The Specker–Blatter theorem does not hold for quaternary relations

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of \emph{additive} classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder.

An abstract notion of action of structures on one another was established by Marcev [2]: Let L and M be two first order languages having only the logical constants in common. Let R be a relational symbol not in LuM, and F a set of sentences in L u M u { R } (which is a two-sorted first order language). Then a s t ructure /~ for L is said to act on a structure 313 for M (in the sense of F) iff the disjoint union & o ] 3 constitutes a model of F under a suitable interpretation of R, where the range of the variables of L is g~_ and the range of the variables of M is Examples abound; we mention only a few: (i) L e t / k be a commutative ring with identity, ]3 an abelian group, and let R stand for unitary scalar multiplication, whose axioms constitute F. (ii) Let & be a group,]3 arbitrary, and let R stand for injective representation in the automorphism group of ]3 (cf. Rabin [3]). (iii) Let & be a lattice, ]3 arbitrary, and let F describe the embeddability of A into the lattice of all substructures (or congruence relations) of (iv) Let & be a field, ]3 an ordered abelian group, and let R stand for nontrivial valuation, whose axioms make up F. If 9.1 is a class of structures for L and ~5 a class of structures for M, let 9JF~B denote the class of all structures in 92 that act on at least one structure in ~ . Then we have the following theorem:

On the expressiveness of second-order spider diagrams

The paper deals with logically definable families of sets (or point-sets) of rational numbers. In particular we are interested whether the families definable over the real line with a unary predicate for the rationals are definable over the rational order alone. Let φ(X,Y ) and ψ(Y ) range over formulas in the first-order monadic language of order. Let Q be the set of rationals and F be the family of subsets J of Q such that φ(Q,J) holds over the real line. The question arises whether, for every φ, F can be defined by means of an appropriate ψ(Y ) interpreted over the rational order. We answer the question negatively. The answer remains negative if the firstorder logic is strengthen to weak monadic second-order logic. The answer is positive for the restricted version of monadic second-order logic where set quantifiers range over open sets. The case of full monadic second-order logic remains open.

In this chapter we present a method for obtaining lower bounds on the computational complexity of logical theories, and give several illustrations of its use. This method is an extension of widely used procedures for proving the recursive undecidability of logical theories. (See Rabin [1965] and Eršov et al. [1965].) One important aspect of this method is that it is based on a family of inseparability results for certain logical problems, closely related to the well-known inseparability result of Trakhtenbrot (as refined by Vaught), that no recursive set separates the logically valid sentences from those which are false in some finite model, as long as the underlying language has at least one non-unary relation symbol. By using these inseparability results as a foundation, we are able to obtain hereditary lower bounds, i.e., bounds which apply uniformly to all subtheories of the theory. The second important aspect of this method is that we use interpretations to transfer lower bounds from one theory to another. By doing this we eliminate the need to code machine computations into the models of the theory being studied. (The coding of computations is done once and for all in proving the inseparability results.) By using interpretations, attention is centred on much simpler definability considerations, viz., what kinds of binary relations on large finite sets can be defined using short formulas in models of the theory. This is conceptually much simpler than other approaches that have been proposed for obtaining lower bounds, such as the method of bounded concatenations of Fleischmann et al. [1977]. We will deal primarily with theories in first-order logic and monadic second-order logic.

The Tarskian classical relevant logic \(\mathsf {TR}\) arises from Tarski’s work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in the first nine sections. \(\mathsf {TR}\) is closely related to the well-known logic \(\mathsf {KR}\). Every formula of relevance logic has a corresponding sentence in Tarski’s extended first-order logic of binary relations with operators on the relation symbols. A formula is in \(\mathsf {TR}\) (by definition), or in \(\mathsf {KR}\) (by a theorem), if and only if its corresponding sentence can be proved in first-order logic, using at most four variables, from the assumptions that all binary relations are dense and, for \(\mathsf {TR}\), commute under composition, or, for \(\mathsf {KR}\), are symmetric. The vocabulary of \(\mathsf {TR}\) is the same as the classical relevant logic \(\mathsf {CR^*}\) proposed by Meyer and Routley but \(\mathsf {TR}\) properly contains \(\mathsf {CR^*}\). The frames characteristic for \(\mathsf {TR}\) are the ones that are characteristic for \(\mathsf {CR^*}\) and satisfy an extra frame condition. There are formulas in \(\mathsf {TR}\) (but not in \(\mathsf {CR^*}\)) that correspond to this frame condition and provide a counterexample to a theorem of T. Kowalski. The frames characteristic for \(\mathsf {TR}\), or \(\mathsf {KR}\), are the ones whose complex algebras are integral dense relation algebras that are commutative, or symmetric, respectively. For both classes, the number of isomorphism types grows like the number of isomorphism types of ternary relations. Asymptotic formulas are obtained for both classes. Similar results apply to a hierarchy of logics defined by the number of variables used in the first-order proofs of their corresponding sentences.

We establish zero-one laws and convergence laws for monadic second-order logic (MSO) (and, a fortiori, first-order logic) on a number of interesting graph classes. In particular, we show that MSO obeys a zero-one law on the class of connected planar graphs, the class of connected graphs of tree-width at most $k$ and the class of connected graphs excluding the $k$-clique as a minor. In each of these cases, dropping the connectivity requirement leads to a class where the zero-one law fails but a convergence law for MSO still holds.

