Atomic Formulas Research Articles

Predictable Synthesis of 3D Polymer Networks Using Crystal Component-Linking.

Controlled synthesis of 3D polymer networks presents a significant challenge because of the complexity of the polymerization reaction in solution. In this study, a polymerization system that facilitates the prediction of a polymer network structure via percolation simulations is realized. The most significant difference between general percolation simulations and experimental polymerization systems is the mobility of the molecules during the reaction. A crystal component-linking method that connects the precisely arranged monomer as a supramolecular crystalline state to imitate the simple percolation theory is adopted. The percolation simulation based on the crystal structure of the arranged monomers is used to accurately calculate the gelation point, gel fraction, degree of swelling, and atomic formula, which correspond with the experimental results. This suggests that the network structures polymerized via the crystal component-linking method can be predicted precisely by a simple percolation simulation. Further, the percolation simulation predicts the structures of the loop, branched polymer, and crosslinking point, which are difficult to measure experimentally. The polymerization of precisely-arranged immobilized monomers in supramolecular structures is promising in synthesizing precisely controlled polymer networks.

Variations on the Kripke Trick

AbstractIn the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic $$\textbf{QS5}$$ QS 5 that include the classical predicate logic $$\textbf{QCl}$$ QCl , Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke’s simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does not work and where, as a result, decidability of monadic modal predicate logics can be obtained.

Realizability problem for constraint LTL

Constraint linear-time temporal logic (CLTL) is an extension of LTL that is interpreted on sequences of valuations of variables over an infinite domain. Atomic formulas can constrain valuations over a range of positions along a sequence, with the range being bounded by a parameter depending on the formula. The satisfiability and model checking problems for CLTL have been studied before. We consider the realizability problem for CLTL. The set of variables is partitioned into two parts, with each part controlled by a player. Players take turns to choose valuations for their variables, generating a sequence of valuations. The winning condition is specified by a CLTL formula — the first player wins if the sequence of valuations satisfies the specified formula. We prove that checking whether the first player has a winning strategy in the realizability game for a given CLTL formula is undecidable in general and identify decidable fragments.

Using granular computing to measure the similarity of sustainable development in China: Addressing goals 1, 3, 8, 10 and 15 of the SDGs

There are regional imbalances in the construction of sustainable development in China, and a framework needs to be proposed to measure it. In this study, a similarity measurement framework based on granular computing is prosed to measure the imbalances of China's provinces (autonomous regions, municipalities) oriented to goals 1, 3, 8, 10, and 15 of the Sustainable Development Goals (SDGs). The framework takes each county as the calculation sample and uses atomic formulas to granulate information for each index of the index system, and the similarity problem among different provinces (autonomous regions, municipalities) is evaluated from the bottom to the top through scale transformation. The framework is a new paradigm for analysing social system research problems from a part to the whole and from the details to the abstract. Based on similarity measurement calculations, most of China's provinces (autonomous regions, municipalities) have little similarity in sustainable development, China's sustainable development exhibits regional imbalance.

Causal modeling semantics for counterfactuals with disjunctive antecedents

Causal Modeling Semantics (CMS, e.g., [6,22,12]) is a powerful framework for evaluating counterfactuals whose antecedent is a conjunction of atomic formulas. We extend CMS to an evaluation of the probability of counterfactuals with disjunctive antecedents, and more generally, to counterfactuals whose antecedent is an arbitrary Boolean combination of atomic formulas. Our main idea is to assign a probability to a counterfactual ▪ at a causal model M as a weighted average of the probability of C in those submodels that truthmakeA∨B[1,3,4]. The weights of the submodels are given by the inverse distance to the original model M, based on a distance metric proposed by Eva et al. [2]. Apart from solving a major problem in the epistemology of counterfactuals, our paper shows how work in semantics, causal inference and formal epistemology can be fruitfully combined.

Something Valid This Way Comes: A Study of Neologicism and Proof-Theoretic Validity

AbstractThe interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects.Chapter 1. Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined.Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.Abstract prepared by Will Stafford extracted partially from the dissertation.E-mail: stafford@flu.cas.czURL: https://escholarship.org/uc/item/33c6h00c

Neutral Free Logic: Motivation, Proof Theory and Models

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka The Journal of Philosophy, 56, 125–137 1959; Lambert Notre Dame Journal of Formal Logic, 8, 133–144 1967, 1997, 2001). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics reject the claim that names need to denote in (ii). Positive free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, negative free logic treats them as uniformly false, and neutral free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (Bulletin of the Section of Logic, 48(2), 137–158 2019) as well as classical logic in Pavlović and Gratzl (Journal of Philosophical Logic, 50, 117–148 2021), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (Studia Logica, 105(1), 93–119 2017, Journal of Applied Non-Classical Logics, 30(3), 272–289 2020), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties.

Structure of a subnanometer-sized semiconductor Cd14Se13 cluster

Structure of a subnanometer-sized semiconductor Cd14Se13 cluster

On definability of team relations with k-invariant atoms

We study the expressive power of logics whose truth is defined over sets of assignments, called teams, instead of single assignments. Given a team X, any k-tuple of variables in the domain of X defines a corresponding k-ary team relation. Thus the expressive power of a logic L with team semantics amounts to the set of properties of team relations which L-formulas can define. We introduce a concept of k-invariance which is a natural semantic restriction on any atomic formulae with team semantics. Then we develop a novel proof method to show that, if L is an extension of FO with any k-invariant atoms, then there are such properties of (k+1)-ary team relations which cannot be defined in L. This method can be applied e.g. for arity fragments of various logics with team semantics to prove undefinability results. In particular, we make some interesting observations on the definability of binary team relations with unary inclusion-exclusion logic.

