First-order Axioms Research Articles

Encoding TLA+ proof obligations safely for SMT

The TLA+ Proof System (TLAPS) allows users to verify proofs with the support of automated theorem provers, including SMT solvers. To increase trust in TLAPS, we revisited the encoding of TLA+ for SMT, whose implementation had become too complex. Our approach is based on a first-order axiomatization with E-matching patterns. The new encoding is available with TLAPS and achieves performances similar to the previous version, despite its simpler design.

A formalization of Descartes’ causal argument for existence of God

In this paper I attempt a formalization of Descartes’ causal argument for the existence of God presented in the Third Meditation. By taking into account the philosophical setting of Descartes’ Meditations, I argue that the best logical theory to carry out the formalization is quantified modal logic where free logic is used for the first-order axioms. This fact is related to Descartes’ distinction between formal and objective reality which — to my mind — suggests both the adoption of modality (e.g. true in formal vs true in objective reality) and the possibility of using empty names (i.e. names which lack reference in “formal reality”). After presenting the formal system and the reconstruction of Descartes’ proof, I will discuss some issues related to the philosophical import of the formalization, such as the understanding of rigid designation and lack of reference. I will argue that the formalization of Descartes’ argument has an intrinsic philosophical value for it sheds light on the structure of argument, its implicit difficulties, and the use of logic itself to understand the philosophical tradition.

On Program Completion, with an Application to the Sum and Product Puzzle

AbstractThis paper describes a generalization of Clark’s completion that is applicable to logic programs containing arithmetic operations and produces syntactically simple, natural looking formulas. If a set of first-order axioms is equivalent to the completion of a program, then we may be able to find standard models of these axioms by running an answer set solver. As an example, we apply this “reverse completion” procedure to the Sum and Product Puzzle.

Simpler foundations for the hyperbolic plane

Abstract H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq

We mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments of Peano arithmetic (PA) as well as ZF and related finitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e. Hilbert’s tenth problem (H10), and the Post correspondence problem (PCP), respectively. In the synthetic setting based on the computability of all functions definable in a constructive foundation, such as Coq’s type theory, it suffices to define these reductions as meta-level functions with no need for further encoding in a formalised model of computation. The concrete cases of PA and the considered set theories are supplemented by a general synthetic theory of undecidable axiomatisations, focusing on well-known connections to consistency and incompleteness. Specifically, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic extensions still justified by such standard models are shown incomplete. As a by-product of the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.

Decidability of a Sound Set of Inference Rules for Computational Indistinguishability

Computational indistinguishability is a key property in cryptography and verification of security protocols. Current tools for proving it rely on cryptographic game transformations. We follow Bana and Comon’s approach [7, 8], axiomatizing what an adversary cannot distinguish. We prove the decidability of a set of first-order axioms that are computationally sound, though incomplete, for protocols with a bounded number of sessions whose security is based on an <small>IND-CCA 2 </small> encryption scheme. Alternatively, our result can be viewed as the decidability of a family of cryptographic game transformations. Our proof relies on term rewriting and automated deduction techniques.

Recursive axiomatizations for representable posets

We use model theoretic techniques and games to construct explicit first-order axiomatizations for the classes of posets that can be represented as systems of sets, where the order relation is given by inclusion, and existing meets and joins of specified countable cardinalities correspond to intersections and unions, respectively.

Syntactic characterizations of classes of first-order structures in mathematical fuzzy logic

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal–existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.

Algebro-geometric axioms for $\operatorname {DCF}_{0,m}$

We give an algebro-geometric first-order axiomatization of $\operatorname {DCF}_{0,m}$, the theory of differentially closed fields of characteristic zero with $m$ commuting derivations, in the spirit of the classical geometric axioms of $\operatorname {DC

Distributive mereotopology: extended distributive contact lattices

Contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with a relation called contact. The elements of the Boolean algebra are considered as formal representations of physical bodies. The contact relation is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of this operation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement. In this paper we consider as non-definable primitives the relations of contact, nontangential inclusion and dual contact. Part I of the paper is devoted to a suitable axiomatization called extended distributive contact lattice (EDCL) by means of universal first-order axioms true in all contact algebras. EDCL may be considered also as an algebraic tool for certain subarea of mereotopology, called in this paper distributive mereotopology. The main result of Part I of the paper is a representation theorem, stating that each EDCL can be embedded into a contact algebra, showing in this way that the presented axiomatization preserves the meaning of mereotopological relations without considering Boolean complementation. Part II of the paper is devoted to topological representation theory of EDCL, transferring into the distributive case important results from the topological representation theory of contact algebras. It is shown that under minor additional assumptions on distributive lattices as extensionality of the definable relations of overlap or underlap one can preserve the good topological interpretations of regions as regular closed or regular open sets in topological space.

The principle of duality in Euclidean and in absolute geometry

In Euclidean geometry and in absolute geometry fragments of the principle of duality hold. Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 1973, §3.9) posed the problem to find a general theorem which describes the extent of an allowed dualization. It is the aim of this paper to solve this problem. To this end a first-order axiomatization of Euclidean (resp. absolute) geometry is provided which allows the application of Godel’s Completeness Theorem for first-order logic and the solution of Bachmann’s problem.