We consider monadic second order logic (MSO) and the modal μ-calculus (Lμ) over transition systems (Kripke structures). It is well known that every class of transition systems which is definable by a sentence of Lμ is definable by a sentence of MSO as well. It will be shown that the converse is also true for an important fragment of MSO: every class of transition systems which is MSO-definable and which is closed under bisimulation - i.e., the sentence does not distinguish between bisimilar models - is also Lμ-definable. Hence we obtain the following expressive completeness result: the bisimulation invariant fragment of MSO and Lμ are equivalent. The result was proved by David Janin and Igor Walukiewicz. Our presentation is based on their article [91]. The main step is the development of automata-based characterizations of Lμ over arbitrary transition systems and of MSO over transition trees (see also Chapter 16). It turns out that there is a general notion of automaton subsuming both characterizations, so we obtain a common ground to compare these two logics. Moreover we need the notion of the ω-unravelling for a transition system, on the one hand to obtain a bisimilar transition tree and on the other hand to increase the possibilities of choosing successors.

We study the language-theoretical aspects of parameterized communicating automata (PCAs), in which processes communicate via rendez-vous. A given PCA can be run on any topology of bounded degree such as pipelines, rings, ranked trees, and grids. We show that, under a context bound, which restricts the local behavior of each process, PCAs are effectively complementable. Complementability is considered a key aspect of robust automata models and can, in particular, be exploited for verification. In this paper, we use it to obtain a characterization of context-bounded PCAs in terms of monadic second-order (MSO) logic. As the emptiness problem for context-bounded PCAs is decidable for the classes of pipelines, rings, and trees, their model-checking problem wrt. MSO properties also becomes decidable. While previous work on model checking parameterized systems typically uses temporal logics without next operator, our MSO logic allows one to express several natural next modalities.

UDC 510.67 We present a complete description of ℵ0-categorical quite o-minimal theories introduced earlier as an extension of the class of o-minimal theories but “quite” inherited various their properties. Bibliography :6 titles. Let L be a countable first order language. Throughout the paper, we consider L-structures and assume that L contains a binary relation symbol < interpreted as a linear ordering in these structures. For arbitrary subsets A and B of a structure M we write A<B if a<b for a ∈ A and b ∈ B. For an arbitrary complete type p we denote by p(M ) the set of realizations of p in M . (∗)

We consider the first-order theory whose language has as nonlogical symbols the constant symbols 0 and 1, the binary relation symbols = and This theory of integers under addition is commonly called the 'Presburger Arithmetic' and is known to be decidable for truth [Presburger (1929), Hilbert and Bernays (1968)]. We prove here that there exists a decision procedure for this theory, involving quantifier elimination, for which there is a superexponential upper bound on the size of formula produced when all variables have been eliminated.

Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.

We investigate local and global paradigms of reasoning about distributed behaviours, modelled as Mazurkiewicz traces, in the context of first-order and monadic second-order logics. We describe new normal forms for properties expressible in these logics. The first normal form, surprisingly, yields a decomposition of a global property as a boolean combination of local properties. The second normal form strengthens McNaughton's theorem and states that global properties of infinite behaviours may also be described as boolean combinations of recurring properties of finite local histories of the behaviours. We briefly touch upon some of the interesting applications of these normal forms.

We establish that every monadic second-order logic (MSO) formula on graphs with bounded treedepth is decidable in a constant number of rounds within the CONGEST model. To our knowledge, this marks the first meta-theorem regarding distributed model-checking. Various optimization problems on graphs are expressible in MSO. Examples include determining whether a graph G has a clique of size k, whether it admits a coloring with k colors, whether it contains a graph H as a subgraph or minor, or whether terminal vertices in G could be connected via vertex-disjoint paths. Our meta-theorem significantly enhances the work of Bousquet et al. [PODC 2022], which was focused on distributed certification of MSO on graphs with bounded treedepth. Moreover, our results can be extended to solving optimization and counting problems expressible in MSO, in graphs of bounded treedepth.

A successor-invariant first-order formula is a formula that has access to an\nauxiliary successor relation on a structure's universe, but the model relation\nis independent of the particular interpretation of this relation. It is well\nknown that successor-invariant formulas are more expressive on finite\nstructures than plain first-order formulas without a successor relation. This\nnaturally raises the question whether this increase in expressive power comes\nat an extra cost to solve the model-checking problem, that is, the problem to\ndecide whether a given structure together with some (and hence every) successor\nrelation is a model of a given formula. It was shown earlier that adding\nsuccessor-invariance to first-order logic essentially comes at no extra cost\nfor the model-checking problem on classes of finite structures whose underlying\nGaifman graph is planar [Engelmann et al., 2012], excludes a fixed minor\n[Eickmeyer et al., 2013] or a fixed topological minor [Eickmeyer and\nKawarabayashi, 2016; Kreutzer et al., 2016]. In this work we show that the\nmodel-checking problem for successor-invariant formulas is fixed-parameter\ntractable on any class of finite structures whose underlying Gaifman graphs\nform a class of bounded expansion. Our result generalises all earlier results\nand comes close to the best tractability results on nowhere dense classes of\ngraphs currently known for plain first-order logic.\n