scholarly journals Undecidable relativizations of algebras of relations

1999 ◽  
Vol 64 (2) ◽  
pp. 747-760 ◽  
Author(s):  
Szabolcs Mikulás ◽  
Maarten Marx
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Similarity Type ◽  
First Order ◽  
Equational Theories ◽  
Algebras Of Relations ◽  
Guarded Fragment ◽  
Cylindric Set Algebras

AbstractIn this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

On the number of generators of cylindric algebras

1985 ◽  
Vol 50 (4) ◽  
pp. 865-873
Author(s):  
H. Andréka ◽  
I. Németi
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Single Element ◽  
Abstract Model ◽  
Finitely Generated ◽  
Cylindric Algebras ◽  
Abstract Model Theory ◽  
First Order ◽  
Cylindric Set Algebras

The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]–[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. ω denotes the set of natural numbers. CAα denotes the class of all cylindric algebras of dimension α; by “a CAα” we shall understand an element of the class CAα. The class Dcα ⊆ CAα was defined in [7]. Note that Dcα = 0 for α ∈ ω. The classes Wsα, and Csα were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that ⊆ CAα. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dcα is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: for which simple CAα's does “finitely generated” imply “generated by a single element” (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.

scholarly journals The Triguarded Fragment with Transitivity

10.29007/z359 ◽  
2020 ◽  
Author(s):  
Emanuel Kieronski ◽  
Adam Malinowski
Keyword(s):  
Decision Procedure ◽  
Order Logic ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
First Order ◽  
Free Variables ◽  
Guarded Fragment

The triguarded fragment of first-order logic is an extension of the guarded fragment in which quantification for subformulas with at most two free variables need not be guarded. Thus, it unifies two prominent decidable logics: the guarded fragment and the two-variable fragment. Its satisfiability problem is known to be undecidable in the presence of equality, but becomes decidable when equality is forbidden. We consider an extension of the tri- guarded fragment without equality by transitive relations, allowing them to be used only as guards. We show that the satisfiability problem for the obtained formalism is decidable and 2-ExpTime-complete, that is, it is of the same complexity as for the analogous exten- sion of the classical guarded fragment. In fact, in our satisfiability test we use a decision procedure for the latter as a subroutine. We also show how our approach, consisting in exploiting some existing results on guarded logics, can be used to reprove some known facts, as well as to derive some other new results on triguarded logics.

Interpolation for extended modal languages

2005 ◽  
Vol 70 (1) ◽  
pp. 223-234 ◽  
Author(s):  
Balder ten Cate
Keyword(s):  
Difference Operator ◽  
Relation Algebra ◽  
Order Logic ◽  
Hybrid Logic ◽  
First Order Logic ◽  
Modal Language ◽  
First Order ◽  
Language L ◽  
Guarded Fragment

AbstractSeveral extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language L′ is the least expressive extension of L with interpolation. For instance, let ℳ(D) be the extension of the basic modal language with a difference operator [7], First-order logic is the least expressive extension of ℳ(D) with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.

Decidability of cylindric set algebras of dimension two and first-order logic with two variables

1999 ◽  
Vol 64 (4) ◽  
pp. 1563-1572 ◽  
Author(s):  
Maarten Marx ◽  
Szabolcs Mikulás
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Universal Theory ◽  
Finite Model ◽  
Finite Model Property ◽  
First Order ◽  
Combinatorial Theorem ◽  
Finite Base ◽  
Dimension 2 ◽  
Cylindric Set Algebras

AbstractThe aim of this paper is to give a new proof for the decidability and finite model property of first-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse2). The new proof also shows the known results that the universal theory of Pse2 is decidable and that every finite Pse2 can be represented on a finite base. Since the class Cs2 of cylindric set algebras of dimension 2 forms a reduct of Pse2, these results extend to Cs2 as well.

scholarly journals The Triguarded Fragment of First-Order Logic

10.29007/m8ts ◽  
2018 ◽  
Author(s):  
Sebastian Rudolph ◽  
Mantas Simkus
Keyword(s):  
Description Logics ◽  
Past Research ◽  
Order Logic ◽  
First Order Logic ◽  
Crucial Importance ◽  
Natural Assumption ◽  
First Order ◽  
Natural Extensions ◽  
Guarded Fragment ◽  
Definition Of

Past research into decidable fragments of first-order logic (FO) has produced two very prominent fragments: the guarded fragment GF, and the two-variable fragment FO2. These fragments are of crucial importance because they provide significant insights into decidabil- ity and expressiveness of other (computational) logics like Modal Logics (MLs) and various Description Logics (DLs), which play a central role in Verification, Knowledge Represen- tation, and other areas. In this paper, we take a closer look at GF and FO2, and present a new fragment that subsumes them both. This fragment, called the triguarded fragment (denoted TGF), is obtained by relaxing the standard definition of GF: quantification is required to be guarded only for subformulae with three or more free variables. We show that, in the absence of equality, satisfiability in TGF is N2ExpTime-complete, but becomes NExpTime-complete if we bound the arity of predicates by a constant (a natural assumption in the context of MLs and DLs). Finally, we observe that many natural extensions of TGF, including the addition of equality, lead to undecidability.

