Existential Formulas Research Articles

On undecidability of unary non-nested PFP-operators for one successor function theory

We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.

Parikh’s Theorem Made Symbolic

Parikh’s Theorem is a fundamental result in automata theory with numerous applications in computer science. These include software verification (e.g. infinite-state verification, string constraints, and theory of arrays), verification of cryptographic protocols (e.g. using Horn clauses modulo equational theories) and database querying (e.g. evaluating path-queries in graph databases), among others. Parikh’s Theorem states that the letter-counting abstraction of a language recognized by finite automata or context-free grammars is definable in Linear Integer Arithmetic (a.k.a. Presburger Arithmetic). In fact, there is a linear-time algorithm computing existential Presburger formulas capturing such abstractions, which enables an efficient analysis via SMT-solvers. Unfortunately, real-world applications typically require large alphabets (e.g. Unicode, containing a ‍million of characters) — which are well-known to be not amenable to explicit treatment of the alphabets — or even worse infinite alphabets. Symbolic automata have proven in the last decade to be an effective algorithmic framework for handling large finite or even infinite alphabets. A symbolic automaton employs an effective boolean algebra, which offers a symbolic representation of character sets (i.e. in terms of predicates) and often lends itself to an exponentially more succinct representation of a language. Instead of letter-counting, Parikh’s Theorem for symbolic automata amounts to counting the number of times different predicates are satisfied by an input sequence. Unfortunately, naively applying Parikh’s Theorem from classical automata theory to symbolic automata yields existential Presburger formulas of exponential size. In this paper, we provide a new construction for Parikh’s Theorem for symbolic automata and grammars, which avoids this exponential blowup: our algorithm computes an existential formula in polynomial-time over (quantifier-free) Presburger and the base theory. In fact, our algorithm extends to the model of parametric symbolic grammars, which are one of the most expressive models of languages over infinite alphabets. We have implemented our algorithm and show it can be used to solve string constraints that are difficult to solve by existing solvers.

Proof-relevance in Bishop-style constructive mathematics

AbstractBishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory$(\mathrm{BST})$, our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas$\phi$of BISH a set${\texttt{Prf}}(\phi)$of “proofs” or witnesses of$\phi$. Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory$(\mathrm{MLTT})$. Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

INTERPRETING A FIELD IN ITS HEISENBERG GROUP

AbstractWe improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$ . Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$ , we give general conditions sufficient to eliminate parameters from interpretations.

Logical Characterizations of Fuzzy Simulations

We provide and prove logical characterizations of fuzzy simulations between fuzzy Kripke models that use a general t-norm-based semantics. We also extend the results for fuzzy Kripke models defined over a general residuated lattice. Our logical characterizations of fuzzy simulations are formulated w.r.t. positive existential fragments of fuzzy propositional dynamic logic without tests and concern fuzzy preservation of positive existential modal formulas under fuzzy simulations as well as the Hennessy-Milner property of fuzzy simulations.

Grounding, Quantifiers, and Paradoxes

The notion of grounding is usually conceived as an objective and explanatory relation. It connects two relata if one—the ground—determines or explains the other—the consequence. In the contemporary literature on grounding, much effort has been devoted to logically characterize the formal aspects of grounding, but a major hard problem remains: defining suitable grounding principles for universal and existential formulae. Indeed, several grounding principles for quantified formulae have been proposed, but all of them are exposed to paradoxes in some very natural contexts of application. We introduce in this paper a first-order formal system that captures the notion of grounding and avoids the paradoxes in a novel and non-trivial way. The system we present formally develops Bolzano’s ideas on grounding by employing Hilbert’s ε-terms and an adapted version of Fine’s theory of arbitrary objects.

Inference from Visible Information and Background Knowledge

We provide a wide-ranging study of the scenario where a subset of the relations in a relational vocabulary is visible to a user—that is, their complete contents are known—while the remaining relations are invisible. We also have a background theory—invariants given by logical sentences—that may relate the visible relations to invisible ones, and also may constrain both the visible and invisible relations in isolation. We want to determine whether some other information, given as a positive existential formula, can be inferred using only the visible information and the background theory. This formula whose inference we are concerned with is denoted as the query . We consider whether positive information about the query can be inferred, and also whether negative information—the sentence does not hold—can be inferred. We further consider both the instance-level version of the problem, where both the query and the visible instance are given, and the schema-level version, where we want to know whether truth or falsity of the query can be inferred in some instance of the schema.

First-order model theory of free projective planes

We prove that the theory of open projective planes is complete and strictly stable, and infer from this that Marshall Hall's free projective planes (πn:4⩽n⩽ω) are all elementary equivalent and that their common theory is strictly stable and decidable, being in fact the theory of open projective planes. We further characterize the elementary substructure relation in the class of open projective planes, and show in particular that (πn:4⩽n⩽ω) is an elementary chain. We then prove that the theory of open projective planes does not have a prime model, that it has elimination of quantifiers down to Boolean combinations of existential formulas, and that it is not model complete. Finally, we characterize the forking independence relation in models of the theory and prove that the πn's (4⩽n⩽ω) are strongly type-homogeneous.

Minimal Predicates for Δ-Definability

We consider two kinds of reducibilities on finite families of predicates on a countable set: the definability of predicates and their complements of one family via another by means of existential formulas with parameters and the same definability on isomorphism types of families. Ordered structures of degrees generated by families of unary predicates are described. It is proved that for both reducibilities, there exist continuum many minimal nonzero degrees.

