First-order Formulas Research Articles

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.

On the Descriptive Complexity of Color Coding

Color coding is an algorithmic technique used in parameterized complexity theory to detect “small” structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing—purely in terms of the syntactic structure of describing formulas—when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in FPT just because of the way they are commonly described using logical formulas.

Limit points of spectra for first-order properties of random hypergraphs

The notion of spectrum for first-order properties introduced by J. Spencer for Erdős–Rényi random graph is considered in relation to random uniform hypergraphs. In this work we study the set of limit points of the spectrum for first-order formulae with bounded quantifier depth and give bounds for its minimum value.

Tableau-based translation from first-order logic to modal logic

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

Формулы и свойства для семейств теорий абелевых групп

First-order formulas reflect an information for semantic and syntactic properties. Links between formulas and properties define their existential and universal interrelations which produce both structural and topological possibilities for characteristics classifying families of semantic and syntactic objects. We adapt general approaches describing links between formulas and properties for families of Abelian groups and their theories defining possibilities for characteristics of formulas and properties including rank values. This adaptation is based on formulas reducing each formula to an appropriate Boolean combination of given ones defining Szmielew invariants for theories of Abelian groups. Using this basedness we describe a trichotomy of possibilities for the rank values of sentences defining neighbourhoods for the set of theories of Abelian groups: the rank can be equal −1, 0, or ∞. Thus the neighbourhoods are either finite or contain continuum many theories. Using the trichotomy we show that each sentence defining a neighbourhood either belongs to finitely many theories or it is generic. We introduce the notion of rich property and generalize the main results for these properties.

Spectral Solutions of Linear and Nonlinear BVPs Using Certain Jacobi Polynomials Generalizing Third- and Fourth-Kinds of Chebyshev Polynomials

This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems. For this purpose, we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials. These two classes generalize the two celebrated non-symmetric classes of polynomials, namely, Chebyshev polynomials of third- and fourth-kinds. The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials. The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved. Furthermore, and based on the first-order derivatives formula of certain Jacobi polynomials, the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method. Convergence analysis of the proposed expansions is investigated. Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Axiomatizing Rectangular Grids with no Extra Non-unary Relations

We construct a first-order formula φ such that all finite models of φ are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆ ℕ is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) ≤ n and t(n); s(n) = Ω(log(n)).

A Novel Categorical Approach to Semantics of Relational First-Order Logic

We present a categorical formalization of a variant of first-order logic. Unlike other texts on this topic, the goal of this paper is to give a very transparent and self-contained account without requiring more background than basic logic and set theory. Our focus is to show how the semantics of first-order formulas can be derived from their usual deduction rules. For understanding the core ideas, it is not necessary to investigate the internal term structure of atomic formulas, thus we abstract atomic formulas to (syntactically opaque) relations; in this sense, our variant of first-order logic is “relational”. While the derived semantics is based on categorical principles (even the duality that arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other), it is nevertheless “constructive” in that it describes explicit computations of the truth values of formulas. We demonstrate this by modeling the categorical semantics in the RISCAL (RISC Algorithm Language) system which allows us to validate the core propositions by automatically checking them in finite models.

INVERSE PROBLEM ALGORITHM APPLICATION TO SEMI-QUANTITATIVE ANALYSIS OF 272 PATIENTS WITH ISCHEMIC STROKE SYMPTOMS: CAROTID STENOSIS RISK ASSESSMENT FOR FIVE RISK FACTORS

This study proposes the inverse problem algorithm (IPA) with five risk factors applied to the semi-quantitative analysis of carotid stenosis 272 patients with suspected ischemic stroke. The IPA is known to provide a substantiated machine learning-based prediction of the expected outcomes by solving an inverse matrix of variable coefficients. In case of carotid stenosis prediction, such risk factors as patient’s age, mean arterial pressure (MAP), glucose AC, low-density lipoprotein-cholesterol (LDL-C), and C-Reactive protein (CRP) were assessed for the main group of 217 patients. Their results were processed by the STATISTICA program with a customized loss function ([Formula: see text]), yielding the first-order nonlinear semi-empirical formula with 16 terms. The loss function was calculated via the total mismatch between the theoretical predictions and true carotid stenosis cases (%) for all 217 patients. Thus, the carotid stenosis (%) compromised solution array [[Formula: see text]] was optimized using [Formula: see text] individual data points via the proposed algorithm. The results showed a complete regression with loss function [Formula: see text]=2.3543, variance [Formula: see text]=87.46%, and correlation coefficient [Formula: see text]. The reference group of 55 more patients with the same preliminary diagnosis and symptoms was selected to validate the method predictive feasibility, which was found quite satisfactory. The decreasing order of three dominant risk factors was as follows: CRP, glucose AC, and MAP, whereas age and LDL-C weakly influenced the program computation results. The IPA showed a strong convergence by its default characteristic. The reduction of the number of variables in computation deteriorated the prediction accuracy, exhibiting the algorithm’s high sensitivity to the number of variables.