Compactly expandable models and stability

1995 ◽  
Vol 60 (2) ◽  
pp. 673-683 ◽  
Author(s):  
Enrique Casanovas
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Unary Predicate ◽  
Similarity Type ◽  
First Order ◽  
Related Notion ◽  
Special Part ◽  
Consistent Extension ◽  
The Relationship ◽  
Close Parallelism

In analogy to ω-logic, one defines M-logic for an arbitrary structure M (see [5],[6]). In M-logic only those structures are considered in which a special part, determined by a fixed unary predicate, is isomorphic to M. Let L be the similarity type of M and T its complete theory. We say that M-logic is κ-compact if it satisfies the compactness theorem for sets of < κ sentences. In this paper we introduce the related notion of compactness for expandability: a model M is κ-compactly expandable if for every extension T′ ⊇ T of cardinality < κ, if every finite subset of T′ can be satisfied in an expansion of M, then T′ can also be satisfied in an expansion of M. Moreover, M is compactly expandable if it is ∥M∥+-compactly expandable. It turns out that M-logic is κ-compact iff M is κ-compactly expandable.Whereas for first-order logic consistency and finite satisfiability are the same, consistency with T and finite satisfiability in M are, in general, no longer the same thing. We call the model Mκ-expandable if every consistent extension T′ ⊇ T of cardinality < κ can be satisfied in an expansion of M. We say that M is expandable if it is ∥M∥+-expandable. Here we study the relationship between saturation, expandability and compactness for expandability. There is a close parallelism between our results about compactly expandable models and some theorems of S. Shelah about expandable models, which are in fact expressed in terms of categoricity of PC-classes (see [7, Th. VI.5.3, VI.5.4 and VI.5.5]). Our results could be obtained directly from these theorems of Shelah if expandability and compactness for expandability were the same notion.

The hierarchy theorem for generalized quantifiers

1996 ◽  
Vol 61 (3) ◽  
pp. 802-817 ◽  
Author(s):  
Lauri Hella ◽  
Kerkko Luosto ◽  
Jouko Väänänen
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
Similarity Type ◽  
First Order ◽  
Generalized Quantifier ◽  
Finite Structures

AbstractThe concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend his method to arbitrary similarity types.

scholarly journals Logical Separability of Incomplete Data under Ontologies

2020 ◽  
Author(s):  
Jean Christoph Jung ◽  
Carsten Lutz ◽  
Hadrien Pulcini ◽  
Frank Wolter
Keyword(s):  
Incomplete Data ◽  
Decision Problem ◽  
Description Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Logical Formula ◽  
Database Queries ◽  
First Order ◽  
Ontology Language ◽  
Guarded Fragment

Finding a logical formula that separates positive and negative examples given in the form of labeled data items is fundamental in applications such as concept learning, reverse engineering of database queries, and generating referring expressions. In this paper, we investigate the existence of a separating formula for incomplete data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and three important fragments thereof: the description logic ALCI, the guarded fragment, and the two-variable fragment. We consider several forms of separability that differ in the treatment of negative examples and in whether or not they admit the use of additional helper symbols to achieve separation. We characterize separability in a model-theoretic way, compare the separating power of the different languages, and determine the computational complexity of separability as a decision problem.

On the Restraining Power of Guards

1999 ◽  
Vol 64 (4) ◽  
pp. 1719-1742 ◽  
Author(s):  
Erich Grädel
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Finite Model ◽  
Model Property ◽  
Tree Model ◽  
First Order ◽  
Bounded Number ◽  
Decidable Classes ◽  
Guarded Fragment ◽  
Almost All

AbstractGuarded fragments of first-order logic were recently introduced by Andréka, van Benthem and Németi; they consist of relational first-order formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful model-theoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of first-order logic.Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of first-order logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is Exptime-complete. We further establish a tree model property for both the guarded fragment and the loosely guarded fragment, and give a proof of the finite model property of the guarded fragment.It is also shown that some natural, modest extensions of the guarded fragments are undecidable.

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