Second-order Logic Research Articles

The progress of science from a computational point of view: the drive towards ever higher solvability

Abstract This essay’s content is rendered by the titles of the successive sections. 1. Effective solvability versus intuitive solvability. — 2. Decidability, i.e. effective solvability, in predicate logic. The speedup phenomenon — 3. Contributions of the second-order logic to the problems of solvability — 4. The infinite progress of science in the light of Turing’s idea of the oracle. The term “oracle” is a technical counterpart of the notion of mathematical intuition. A more detailed summary can be obtained through juxtaposing the textboxes labelled with letters A...F. Conclusion: in the progress of science an essential role is played by the feedback between intellectual intuitions (intuitive solvability) and algorithmic procedures (effective solvability).

Logics for unordered trees with data constraints

We study monadic second-order logics with counting constraints (CMso) for unordered data trees. Our objective is to enhance this logic with data constraints for comparing string data values. Comparisons between data values at arbitrary positions of a data tree quickly lead to undecidability. Therefore, we restrict ourselves to comparing sibling data values of unordered trees. But even in this case CMso remains undecidable when allowing for data comparisons that can check the equality of string factors. However, for more restricted data constraints that can only check the equality of string prefixes, it becomes decidable. This decidability result is obtained by reduction to WSkS. Furthermore, we exhibit a restricted class of constraints which can be used in transitions of tree automata, resulting in a model with tractable complexity, which can be extended with structural equality tests between siblings. This efficient restriction is relevant to applications such as checking well-formedness properties of file system trees.

Metalogic and the Overgeneration Argument

AbstractA prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic.

Elementary Equivalence of Endomorphism Monoids of Almost Free S-Acts

In this paper, we study the connection between elementary equivalence of endomorphism monoids of almost free S-acts (acts that are unions of projective indecomposable cyclic S-acts) and equivalence (in first- or second-order logic) of original monoids.

The Harmony of Identity

The standard natural deduction rules for the identity predicate have seemed to some not to be harmonious. Stephen Read has suggested an alternative introduction rule that restores harmony but presupposes second-order logic. Here it will be shown that the standard rules are in fact harmonious. To this end, natural deduction will be enriched with a theory of definitional identity. This leads to a novel conception of canonical derivation, on the basis of which the identity elimination rule can be justified in a proof-theoretical manner.

Impredicativity and Paradox

Michael Dummett famously asked how the serpent of inconsistency entered Frege’s paradise. He himself blamed the impredicative nature of second-order quantification, while many others focused on the inflationary nature of the axiom. Axiom V is, after all, the denial of a higher-order generalization of Cantor’s theorem. Predicativists do not deny this, but they block the derivation of the relevant generalization in predicative fragments of second-order logic. Unfortunately, there is more than one higher-order generalization of Cantor’s theorem, and one of them remains a theorem in predicative fragments of higher-order logic. Our recommendation to predicativists is to respond that only one of them supports the cardinality gloss we associate with Cantor’s theorem and that it is, in fact, false. The other remains a theorem of predicative fragments of higher-order logic but its derivability seemsmore closely related to the Grelling’s paradox than to cardinality considerations.

Abstraction Principles and the Classification of Second-Order Equivalence Relations

This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation $E$ is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class---those equinumerous elements of the equivalence class with equinumerous complements---can have one of only three profiles. The improvements to Fine's theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan's relative categoricity theorem.

Completeness for μ-calculi: A coalgebraic approach

We set up a generic framework for proving completeness results for variants of the modal mu-calculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus.Besides these main applications, our result covers the Kozen–Walukiewicz completeness theorem for the standard modal mu-calculus, as well as the linear-time mu-calculus and modal fixpoint logics on ranked trees. Completeness of the linear-time mu-calculus is known, but the proof we obtain here is different and places the result under a common roof with Walukiewicz' result.Our approach combines insights from the theory of automata operating on potentially infinite objects, with methods from the categorical framework of coalgebra as a general theory of state-based evolving systems. At the interface of these theories lies the notion of a coalgebraic modal one-step language. One of our main contributions here is the introduction of the novel concept of a disjunctive basis for a modal one-step language. Generalizing earlier work, our main general result states that in case a coalgebraic modal logic admits such a disjunctive basis, then soundness and completeness at the one-step level transfer to the level of the full coalgebraic modal mu-calculus.

Approximating max-cut under graph-MSO constraints

We consider the max-cut and max-k-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a 12-approximation algorithm for this class of problems.

Completeness of Second-Order Intuitionistic Propositional Logic with Respect to Phase Semantics for Proof-Terms

Girard introduced phase semantics as a complete set-theoretic semantics of linear logic, and Okada modified phase-semantic completeness proofs to obtain normal-form theorems. On the basis of these works, Okada and Takemura reformulated Girard’s phase semantics so that it became phase semantics for proof-terms, i.e., lambda-terms. They formulated phase semantics for proof-terms of Laird’s dual affine/intuitionistic lambda-calculus and proved the normal-form theorem for Laird’s calculus via a completeness theorem. Their semantics was obtained by an application of computability predicates. In this paper, we first formulate phase semantics for proof-terms of second-order intuitionistic propositional logic by modifying Tait-Girard’s saturated sets method. Next, we prove the completeness theorem with respect to this semantics, which implies a strong normalization theorem.

AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

AbstractWe develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

Recognizable languages of arrows and cospans

In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

Propositional Quantification in Bimodal $$\mathbf {S5}$$S5

Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics containing the logic of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions.

Automata column

Last January, the column by Wolfgang Thomas shed new light on the origins of the automata and logic connection, by uncovering the role of Ehrenfeucht in showing that weak monadic second-order logic is decidable over the natural numbers (or over finite words). In this column, Howard Straubing examines another famous part of the automata-logic story, namely the correspondence between aperiodic semigroups and word languages definable in first-order logic. As usual, we discover that the habitual citations are inaccurate, and that the greats knew more than we think.

Most Preschoolers Don’t Know Most

ABSTRACTRecently, researchers interested in the nature and origins of semantic representations have investigated an especially informative case study: The acquisition of the word most—a quantifier which by all accounts demands a sophisticated second-order logic, and which therefore poses an interesting challenge to theories of language acquisition. According to some reports, children acquire most as early as three years of age, suggesting that it does not draw on cardinal representations of quantity (contrary to some formal accounts), since adult-like knowledge of counting emerges later in development. Other studies, however, have provided evidence that children acquire most much later—possibly by the age of 6 or 7—thereby drawing this logic into question. Here we explore this issue by conducting a series of experiments that probed children’s knowledge of most in different ways. We conclude that children do not acquire an adult-like meaning for most until very late in development—around the age of 6—and that certain behaviors which appear consistent with earlier knowledge are better explained by children’s well-attested bias to select larger sets (a “more” bias), especially when tested with unfamiliar words.

Program Synthesis for Program Analysis

In this article, we propose a unified framework for designing static analysers based on program synthesis . For this purpose, we identify a fragment of second-order logic with restricted quantification that is expressive enough to model numerous static analysis problems (e.g., safety proving, bug finding, termination and non-termination proving, refactoring). As our focus is on programs that use bit-vectors, we build a decision procedure for this fragment over finite domains in the form of a program synthesiser. We provide instantiations of our framework for solving a diverse range of program verification tasks such as termination, non-termination, safety and bug finding, superoptimisation, and refactoring. Our experimental results show that our program synthesiser compares positively with specialised tools in each area as well as with general-purpose synthesisers.

A Formalized 3D Geovisualization Illustrated to Selectivity Purpose of Virtual 3D City Model

Virtual 3D city models act as valuable central information hubs supporting many aspects of cities, from management to planning and simulation. However, we noted that 3D city models are still underexploited and believe that this is partly due to inefficient visual communication channels across 3D model producers and the end-user. With the development of a formalized 3D geovisualization approach, this paper aims to support and make the visual identification and recognition of specific objects in the 3D models more efficient and useful. The foundation of the proposed solution is a knowledge network of the visualization of 3D geospatial data that gathers and links mapping and rendering techniques. To formalize this knowledge base and make it usable as a decision-making system for the selection of styles, second-order logic is used. It provides a first set of efficient graphic design guidelines, avoiding the creation of graphical conflicts and thus improving visual communication. An interactive tool is implemented and lays the foundation for a suitable solution for assisting the visualization process of 3D geospatial models within CAD and GIS-oriented software. Ultimately, we propose an extension to OGC Symbology Encoding in order to provide suitable graphic design guidelines to web mapping services.

Logical limit laws for minor-closed classes of graphs

Let G be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in G on n vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface S. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of the same length ≈5.39⋅10−6. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on n vertices, even in FO.

Ehrenfeucht, Vaught, and the decidability of the weak monadic theory of successor

The decidability of the weak monadic theory of successor is usually considered as a consequence of the connection between monadic second-order logic and finite automata, as established around 1960 in papers of Büchi, Elgot, and Trakhtenbrot. However, there are several remarks and footnotes in papers of that time indicating that the result is also derivable from a theorem of A. Ehrenfeucht using an unpublished remark of R. L. Vaught. In the present note we review these hints and provide a proof along these lines. This simple argument is of methodological interest since it relies solely on first-order model theory and does not use finite automata.

Weighted register automata and weighted logic on data words

Data words are sequences of pairs where the first element is taken from a finite alphabet and the second element is taken from an infinite data domain. Register automata provide a widely studied model for reasoning on data words. In this paper, we investigate automata models for quantitative aspects of systems with infinite data domains, e.g., the costs of storing data on a remote server or the consumption of resources (e.g., memory, energy, time) during a data analysis. We introduce weighted register automata on data words over commutative data semirings equipped with a collection of binary data functions, and we investigate their closure properties. Unlike the other models considered in the literature, we allow data comparison by means of an arbitrary collection of binary data relations. This enables us to incorporate timed automata and weighted timed automata into our framework. In our main result, we give a logical characterization of weighted register automata by means of weighted existential monadic second-order logic; for the proof we employ a new class of determinizable visibly register automata.