The uniform one-dimensional fragment of first-order logic was introduced a few years ago as a generalization of the two-variable fragment to contexts involving relations of arity greater than two. Quantifiers in this logic are used in blocks, each block consisting only of existential quantifiers or only of universal quantifiers. In this paper we consider the possibility of mixing both types of quantifiers in blocks. We show the finite (exponential) model property and NExpTime-completeness of the satisfiability problem for two restrictions of the resulting formalism: in the first we require that every block of quantifiers is either purely universal or ends with the existential quantifier, in the second we restrict the number of variables to three; in both equality is not allowed. We also extend the second variation to a rich subfragment of the three-variable fragment (without equality) that still has the finite model property and decidable, NExpTime-complete satisfiability.
This paper explores the computational complexity of various natural one-variable fragments of first-order modal logics with the addition of counting quantifiers, over both constant and varying domains. The addition of counting quantifiers provides us a rich language with which to succinctly express statements about the quantity of objects satisfying a given first-order property, using a single variable. Optimal NExpTime upper-bounds are provided for the satisfiability problems of the one-variable fragment of the minimal first-order modal logic QK, over both constant and expanding/decreasing do…
During the last decades, a lot of effort was put into identifying decidable fragments of first-order logic. Such efforts gave birth, among the others, to the two-variable fragment and the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in areas like formal verification and knowledge representation, such first-order fragments are too weak to express even the simplest statistical constraints, required for modelling of influence networks or in statistical reasoning. In this work we investigate the extensions…
It is known due to the work of Van den Broeck et al [KR, 2014] that weighted first-order model counting (WFOMC) in the two-variable fragment of first-order logic can be solved in time polynomial in the number of domain elements. In this paper we extend this result to the two-variable fragment with counting quantifiers.
Separation Logic (SL) is a well-known assertion language used in Hoare-style modular proof systems for programs with dynamically allocated data structures. In this paper we investigate the fragment of first-order SL restricted to the Bernays-Schoenfinkel-Ramsey quantifier prefix $\exists^*\forall^*$, where the quantified variables range over the set of memory locations. When this set is uninterpreted (has no associated theory) the fragment is PSPACE-complete, which matches the complexity of the quantifier-free fragment. However, SL becomes undecidable when the quantifier prefix belongs to $\ex…
Craig interpolation has emerged as an effective means of generating candidate program invariants. We present interpolation procedures for the theories of Presburger arithmetic combined with (i) uninterpreted predicates (QPA+UP), (ii) uninterpreted functions (QPA+UF) and (iii) extensional arrays (QPA+AR). We prove that none of these combinations can be effectively interpolated without the use of quantifiers, even if the input formulae are quantifier-free. We go on to identify fragments of QPA+UP and QPA+UF with restricted forms of guarded quantification that are closed under interpolation. Form…
The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We consider two extensions of this logic. We show that adding either additional binary symbols or counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem. However, combining the two extensions and adding both binary symbols and counting quantifiers leads to an explosion of this complexity. We also compare the expressive power of the two-variable fragment over trees with its extension with counting q…
The list segment predicate ls used in separation logic for verifying programs with pointers is well-suited to express properties on singly-linked lists. We study the effects of adding ls to the full quantifier-free separation logic with the separating conjunction and implication, which is motivated by the recent design of new fragments in which all these ingredients are used indifferently and verification tools start to handle the magic wand connective. This is a very natural extension that has not been studied so far. We show that the restriction without the separating implication can be solv…
The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment $Σ_2$ over infinite words. Here, $Σ_2$ consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is $\mathbb{B}Σ_2$. Our first main result …
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field of knowledge representation. We will also show that the decision problem for our language has a non-deterministic exponential time complexity. However, for the restricted case of formulae whose quantifier prefixes have length bounded by a constant, the decision problem becomes NP-complete. We also observe that in spite of such restriction, several useful set…
Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent real-time extensions of Linear Temporal Logic (LTL). In general, the satisfiability checking problem for these extensions is undecidable when both the future U and the past S modalities are used. In a classical result, the satisfiability checking for MITL[U,S], a non punctual fragment of MTL[U,S], is shown to be decidable with EXPSPACE complete complexity. Given that this notion of non punctuality does not recover decidability in the case of TPTL[U,S], we propose a generalization of non punctuality called \…
Recently, the separated fragment (SF) has been introduced and proved to be decidable. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. The known upper bound on the time required to decide SF's satisfiability problem is formulated in terms of quantifier alternations: Given an SF sentence $\exists \vec{z} \forall \vec{x}_1 \exists \vec{y}_1 \ldots \forall \vec{x}_n \exists \vec{y}_n . ψ$ in which $ψ$ is quantifier free, satisfiability can be decided in nondeterministic $n$-fold exponential time. In the present paper, we conduct a …
We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones -- the Bernays-Schönfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property …
We show that each level of the quantifier alternation hierarchy within FO^2[<] -- the 2-variable fragment of the first order logic of order on words -- is a variety of languages. We then use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy -- and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one can effectively compute an integer m such that alpha is equivalent to a…
We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as boolean closure of existential fragment of FO via a strengthening of Simon's theorem about piecewise testable languages. We propose a new extension of FO which admits infinitary quantifiers to reason about the inherent infinitary properties of countable words. We provide a very natural and hierarchical block-product based characterization of the new extension. …
Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison $x<y$ and predicates for $x \equiv i \mod n$ has been investigated by Barrington, Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that definability in the two-variable fragment FO2[<,MOD] is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in Sigma2[<,MOD]. The fragment Sigm…
Uniform one-dimensional fragment UF1^= is a formalism obtained from first-order logic by limiting quantification to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulas with two or more variables. The fragment can be seen as a canonical generalization of two-variable logic, defined in order to be able to deal with relations of arbitrary arities. The fragment was introd…
We study fragments of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic. We also show that these both ways of defining fragments of dependence logic give rise to a hierarchy in expressive power with respect to k.
