First-order Logic Formulas Research Articles

Proving Query Equivalence Using Linear Integer Arithmetic

Proving the equivalence between SQL queries is a fundamental problem in database research. Existing solvers model queries using algebraic representations and convert such representations into first-order logic formulas so that query equivalence can be verified by solving a satisfiability problem. The main challenge lies in "unbounded summations", which appear commonly in a query's algebraic representation in order to model common SQL features, such as projection and aggregate functions. Unfortunately, existing solvers handle unbounded summations in an ad-hoc manner based on heuristics or syntax comparison, which severely limits the set of queries that can be supported. This paper develops a new SQL equivalence prover called SQLSolver, which can handle unbounded summations in a principled way. Our key insight is to use the theory of LIA^*, which extends linear integer arithmetic formulas with unbounded sums and provides algorithms to translate a LIA^* formula to a LIA formula that can be decided using existing SMT solvers. We augment the basic LIA^* theory to handle several complex scenarios (such as nested unbounded summations) that arise from modeling real-world queries. We evaluate SQLSolver with 359 equivalent query pairs derived from the SQL rewrite rules in Calcite and Spark SQL. SQLSolver successfully proves 346 pairs of them, which significantly outperforms existing provers.

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

A Hybrid Driving Decision-Making System Integrating Markov Logic Networks and Connectionist AI

Connectionist artificial intelligence (AI) can power many critical tasks for connected and autonomous vehicles (CAVs). However, connectionist AI lacks interpretability and usually needs large amount of data for learning. A Markov logic network (MLN), which combines first-order logic (FOL) with statistical learning, learns weighted FOL formulas for inference. MLNs can incorporate domain expert knowledge in the form of FOL formulas to achieve data-efficient learning and transparent decision process. In this paper, we propose a hybrid driving decision-making system, which integrates a MLN module and a deep Q-network (DQN) for enhanced driving safety. The MLN module evaluates the safety of ranked actions from DQN to reduce potential collisions. A collective MLN (Co-MLN) learning algorithm is proposed and it enables CAVs collectively learn a global MLN model for safe state transitions, given distributed small amount of noisy data. A hybrid DQN-MLN learning algorithm is also developed for CAVs to collectively learn to drive in new driving environments. Simulations performed using a highway driving simulator show that the proposed Co-MLN algorithm is highly data-efficient and the learned hybrid driving system can effectively reduce collisions. In addition, the learned MLN module provides transparency for safety-critical driving decisions.

T-norms driven loss functions for machine learning

Injecting prior knowledge into the learning process of a neural architecture is one of the main challenges currently faced by the artificial intelligence community, which also motivated the emergence of neural-symbolic models. One of the main advantages of these approaches is their capacity to learn competitive solutions with a significant reduction of the amount of supervised data. In this regard, a commonly adopted solution consists of representing the prior knowledge via first-order logic formulas, then relaxing the formulas into a set of differentiable constraints by using a t-norm fuzzy logic. This paper shows that this relaxation, together with the choice of the penalty terms enforcing the constraint satisfaction, can be unambiguously determined by the selection of a t-norm generator, providing numerical simplification properties and a tighter integration between the logic knowledge and the learning objective. When restricted to supervised learning, the presented theoretical framework provides a straight derivation of the popular cross-entropy loss, which has been shown to provide faster convergence and to reduce the vanishing gradient problem in very deep structures. However, the proposed learning formulation extends the advantages of the cross-entropy loss to the general knowledge that can be represented by neural-symbolic methods. In addition, the presented methodology allows the development of novel classes of loss functions, which are shown in the experimental results to lead to faster convergence rates than the approaches previously proposed in the literature.

Modular Primal-Dual Fixpoint Logic Solving for Temporal Verification

We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novel modular primal-dual method. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the original primal instance has a corresponding (co-)inductive predicate representing its complement in the dual instance obtained by taking the standard De Morgan’s dual of the primal instance, and (2) partial solutions for (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is also modular in that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems.