Sensitivity and uncertainty analysis for the UAM-SFR sub-exercises with linear regression from random sampling

The paper presents solutions for the neutron transport sub-exercises regarding pin cells, fuel assemblies, and supercells of the OECD/NEA UAM-SFR Benchmark. For this, the GRS XSUSA methodology is applied with a recently introduced approach performing linear regression analysis on the output samples, denoted as XSUSA(LR). The main purpose for this is to calculate sensitivity profiles and the main contributions to the total output uncertainties via the first-order uncertainty propagation formula, i.e. by multiplying the corresponding total/partial nuclear data covariance matrices with the total/partial sensitivity vectors. Confidence intervals were estimated by applying Jackknife resampling. Implicit effects are considered. All output uncertainties are compared to the corresponding values calculated with deterministic linear perturbation theory using TSUNAMI from SCALE 6.2.3. Various optimizations were applied with respect to nuclear data and problem geometry. The output uncertainties are significant – around 1.5% for multiplication factors and 5% for Doppler and sodium void reactivity effects.

Verifying Tight Logic Programs with anthem and vampire

AbstractThis paper continues the line of research aimed at investigating the relationship between logic programs and first-order theories. We extend the definition of program completion to programs with input and output in a subset of the input language of the ASP grounder gringo, study the relationship between stable models and completion in this context, and describe preliminary experiments with the use of two software tools, anthem and vampire, for verifying the correctness of programs with input and output. Proofs of theorems are based on a lemma that relates the semantics of programs studied in this paper to stable models of first-order formulas.

An optimal cut-off algorithm for parameterised refinement checking

• The cut-off set problem of a parameterised system is expressed in first-order logic. • An SMT-based semi-algorithm for computing a cut-off set is presented. • The algorithm terminates for topologies expressed in the E*A* fragment of FOL. • Upon termination, the algorithm produces the optimal cut-off set. • The algorithm is applied to the leader election phase of (Byzantine) Raft. The verification of contemporary distributed software systems is challenging, because they are heavily parameterised, containing components whose number and connections cannot be a priori fixed. In this work, we consider the multi-parameterised verification of safety properties by refinement checking in the context of labelled transition systems (LTSs). The LTSs are parameterised by using first-order constructs, sorts, variables, and predicates, while preserving compositionality. This allows us to parameterise not only the number of replicated components but also the communication topology of the system. Our approach to solving a verification task in the parameterised LTS formalism is to determine a finite cut-off set of parameter values such that in order to prove a parameterised system implementation correct with respect to its specification, it is sufficient to consider only finitely many instances of the parameterised system generated by the parameter values in the cut-off set. In the conference version of this work, we converted the problem of determining a finite cut-off set into the unsatisfiability of a first-order formula and provided a satisfiability modulo theories (SMT)-based semi-algorithm for dynamically, i.e., iteratively, computing a cut-off set. In this article, we present a new version of the algorithm and prove that the cut-off sets computed by this new algorithm are optimal. Hence, we call the new version the optimal cut-off algorithm. The algorithm will always terminate for system topologies expressible in the ∃ ⁎ ∀ ⁎ fragment of first-order logic. It also enables us to consider systems with topologies beyond this fragment, but for these systems, the algorithm is not guaranteed to terminate. We have implemented the approach on top of the Z3 SMT solver and successfully applied it to several system models. As a running example, we consider the leader election phase of the generalised (Byzantine) Raft consensus algorithm and prove the optimal cut-off set of six (respectively, thirteen) parameter values corresponding to instances up to three (respectively, four) servers. To the best of our knowledge, this is the first time a Byzantine variant of the parameterised Raft leader election is automatically verified.