We show that the satisfiability problem for the quantifier-free theory of product structures with the equicardinality relation is in NP. As an application, we extend the combinatory array logic fragment to handle cardinality constraints. The resulting fragment is independent of the base element and index set theories.
Let $\mathrm{SO}^{\mathit{plog}}$ denote the restriction of second-order logic, where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. In this article we investigate the problem, which Turing machine complexity class is captured by Boolean queries over ordered relational structures that can be expressed in this logic. For this we define a hierarchy of fragments $Σ^{\mathit{plog}}_m$ (and $Π^{\mathit{plog}}_m$) defined by formulae with alternating blocks of existential and universal second-order quantifiers in quantifier-prenex nor…
We study the expressive power of fragments of inclusion logic under the so-called lax team semantics. The fragments are defined either by restricting the number of universal quantifiers or the arity of inclusion atoms in formulae. In case of universal quantifiers, the corresponding hierarchy collapses at the first level. Arity hierarchy is shown to be strict by relating the question to the study of arity hierarchies in fixed point logics.
The subsumption problem with respect to terminologies in the description logic ALC is EXPTIME-complete. We investigate the computational complexity of fragments of this problem by means of allowed Boolean operators. Hereto we make use of the notion of clones in the context of Post's lattice. Furthermore we consider all four possible quantifier combinations for each fragment parameterized by a clone. We will see that depending on what quantifiers are available the classification will be either tripartite or a quartering.
We stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the $Δ_2$ level is undecidable and that $Σ_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $Σ_1$ is complete for co-Nexptime.
We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their posit…
The satisfiability problem of the branching time logic CTL is studied in terms of computational complexity. Tight upper and lower bounds are provided for each temporal operator fragment. In parallel, the minimal model size is studied with a suitable notion of minimality. Thirdly, flat CTL is investigated, i.e., formulas with very low temporal operator nesting depth. A sharp dichotomy is shown in terms of complexity and minimal models: Temporal depth one has low expressive power, while temporal depth two is equivalent to full CTL.
This paper establishes model-theoretic properties of $\mathrm{FOE}^{\infty}$, a variation of monadic first-order logic that features the generalised quantifier $\exists^\infty$ (`there are infinitely many'). We provide syntactically defined fragments of $\mathrm{FOE}^{\infty}$ characterising four different semantic properties of $\mathrm{FOE}^{\infty}$-sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) invariant under taking quotients. In each case, we produce an effectively defined map that tr…
In this paper we consider Modal Team Logic, a generalization of Classical Modal Logic in which it is possible to describe dependence phenomena between data. We prove that most known fragment of Full Modal Team Logic allow the elimination of the so called "existential bisimulation quantifiers", where the existence of a certain set is made modulo bisimulation. As a consequence, we prove that these fragments enjoy the Uniform Interpolation Property.
We study the expressive power of the two-variable fragment of order-invariant first-order logic. This logic departs from first-order logic in two ways: first, formulas are only allowed to quantify over two variables. Second, formulas can use an additional binary relation, which is interpreted in the structures under scrutiny as a linear order, provided that the truth value of a sentence over a finite structure never depends on which linear order is chosen on its domain. We prove that on classes of structures of bounded degree, any property expressible in this logic is definable in first-order …
Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard translation of modal logic to first-order logic. This applies most notably to the guarded fragment, where quantifiers are appropriately relativized by atoms, and the fragment defined by restricting the number of variables to two. The aim of this talk is to review recent work concerning these fragments and their popular extensions. When presenting the material spe…
It is shown that for any fixed $i>0$, the $Σ_{i+1}$-fragment of Presburger arithmetic, i.e., its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\mathsfΣ^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between $\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques develope…
We study the satisfiability problem for the two-variable first-order logic over structures with one transitive relation. % We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model nor the tree model property, to show decidability we introduce novel model construction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decidability for the full two-variable logic with one transitive relation…
We study the logic FO(~), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO2(~) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(~) with respect to the number of variables and the quantifier rank, and prove a dichotomy betwee…
We call a first-order formula one-dimensional if its every maximal block of existential (universal) quantifiers leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO2. We denote the resulting formalisms, resp., GF1, and TGF1. We show that GF1 has an exponential model property and NExpTime-co…
We develop a doubly-exponential decision procedure for the satisfiability problem of guarded separation logic -- a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the above connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly-ex…
In a recently launched research program for developing logic as a formal theory of (interactive) computability, several very interesting logics have been introduced and axiomatized. These fragments of the larger Computability Logic aim not only to describe "what" can be computed, but also provide a mechanism for extracting computational algorithms from proofs. Among the most expressive and fundamental of these is CL4, known to be (constructively) sound and complete with respect to the underlying computational semantics. Furthermore, the fragment of CL4 not containing blind quantifiers was show…
A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. To reason about data words, linear temporal logic is extended by the freeze quantifier, which stores the element at the current word position into a register, for equality comparisons deeper in the formula. By translations from the logic to alternating automata with registers and then to faulty counter automata whose counters may erroneously increase at any time, and from faulty and error-free counter automata to the logic, we obtain a c…