Logic Explained Networks

The large and still increasing popularity of deep learning clashes with a major limit of neural network architectures, that consists in their lack of capability in providing human-understandable motivations of their decisions. In situations in which the machine is expected to support the decision of human experts, providing a comprehensible explanation is a feature of crucial importance. The language used to communicate the explanations must be formal enough to be implementable in a machine and friendly enough to be understandable by a wide audience. In this paper, we propose a general approach to Explainable Artificial Intelligence in the case of neural architectures, showing how a mindful design of the networks leads to a family of interpretable deep learning models called Logic Explained Networks (LENs). LENs only require their inputs to be human-understandable predicates, and they provide explanations in terms of simple First-Order Logic (FOL) formulas involving such predicates. LENs are general enough to cover a large number of scenarios. Amongst them, we consider the case in which LENs are directly used as special classifiers with the capability of being explainable, or when they act as additional networks with the role of creating the conditions for making a black-box classifier explainable by FOL formulas. Despite supervised learning problems are mostly emphasized, we also show that LENs can learn and provide explanations in unsupervised learning settings. Experimental results on several datasets and tasks show that LENs may yield better classifications than established white-box models, such as decision trees and Bayesian rule lists, while providing more compact and meaningful explanations.

Parameterized Complexity of Elimination Distance to First-Order Logic Properties

The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem’s fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph property P expressible by a first order-logic formula \( \varphi \in \Sigma _3 \) , that is, of the form \( \begin{equation*} \varphi =\exists x_1\exists x_2\cdots \exists x_r\ \ \forall y_{1}\forall y_{2}\cdots \forall y_{s}\ \ \exists z_1\exists z_2\cdots \exists z_t~~ \psi ,\end{equation*} \) where \( \psi \) is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed \( k \) , is fixed-parameter tractable parameterized by \( k \) . Properties of graphs expressible by formulas from \( \Sigma _3 \) include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas \( \varphi \in \Pi _3 \) , for which computing elimination distance is \( {\sf W}[2] \) -hard.

Architectures in parametric component-based systems: Qualitative and quantitative modelling

One of the key aspects in component-based design is specifying the software architecture that characterizes the topology and the permissible interactions of the components of a system. To achieve well-founded design there is need to address both the qualitative and non-functional aspects of architectures. In this paper we study the qualitative and quantitative formal modelling of architectures applied on parametric component-based systems, that consist of an unknown number of instances of each component. Specifically, we introduce an extended propositional interaction logic and investigate its first-order level which serves as a formal language for the interactions of parametric systems. Our logics achieve to encode the execution order of interactions, which is a main feature in several important architectures, as well as to model recursive interactions. Moreover, we prove the decidability of equivalence, satisfiability, and validity of first-order extended interaction logic formulas, and provide several examples of formulas describing well-known architectures. We show the robustness of our theory by effectively extending our results for parametric weighted architectures. For this, we study the weighted counterparts of our logics over a commutative semiring, and we apply them for modelling the quantitative aspects of concrete architectures. Finally, we prove that the equivalence problem of weighted first-order extended interaction logic formulas is decidable in a large class of semirings, namely the class (of subsemirings) of skew fields.

Vapnik–Chervonenkis dimension and density on Johnson and Hamming graphs

VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of dependent theories. A set system that is definable by a formula of first-order logic with parameters has finite VC-dimension if and only if the formula is a dependent formula.In this paper we study the VC-dimension and the VC-density of the edge relation Exy on Johnson graphs and on Hamming graphs. On a graph G, the set system defined by the formula Exy is the vertex set of G along with the collection of all open neighborhoods of G. We show that the edge relation has VC-dimension at most 4 on Johnson graphs and at most 3 on Hamming graphs and these bounds are optimal. We furthermore show that the VC-density of the edge relation on the class of all Johnson graphs is 2, and on the class of all Hamming graphs the VC-density is 2 as well. Moreover, we show that our bounds on the VC-dimension carry over to the class of all induced subgraphs of Johnson graphs, and to the class of all induced subgraphs of Hamming graphs, respectively. It also follows that the VC-dimension of the set systems of closed neighborhoods in Johnson graphs and Hamming graphs is bounded.Johnson graphs and Hamming graphs are well known examples of distance transitive graphs. Neither of these graph classes is nowhere dense nor is there a bound on their (local) clique-width. Our results contrast this by giving evidence of structural tameness of the graph classes.

