Undecidable First Order Theories of Affine Geometries

Studia Logica, 1977

Bulletin of Symbolic Logic, 2004

1980

Annals of Pure and Applied Logic, 1989

The complexity of subclasses of Magical theories (mainly Presburger and Skolem arithmetic) is studied. The subclasses are defined by the structure of the quantifier prefix.

Mathematical Logic Quarterly, 1999

In this paper we address our efforts to extend the well-known connection in equational logic between equational theories and fully invariant congruences to otherpossibly infinitarylogics. In the special case of algebras, this problem has been formerly treated by H. J. Hoehnke [lo] and R. W. Quackenbush . Here we show that the connection extends at least up to the universal fragment of logic. Namely, we establish that the concept of (infinitary) universal theory matches the abstract notion of fully invariant system. We also prove that, inside this wide group of theories, the ones which are strict universal Horn correspond to fully invariant closure systems, whereas those which are universal atomic can be characterized as principal fully invariant systems.

DOAJ (DOAJ: Directory of Open Access Journals), 2020

We study two extensions of FO 2 [<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y' and 'the factor u appears between positions x and y'. These are, in a sense, the simplest properties that are not expressible using only two variables. We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We give effective conditions, in terms of the syntactic monoid of a regular language, for a property to be expressible in these logics. This algebraic analysis allows us to prove, among other things, that our new logics have strictly less expressive power than full first-order logic FO[<]. Our proofs required the development of novel techniques concerning factorizations of words.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1980

Lecture Notes in Computer Science, 2004

In this paper we provide an automaton-based solution to the decision problem for a large set of monadic second-order theories of deterministic tree structures. We achieve it in two steps: first, we reduce the considered problem to the problem of determining, for any Rabin tree automaton, whether it accepts a given tree; then, we exploit a suitable notion of tree equivalence to reduce (a number of instances of) the latter problem to the decidable case of regular trees. We prove that such a reduction works for a large class of trees, that we call residually regular trees. We conclude the paper with a short discussion of related work.

1972

A. N. Whitehead, in two basic books, considers two different approaches to point-free geometry: the inclusion-based approach, whose primitive notions are regions and inclusion relation between regions and the connection-based approach, where the connection relation is considered instead of the inclusion. We show that the latter cannot be reduced to the first one, although this can be done in the framework of multivalued logics.

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Annals of the Japan Association for Philosophy of Science, 1989

2018

We prove a weighted Feferman-Vaught decomposition theorem for disjoint unions and products of finite structures. The classical Feferman-Vaught Theorem describes how the evaluation of a first order sentence in a generalized product of relational structures can be reduced to the evaluation of sentences in the contributing structures and the index structure. The logic we employ for our weighted extension is based on the weighted MSO logic introduced by Droste and Gastin to obtain a Buchi-type result for weighted automata. We show that for disjoint unions and products of structures, the evaluation of formulas from two respective fragments of the logic can be reduced to the evaluation of formulas in the contributing structures. We also prove that the respective restrictions are necessary. Surprisingly, for the case of disjoint unions, the fragment is the same as the one used in the Buchi-type result of weighted automata. In fact, even the formulas used to show that the respective restric...

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1971

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.

2014

We introduce a new sub logic of third order logic (TO), the logic TO ω , as a semantic restriction of TO. We focus on the existential fragment of TO ω , which we denote Σ 2,ω 1 , and we study its relational complexity by introducing a variation of the non deterministic relational machine, which we denote 3-NRM, where we allow third order relations in the relational store of the machine. We then prove that Σ 2,ω 1 characterizes exactly NEXPTIME3,r. 2 Preliminaries We assume a basic knowledge of Logic and Model Theory (refer to [Lib,04]). We only consider vocabularies of the form σ = R 1 ,. .. , R s (i.e., purely relational), where the arities of the relation symbols are r 1 ,. .. , r s ≥ 1, respectively. We assume that they also contain equality. And we consider only finite σ-structures, denoted as A = A, R A 1 ,. .. , R A s , where A is the domain, also denoted dom(A), and R A 1 ,. .. , R A s are (second order) relations in A r1 ,. .. , A rs , respectively. If γ(x 1 ,. .. , x l) is a formula of some logic with free F O variables {x 1 ,. .. , x l }, for some l ≥ 1, with γ A we denote the l-ary relation defined by γ in A, i.e., the set {(a 1 ,. .. , a l) : a 1 ,. .. , a l ∈ A ∧ A |= γ(x 1 ,. .. , x l)[a 1 ,. .. , a l ]}. For any l-tuplē a = (a 1 ,. .. , a l) of elements in A, with 1 ≤ l ≤ k, we define the F O k type ofā, denoted T ype k (A,ā), to be the set of F O k formulas ϕ ∈ F O k with free variables among x 1 ,. .. , x l , such that A |= ϕ[a 1 ,. .. , a l ]. If τ is an F O k type, we say that the tupleā realizes τ in A, if and only if, τ = T ype k (A,ā). Let A and B be σ-structures and letā andb be two l-tuples on A and B respectively, we write (A,ā) ≡ k (B,b), to denote that T ype k (A,ā) = T ype k (B,b). If A = B, we also writeā ≡ kb. We denote as size k (A) the number of equivalence classes in ≡ k in A. An l-ary relation R in A is closed under ≡ k if for any two l-tuplesā,b in A l , a ∈ R ∧ā ≡ kb ⇒b ∈ R. Let S be a set, a binary relation R is a pre-order on S if it satisfies: 1) ∀a ∈ S (a, a) ∈ R (reflexive). 2) ∀a, b, c ∈ S (a, b) ∈ R ∧ (b, c) ∈ R ⇒ (a, c) ∈ R (transitive). 3) ∀a, b ∈ S (a, b) ∈ R ∨ (b, a) ∈ R (conex). A preorder on S induces an equivalence relation ≡ on S (i.e., a ≡ b ⇔ a b∧b a), and also induces a total order over the set of equivalence classes of ≡. When the equivalence classes induced by a pre-order on k-tuples from some structure A agree with the equivalence classes of ≡ k , then the pre-order establishes a total order over the FO k types for k-tuples which are realized on A .