This article shows that there exist two particular linear orders such that first-order logic with these two linear orders has the same expressive power as first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that there also exists a built-in permutation such that first-order logic with a linear order and this permutation is as expressive as FO(Bit).
Recent work on combinatory logic demonstrates a compositional translation from lambda calculus that gives meaning to open terms. As the meaning of open terms is a key difficulty in the study of metaprogramming, we investigate whether this idea can be extended to metaprogramming systems with hygienic and unhygienic substitution, specifically $λ_{e}^{\rightarrow \Box}$ and $λ_{\mathcal{S}}$. We conclude that there is quantitative evidence to support the claim that metaprogramming with unhygienic substitution is more powerful than metaprogramming with hygienic substitution.
We investigate the expressive power of the two main kinds of program logics for complex, non-regular program properties found in the literature: those extending propositional dynamic logic (PDL), and those extending the modal mu-calculus. This is inspired by the recent discovery of a decidable program logic called Visibly Pushdown Fixpoint Logic with Chop which extends both the modal mu-calculus and PDL over visibly pushdown languages, which, so far, constituted the ends of two pillars of decidable program logics. Here we show that this logic is not only more expressive than either of its two …
We investigate a family of rule-based logics. The focus is on very expressive languages. We provide a range of characterization results for the expressive powers of the logics and relate them with corresponding game systems.
Recent works by Asada, Ong and Tsukada have championed a rigid description of resources. Whereas in non-rigid paradigms (e.g., standard Taylor expansion or non-idempotent intersection types), bags of resources are multisets and invariant under permutation, in the rigid ones, permutations must be processed explicitly and can be allowed or disallowed. Rigidity enables a fine-grained control of reduction paths and their effects on e.g., typing derivations. We previously introduced a very constrained coinductive type system (system S) in which permutation is completely disallowed. One may wonder i…
College of Intelligence and Computing, Tianjin University（天津大学智能与计算学部）
Existential rule languages are a family of ontology languages that have been widely used in ontology-mediated query answering (OMQA). However, for most of them, the expressive power of representing domain knowledge for OMQA, known as the program expressive power, is not well-understood yet. In this paper, we establish a number of novel characterizations for the program expressive power of several important existential rule languages, including tuple-generating dependencies (TGDs), linear TGDs, as well as disjunctive TGDs. The characterizations employ natural model-theoretic properties, and aut…
These notes present the essentials of first- and second-order monadic logics on strings with introductory purposes. We discuss Monadic First-Order logic and show that it is strictly less expressive than Finite-State Automata, in that it only captures a strict subset of Regular Languages -- the non-counting ones. We then introduce Monadic Second-Order logic; such a logic is, syntactically, a superset of Monadic First-Order logic and captures Regular Languages exactly. We also show how to transform an automaton into a corresponding formula and vice versa. Finally, we discuss the use of logical c…
In this article we study linear temporal logics with team semantics (TeamLTL) that are novel logics for defining hyperproperties. We define Kamp-type translations of these logics into fragments of first-order team logic and second-order logic. We also characterize the expressive power and the complexity of model-checking and satisfiability of team logic and second-order logic by relating them to second- and third-order arithmetic. Our results set in a larger context the recent results of Lück showing that the extension of TeamLTL by the Boolean negation is highly undecidable under the so-calle…
Most graph query languages are rooted in logic. By contrast, in this paper we consider graph query languages rooted in linear algebra. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. A complete picture is painted of the impact of the linear algebra operations in MATLANG on their ability to distinguish graphs.
With the emerging applications that involve complex distributed systems branching-time specifications are specifically important as they reflect dynamic and non-deterministic nature of such applications. We describe the expressive power of a simple yet powerful branching-time specification framework -- branching-time normal form (BNF), which has been developed as part of clausal resolution for branching-time temporal logics. We show the encoding of Buchi Tree Automata in the language of the normal form, thus representing, syntactically, tree automata in a high-level way. Thus we can treat BNF …
In this paper we study the expressive power of k-ary exclusion logic, EXC[k], that is obtained by extending first order logic with k-ary exclusion atoms. It is known that without arity bounds exclusion logic is equivalent with dependence logic. By observing the translations, we see that the expressive power of EXC[k] lies in between k-ary and (k+1)-ary dependence logics. We will show that, at least in the case of k=1, the both of these inclusions are proper. In a recent work by the author it was shown that k-ary inclusion-exclusion logic is equivalent with k-ary existential second order logic,…
A classical result in descriptive complexity theory states that Datalog expresses exactly the class of polynomially computable queries on ordered databases. In this paper we extend this result to the case of higher-order Datalog. In particular, we demonstrate that on ordered databases, for all $k\geq2$, $k$-order Datalog captures $(k-1)$-EXPTIME. This result suggests that higher-order extensions of Datalog possess superior expressive power and they are worthwhile of further investigation both in theory and in practice. This paper is under consideration for acceptance in TPLP.
We study the expressive power of the LARA language -- a recently proposed unified model for expressing relational and linear algebra operations -- both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. We start by showing LARA to be expressive complete with respect to first-order logic with aggregation. Since LARA is parameterized by a set of user-defined functions which allow to transform values in tables, the exact expressive power of the language depends on how these functions are defined. We distinguish two main cases de…
How Expressive Are Graph Neural Networks in the Presence of Node Identifiers?
Institute for Logic Language and Computation（逻辑语言与计算研究所）；University of Amsterdam（阿姆斯特丹大学）；Department of Computer Science（计算机科学系）；University of Oxford（牛津大学）；Lehrstuhl Informatik 7（信息学7系）；RWTH Aachen University（亚琛工业大学）
The verification of systems combining hard timing constraints with concurrency is challenging. This challenge becomes even harder when some timing constants are missing or unknown. Parametric timed formalisms, such as parametric timed automata (PTAs), tackle the synthesis of such timing constants (seen as parameters) for which a property holds. Such formalisms are highly expressive, but also undecidable, and few decidable subclasses were proposed. We propose here a syntactic restriction on PTAs consisting in removing guards (constraints on transitions) to keep only invariants (constraints on l…