Model theoretic dynamics in Galois fashion

We fix a monster model D of some stable theory and investigate substructures of D which are existentially closed within the class of substructures equipped with an action of a fixed group G. We describe them as PAC substructures of D and obtain results related to Galois theory.Assuming that the class of these existentially closed substructures is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas. Moreover, this theory codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imaginaries.

Denial of the Le Barse conjecture on the law of zero or one for existential monadic formulas

Denial of the Le Barse conjecture on the law of zero or one for existential monadic formulas

SOME MODEL THEORY OF GUARDED NEGATION

AbstractThe Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all positive existential formulas, can express the first-order translations of basic modal logic and of many description logics, along with many sentences that arise in databases. It has been shown that the syntax of GNFO is restrictive enough so that computational problems such as validity and satisfiability are still decidable. This suggests that, in spite of its expressive power, GNFO formulas are amenable to novel optimizations. In this article we study the model theory of GNFO formulas. Our results include effective preservation theorems for GNFO, effective Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to a large class of GNFO sentences within very restricted logics.

On the complexity of computation maximal exponent of periodicity of word equations and expressible relations (note)

Denote by SWE the satisfiability problem for word equations. Consider the problem of computability of the maximal exponent of periodicity for word equations MaxExpProblem(e,k) in which given positive integer k and a word equation e one is to check whether maximal exponent of periodicity of a solution of a word equation is at least k. We present both a deterministic polynomial time reduction of the MaxExpProblem(e,k) problem to the satisfiability problem for word equations and a deterministic polynomial time reduction in the opposite direction. Consequently, if the satisfiability problem is in NP then the other problem is NP-complete. As a simple consequence we get that computation of maximal exponent of periodicity of a solution set for a word equation is in PSWE (SWE is an oracle) and, consequently in PSPACE.Let ϕ be an existential boolean formula built on word equations expressing a relation on words by means of its unknowns. Consider the problem of checking the maximal exponent of periodicity of a component of a relation CMaxExpProblem(ϕ,k) in which given positive integer k and a formula ϕ one is to check whether maximal exponent of periodicity of a component of the relation is at least k.We give nondeterministic polynomial time reduction of the described above CMaxExpProblem(ϕ,k) problem to the satisfiability problem for word equations and deterministic polynomial time reduction in the other direction. Consequently, if the satisfiability problem is in NP then the other problem is NP-complete.Denote by SAT(WE) the satisfiability problem of closed existential formula built on word equations. As a simple consequence of our considerations we get that computation of maximal exponent of periodicity of a component of expressible relation is in PSAT(WE) (SAT(WE) is an oracle), and, consequently, in PSPACE. Another consequence of that is that if SWE is in NP, then the computability problem is in PNP. The above results hold also for semigroups with involution.If the expressible relation is defined by a formula using word equations and regular constraints we prove that the problem is PSPACE-complete. The same holds for free semigroups with involution and free groups.

Undecidability of equations in free Lie algebras

In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.

Strong jump inversion

We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion} provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some $\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some $\mathcal{B}\cong\mathcal{A}$. Jockusch and Soare \cite{JS} showed that there are low linear orderings without computable copies, but Downey and Jockusch \cite{DJ} showed that every Boolean algebra admits strong jump inversion. More recently, D.\ Marker and R.\ Miller \cite{MM} have shown that all countable models of $DCF_0$ (the theory of differentially closed fields of characteristic $0$) admit strong jump inversion. We establish a general result with sufficient conditions for a structure $\mathcal{A}$ to admit strong jump inversion. Our conditions involve an enumeration of $B_1$-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no $1$-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of $DCF_0$. This also yields the fact that the saturated model of $DCF_0$ has a decidable copy.

Algebraically closed and existentially closed Abelian lattice-ordered groups

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensatze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.

Defining \mathbb Z in \mathbb Q

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for ${\mathbb Z}$ in ${\mathbb Q}$, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over ${\mathbb Q}$ with many ${\mathbb Q}$-rational points.

Mamardašvilis, Proustas ir savęs realizavimo problema

Article deals with Merab Mamardashvili’s suggested view to Marcel Proust’s literature activity. According to Mamardashvili, main interest and creativity pathos of Proust can be expressed by formula “to realize yourself”. But Mamardashvili explicates this personalistic and existential formula of self-realization in concepts, unusual for personalism and existentialism. Mamardashvili says that “to realize yourself” means “to realize impression”. The aim of this article is to explore the conceptual relation between self-realization and realization of impression.

The field of reals with a predicate for the real algebraic numbers and a predicate for the integer powers of two

Given a theory T of a polynomially bounded o-minimal expansion R of $${\bar{\mathbb{R}} = \langle\mathbb{R}, +, ., 0, 1, < \rangle}$$R¯=?R,+,.,0,1,<? with field of exponents $${\mathbb{Q}}$$Q, we introduce a theory $${\mathbb{T}}$$T whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that $${\mathbb{T}}$$T is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then $${\mathbb{T}}$$T axiomatises $${\langle\bar{\mathbb{R}}, \mathbb{R}_{alg}, 2^{\mathbb{Z}}\rangle}$$?R¯,Ralg,2Z?, where $${\mathbb{R}_{alg}}$$Ralg denotes the real algebraic numbers. We describe types and definable sets in our models and prove that $${\mathbb{T}}$$T is dependent.

The Universal Theory of First Order Algebras and Various Reducts

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras (where we restrict attention to the positive existential formulas) and the quantifier-free algebras. We also explain the relationship to theories, and indicate how to add in function symbols.