A Fixed-point Theorem for Horn Formula Equations

We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn formula equations which is based on expressing the fixed-point computation of a minimal model of a set of Horn clauses on the object level as a formula in first-order logic with a least fixed point operator. We describe several corollaries of this fixed-point theorem, in particular concerning the logical foundations of program verification, and sketch how to generalise it to incorporate abstract interpretations.

Temporal prophecy for proving temporal properties of infinite-state systems

Various verification techniques for temporal properties transform temporal verification to safety verification. For infinite-state systems, these transformations are inherently imprecise. That is, for some instances, the temporal property holds, but the resulting safety property does not. This paper introduces a mechanism for tackling this imprecision. This mechanism, which we call temporal prophecy, is inspired by prophecy variables. Temporal prophecy refines an infinite-state system using first-order linear temporal logic formulas, via a suitable tableau construction. For a specific liveness-to-safety transformation based on first-order logic, we show that using temporal prophecy strictly increases the precision. Furthermore, temporal prophecy leads to robustness of the proof method, which is manifested by a cut elimination theorem. We integrate our approach into the Ivy deductive verification system, and show that it can handle challenging temporal verification examples.

A clock-based dynamic logic for the verification of CCSL specifications in synchronous systems

The Clock Constraint Specification Language (CCSL) is a clock-based specification language for real-time embedded systems. With logical clocks defined as first-class citizens, CCSL provides a natural way for describing clock constraints in synchronous systems — a classical model of concurrency for real-time embedded systems. In this paper, we propose a clock-based dynamic logic called CCSL Dynamic Logic (CDL) for the verification of CCSL specifications in synchronous systems. It extends the first-order dynamic logic with a synchronous execution mechanism in its program model and with CCSL primitives as terms in its logical formulae. We build a sound and relatively complete proof system for CDL to support the verification. Compared with previous approaches for verifying CCSL specifications, which are based on model checking and SMT checking techniques, our approach, which is based on theorem-proving, offers a unified verification framework in which both bounded and unbounded CCSL specifications can be verified. Technically, with the proof system of CDL, a complex CDL formula can be semi-automatically transformed into a set of quantifier-free, arithmetical first-order logic (QF-AFOL) formulae which can be checked by an SMT solver in an efficient way. As a case study, we analyze a simple synchronous system throughout the paper to illustrate how CDL works. We analyze and prove the soundness and completeness of the proof system for CDL. Currently, CDL is partially mechanized in Coq. • Dynamic Logic to deal with Logical Clocked Systems. • Proof system for CCSL and synchronous systems. • Rely on a decidable subset theory for efficient use of SMT solvers. • Proof of soundness and relative completeness.

Measuring Inconsistency in Some Logics with Modal Operators

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

WITTGENSTEIN’S ELIMINATION OF IDENTITY FOR QUANTIFIER-FREE LOGIC

AbstractOne of the central logical ideas in Wittgenstein’sTractatus logico-philosophicusis the elimination of the identity sign in favor of the so-called “exclusive interpretation” of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier’s account, the first concerning the treatment of individual constants, the second concerning so-called “pseudo-propositions” (Scheinsätze) of classical logic such as$a=a$or$a=b \wedge b=c \rightarrow a=c$. We argue that overcoming these problems requires two fairly drastic departures from Wehmeier’s account: (1) Not every formula of classical first-order logic will be translatable into asingleformula of Wittgenstein’s exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original “inclusive” formulas to beambiguous. (2) Certain formulas of first-order logic such as$a=a$will not be translatable into Wittgenstein’s notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a “correct” conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations.

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.