Adding Decision Procedures to SMT Solvers Using Axioms with Triggers

Satisfiability modulo theories (SMT) solvers are efficient tools to decide the satisfiability of ground formulas, including a number of built-in theories such as congruence, linear arithmetic, arrays, and bit-vectors. Adding a theory to that list requires delving into the implementation details of a given SMT solver, and is done mainly by the developers of the solver itself. For many useful theories, one can alternatively provide a first-order axiomatization. However, in the presence of quantifiers, SMT solvers are incomplete and exhibit unpredictable behavior. Consequently, this approach can not provide us with a complete and terminating treatment of the theory of interest. In this paper, we propose a framework to solve this problem, based on the notion of instantiation patterns, also known as triggers. Triggers are annotations that suggest instances which are more likely to be useful in proof search. They are implemented in all SMT solvers that handle first-order logic and are included in the SMT-LIB format. In our framework, the user provides a theory axiomatization with triggers, along with a proof of completeness and termination properties of this axiomatization, and obtains a sound, complete, and terminating solver for her theory in return. We describe and prove a corresponding extension of the traditional Abstract DPLL Modulo Theory framework. Implementing this mechanism in a given SMT solver requires a one-time development effort. We have implemented the proposed extension in the Alt-Ergo prover and we discuss some implementation details in the paper. To show that our framework can handle complex theories, we prove completeness and termination of a feature-rich axiomatization of doubly-linked lists. Our tests show that our approach results in a better performance of the solver on goals that stem from the verification of programs manipulating doubly-linked lists and sets.

A Superposition Calculus for Abductive Reasoning

We present a modification of the Superposition Calculus that is meant to generate consequences of sets of first-order axioms. This approach is proven to be sound and deductive-complete in the presence of redundancy elimination rules, provided the considered consequences are built on a given finite set of ground terms, represented by constant symbols. In contrast to other approaches, most existing results about the termination of the Superposition calculus can be carried over to our procedure. This ensures in particular that the calculus is terminating for many theories of interest to the SMT community.

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class 𝒞 of algebras let G(𝒞)={G(A)∣A∈𝒞}. Assume that 𝒞 is a class of semigroups possessing a nontrivial member with a neutral element and let ℋ be the universal Horn class generated by G(𝒞). We prove that the Boolean core of ℋ, i.e., the topological prevariety generated by finite members of ℋ equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ℋ. We derive analogous results when 𝒞 is a class of monoids or groups with a nontrivial member.

The Simplest Axiom System for Hyperbolic Geometry Revisited, Again

Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.

Geometric axioms for differentially closed fields with several commuting derivations

A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce and Pillay (1998) [13], is formulated in terms of a relative notion of prolongation for Kolchin-closed sets.

A framework for testing first-order logic axioms in program verification

Program verification systems based on automated theorem provers rely on user-provided axioms in order to verify domain-specific properties of code. However, formulating axioms correctly (that is, formalizing properties of an intended mathematical interpretation) is non-trivial in practice, and avoiding or even detecting unsoundness can sometimes be difficult to achieve. Moreover, speculating soundness of axioms based on the output of the provers themselves is not easy since they do not typically give counterexamples. We adopt the idea of model-based testing to aid axiom authors in discovering errors in axiomatizations. To test the validity of axioms, users define a computational model of the axiomatized logic by giving interpretations to the function symbols and constants in a simple declarative programming language. We have developed an axiom testing framework that helps automate model definition and test generation using off-the-shelf tools for meta-programming, property-based random testing, and constraint solving. We have experimented with our tool to test the axioms used in Auto-Cert, a program verification system that has been applied to verify aerospace flight code using a first-order axiomatization of navigational concepts, and were able to find counterexamples for a number of axioms.

Division rings whose vector spaces are pseudofinite

AbstractVector spaces over fields are pseudofinite, and this remains true for vector spaces over division rings that are finite-dimensional over their center. We also construct a division ring such that the nontrivial vector spaces over it are not pseudofinite, using Richard Thompson's group F. The idea behind the construction comes from a first-order axiomatization of the class of division rings all whose nontrivial vector spaces are pseudofinite.

Theory of finite or infinite trees revisited

AbstractWe present in this paper a first-order axiomatization of an extended theoryTof finite or infinite trees, built on a signature containing an infinite set of function symbols and a relationfinite(t), which enables to distinguish between finite and infinite trees. We show thatThas at least one model and prove its completeness by giving not only a decision procedure, but a full first-order constraint solver that gives clear and explicit solutions for any first-order constraint satisfaction problem inT. The solver is given in the form of 16 rewriting rules that transform any first-order constraintinto an equivalent disjunction φ of simple formulas such that φ is either the formulatrueor the formulafalseor a formula having at least one free variable, being equivalent neither totruenor tofalseand where the solutions of the free variables are expressed in a clear and explicit way. The correctness of our rules implies the completeness ofT. We also describe an implementation of our algorithm in CHR (Constraint Handling Rules) and compare the performance with an implementation in C++ and that of a recent decision procedure for decomposable theories.

Representable semilattice-ordered monoids

We show that no finite set of first-order axioms can define the class of representable semilattice-ordered monoids.