The uniform measure of simple regular sets of infinite trees

We consider the problem of computing the measure of a regular set of infinite binary trees. While the general case remains unsolved, we show that the measure of a language can be computed when the set is given in one of the following three formalisms: a first-order formula with no descendant relation; a Boolean combination of conjunctive queries (with descendant relation); or by a non-deterministic safety tree automaton. Additionally, in the first two cases the measure of the set is always rational, while in the third it is an algebraic number. Moreover, we provide an example of a first-order formula that uses descendant relation and defines a language of infinite trees having an irrational (but algebraic) measure.

Destructive Error Interference in Product-Formula Lattice Simulation.

Quantum computers can efficiently simulate the dynamics of quantum systems. In this Letter, we study the cost of digitally simulating the dynamics of several physically relevant systems using the first-order product-formula algorithm. We show that the errors from different Trotterization steps in the algorithm can interfere destructively, yielding a much smaller error than previously estimated. In particular, we prove that the total error in simulating a nearest-neighbor interacting system of n sites for time t using the first-order product formula with r time slices is O(nt/r+nt^{3}/r^{2}) when nt^{2}/r is less than a small constant. Given an error tolerance ϵ, the error bound yields an estimate of max{O(n^{2}t/ϵ),O(n^{2}t^{3/2}/ϵ^{1/2})} for the total gate count of the simulation. The estimate is tighter than previous bounds and matches the empirical performance observed in Childs etal. [Proc. Natl. Acad. Sci. U.S.A. 115, 9456 (2018)PNASA60027-842410.1073/pnas.1801723115]. We also provide numerical evidence for potential improvements and conjecture an even tighter estimate for the gate count.

The complexity of definability by open first-order formulas

Abstract In this article, we formally define and investigate the computational complexity of the definability problem for open first-order formulas (i.e. quantifier free first-order formulas) with equality. Given a logic $\boldsymbol{\mathcal{L}}$, the $\boldsymbol{\mathcal{L}}$-definability problem for finite structures takes as an input a finite structure $\boldsymbol{A}$ and a target relation $T$ over the domain of $\boldsymbol{A}$ and determines whether there is a formula of $\boldsymbol{\mathcal{L}}$ whose interpretation in $\boldsymbol{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem and prove that if the size and the arity of the target relation $T$ are taken as parameters, then open definability is $\textrm{coW}[1]$-complete for every vocabulary $\tau $ with at least one, at least binary, relation.

The FAN principle and weak König's lemma in herbrandized second-order arithmetic

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

Descriptive complexity of #P functions: A new perspective

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC0 appear as classes of this hierarchy. In this way, we unconditionally place #AC0 properly in a strict hierarchy of arithmetic classes within #P. Furthermore, we show that some of our classes admit efficient approximation in the sense of FPRAS. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. and argue that our approach is better suited to study arithmetic circuit classes such as #AC0 which can be descriptively characterized as a class in our framework.

A Constraint-Based Approach to Learning and Explanation

In the last few years we have seen a remarkable progress from the cultivation of the idea of expressing domain knowledge by the mathematical notion of constraint. However, the progress has mostly involved the process of providing consistent solutions with a given set of constraints, whereas learning “new” constraints, that express new knowledge, is still an open challenge. In this paper we propose a novel approach to learning of constraints which is based on information theoretic principles. The basic idea consists in maximizing the transfer of information between task functions and a set of learnable constraints, implemented using neural networks subject to L1 regularization. This process leads to the unsupervised development of new constraints that are fulfilled in different sub-portions of the input domain. In addition, we define a simple procedure that can explain the behaviour of the newly devised constraints in terms of First-Order Logic formulas, thus extracting novel knowledge on the relationships between the original tasks. An experimental evaluation is provided to support the proposed approach, in which we also explore the regularization effects introduced by the proposed Information-Based Learning of Constraint (IBLC) algorithm.

Model-Checking on Ordered Structures

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this article, we continue this line of research and study model-checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order-invariant case, where the formulas may resort to an ordering, but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of structures. In contrast, in the order-invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.

Deciding Quantifier-free Definability in Finite Algebraic Structures

This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present a decision algorithm based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.