On the Parameterized Complexity of Graph Modification to First-Order Logic Properties

We establish connections between parameterized/kernelization complexity of graph modification problems and expressibility in logic. For a first-order logic formula φ, we consider the problem of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the resulting modification has the property expressible by φ. We provide sufficient and necessary conditions on the structure of the prefix of φ specifying when the corresponding graph modification problem is fixed-parameter tractable (parameterized by k) and when it admits a polynomial kernel.

Bounded model checking of signal temporal logic properties using syntactic separation

Signal temporal logic (STL) is a temporal logic formalism for specifying properties of continuous signals. STL is widely used for analyzing programs in cyber-physical systems (CPS) that interact with physical entities. However, existing methods for analyzing STL properties are incomplete even for bounded signals, and thus cannot guarantee the correctness of CPS programs. This paper presents a new symbolic model checking algorithm for CPS programs that is refutationally complete for general STL properties of bounded signals. To address the difficulties of dealing with an infinite state space over a continuous time domain, we first propose a syntactic separation of STL, which decomposes an STL formula into an equivalent formula so that each subformula depends only on one of the disjoint segments of a signal. Using the syntactic separation, an STL model checking problem can be reduced to the satisfiability of a first-order logic formula, which is decidable for CPS programs with polynomial dynamics using satisfiability modulo theories (SMT). Unlike the previous methods, our method can verify the correctness of CPS programs for STL properties up to given bounds.

John Cage's Silent Piece and the Japanese gardening technique of shakkei: Formalizing Whittington's conjecture through conceptual graphs

This article examines John Cage's Silent Piece (commonly known as ). I analyze the work through formalizing the frame conjecture of Stephen Whittington [2013. “Digging In John Cage's Garden: Cage and Ryōanji.” Malaysian Music Journal 2 (2): 12–21], which suggests that the work's spatial conditions are akin to those found in the Japanese gardening technique of shakkei, or borrowed scenery. By firstly expressing core axioms in natural language that describe the spatial conditions of Cage's Silent Piece, its' 1952 premiere as , the technique of shakkei, and an exemplar manifestation of shakkei at the garden Adachi Teien (Japan), I construct an ontology that enables the fundamental assumptions of Whittington's conjecture to be explicated through conceptual graphs John Sowa [1984. Conceptual Structures: Information Processing in Mind and Machine. Reading, MA: Addison-Wesley]. From the ontology, I create a number of graphs and derive first-order logic formulas that are then used for inference and further reasoning. I conclude by providing a formalism of Whittington's conjecture via a conceptual graph rule that accounts for both auditory and visual conditions that generate shakkei in the cited examples of the premiere of the Silent Piece as and the garden of Adachi Teien.

Parallel-Inference Algorithms and Research of their Efficiency on Computer Systems

This article presents the results of research conducted on cluster systems with multicore nodes to create a method and an algorithm that implement parallel inference in the proof of formulas in first-order logic. The algorithm is based on of J. Robinson’s resolution method of inference. A procedure for its parallelization, which allows one to dynamically control the uniform division of the set of resolvents distributed to the nodes of a computer system for inferring, is proposed. A number of heuristics, which make it possible to significantly reduce the time of implementing inference control decisions, are also introduced. Experiments here show that the effect on acceleration is higher than that of the previously studied parallel-inference algorithms.

Static serializability analysis for causal consistency

Many distributed databases provide only weak consistency guarantees to reduce synchronization overhead and remain available under network partitions. However, this leads to behaviors not possible under stronger guarantees. Such behaviors can easily defy programmer intuition and lead to errors that are notoriously hard to detect. In this paper, we propose a static analysis for detecting non-serializable behaviors of applications running on top of causally-consistent databases. Our technique is based on a novel, local serializability criterion and combines a generalization of graph-based techniques from the database literature with another, complementary analysis technique that encodes our serializability criterion into first-order logic formulas to be checked by an SMT solver. This analysis is more expensive yet more precise and produces concrete counter-examples. We implemented our methods and evaluated them on a number of applications from two different domains: cloud-backed mobile applications and clients of a distributed database. Our experiments demonstrate that our analysis is able to detect harmful serializability violations while producing only a small number of false alarms.