general methods.

The complexity of temporal constraint satisfaction problems

A temporal constraint language is a set of relations that has a first-order definition in(Q;&lt;), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint languages, then the CSP can be solved in polynomial time; otherwise, the CSP is NP-complete. Our proof combines model-theoretic concepts with techniques from universal algebra, and also applies the so-called product Ramsey theorem, which we believe will useful in similar contexts of constraint satisfaction complexity classification. An extended abstract of this article appeared in the proceedings of STOC'08.

Varieties with definable factor congruences

We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first-order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

Defining the integers in large rings of a number field using one universal quantifier

Julia Robinson gave a first-order definition of the set of integers $$ \mathbb{Z} $$ in the rational numbers $$ \mathbb{Q} $$ by a formula (∀∃∀∃)(F = 0) where the ∀-quantifiers run over a total of 8 variables and F is polynomial. We show that for a large class of number fields, not including $$ \mathbb{Q} $$ , for every e > 0 there exist a set of primes $$ \mathcal{S} $$ of natural density exceeding 1 − e such that $$ \mathbb{Z} $$ can be defined as a subset of the “large” subring $$ \left\{ {x \in K:{\text{ord}}_p x \ge 0,\quad \forall \;\mathfrak{p} \notin \mathcal{S}} \right\} $$ of K by a formula where there is only one ∀-quantifier. In the case of $$ \mathbb{Q} $$ , we will need two quantifiers. We also show that in some cases one can define a subfield of a number field using just one universal quantifier. Bibliography: 18 titles.

Uniform first-order definitions in finitely generated fields

We prove that there is a first-order sentence in the language of rings that is true for all finitely generated fields of characteristic 0 and false for all fields of characteristic >0. We also prove that for each n in N, there is a first-order formula psi_n(x_1,...,x_n) that when interpreted in a finitely generated field K is true for elements x_1,...,x_n in K if and only if the elements are algebraically dependent over the prime field in K.

Elliptic divisibility sequences and undecidable problems about rational points

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity \forall \exists, involving only one universally quantified variable. This improves the complexity of defining Z in Q in two ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is undecidable (recall that Hilbert's Tenth Problem for Q is the question whether the \Sigma_1-theory of Q is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over Q for which one cannot decide whether or not they all have a rational point. The conjecture is related to properties of elliptic divisibility sequences on an elliptic curve and its image under rational 2-descent, namely existence of primitive divisors in suitable residue classes, and we discuss how to prove weaker-in-density versions of the conjecture and present some heuristics.