AN AXIOMATIC APPROACH TO FORCING IN A GENERAL SETTING

AbstractThe technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].

Substitution of ‘small’ divalent cations (e.g. Mg) for Si and Al in the nepheline tetrahedral framework: 2. The occurrence of Mg-rich nepheline and kalsilite

AbstractNatural nepheline usually contains very small amounts of MgO (<0.1 wt.%), although these examples are mainly from Mg-poor alkaline igneous rocks such as nepheline syenites. However, this work shows that nepheline and kalsilite with much higher MgO concentrations can occur in the groundmass of strongly SiO2-undersaturated, feldspar-free, mafic volcanic rocks (i.e. olivine-rich foidites). Furthermore, a strong positive correlation is evident between their Mg and Fe contents. The occurrence of Mg-rich nepheline and kalsilite seems to be related to their derivation from Mg-rich magmas when compared to most of the host rocks investigated to date. Additionally, the physicochemical conditions of crystallisation seem to have an important role in the incorporation of ‘small’ divalent cations by these minerals. The prevalence of Mg-rich nepheline and kalsilite as late magmatic phases and the divergent Mg and Fe relationships for phenocrysts and ‘quenched’ groundmass crystals support this hypothesis. The positive correlation between Mg and Fe contents reflects their strong geochemical affinity and the entrance of Fe3+, Fe2+ and Mg2+ cations into the same crystallographic site of nepheline and kalsilite structures. The calculation of atomic formulae and stoichiometry parameters for nepheline-group minerals where data for the T2+ cations (e.g. Mg2+) are incorporated gives more reliable compositional parameters (see Paper 1). Calculated excess silica values (Si′) are affected significantly when the coupled substitution 2Al3+ = Mg2+ + Si4+ is considered. Thus, specific analyses of ‘small’ divalent cations are essential to obtain more realistic values of excess Si′, in particular, for nepheline and kalsilite that crystallised from Mg-rich, Si-poor, mafic–ultramafic alkaline lavas.

Mutual algebraicity and cellularity

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $\omega$-categorical and mutually algebraic. Second, if a countable structure $M$ in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen.

Model checking QCTL plus on quantum Markov chains

Verifying temporal properties of quantum systems, including quantum Markov chains (QMCs), has attracted an increasing interest in the last decade. Typically, the properties are specified by quantum computation tree logic (QCTL), in which reachability analysis plays a central role. However, safety as the dual problem is known little. Motivated by this, we propose a more expressive logic — QCTL+ (QCTL plus), which extends QCTL by allowing the conjunction in path formulas and the negation in the top level of path formulas. The former can be adopted to express conditional events, and the latter can express safety. To deal with conjunction, we present a product construction of classical states in the QMC and the tri-valued truths of atomic path formulas; to deal with negation, we develop an algebraic approach to compute the safety of the bottom strongly connected component subspaces with respect to a super-operator under some necessary and sufficient convergence conditions. Thereby we conditionally decide QCTL+ formulas over QMCs; without the convergence conditions the safety problem still remains open. The complexity of our method is provided in terms of the size of both the input QMC and the QCTL+ formula.

From axioms to synthetic inference rules via focusing

An important application of focused variants of Gentzen's sequent calculus proof rules is the construction of (possibly) large synthetic inference rules. This paper examines the synthetic inference rules that arise when using theories composed of bipolars, and we do this in both classical and intuitionistic logics. A key step in transforming a formula into synthetic inference rules involves attaching a polarity to atomic formulas and some logical connectives. Since there are different choices in how polarity is assigned, it is possible to produce different synthetic inference rules for the same formula. We show that this flexibility allows for the generalization of different approaches for transforming axioms into sequent rules present in the literature. We finish the paper showing how to apply these results to organize the proof theory of labeled sequent systems for several propositional modal logics.

Extraction of common properties of objects for creation of a logic ontology

The paper describes an approach to the formation of ontology based on descriptions of objects in terms of the predicate calculus language. With this approach, an object is represented as a set of its elements, on which a set of predicates is defined that defines the properties of these elements and the relationship between them. A description of an object is a conjunction of literals that are true on elements of an object. In the present work, ontology is understood as an oriented graph with descriptions of subsets as nodes and such that the elements of a set at the end of an oriented edge have the properties of the elements of the set at the beginning of this edge. Three settings of an ontology construction problem are considered: 1) all predicates are binary and subsets of the original set of objects are given; 2) all predicates are binary and it is required to find subsets of the original set; 3) among the predicates there are many-valued ones and subsets of the original set of objects are given. The main tool for construction such a description is to extract an elementary conjunction of literals of predicate formulas that is isomorphic to subformulas of some formulas. The definition of an isomorphism of elementary conjunctions of atomic predicate formulas is given. The method of ontology construction is formulated. An illustrative example is provided.

Proof-relevance in Bishop-style constructive mathematics

AbstractBishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory$(\mathrm{BST})$, our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas$\phi$of BISH a set${\texttt{Prf}}(\phi)$of “proofs” or witnesses of$\phi$. Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory$(\mathrm{MLTT})$. Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

Association Rules Enhanced Knowledge Graph Attention Network

Embedding knowledge graphs into continuous vector spaces has recently attracted increasing interest in knowledge base completion. However, in most existing embedding methods, only fact triplets are utilized, and logical rules have not been thoroughly studied for the knowledge base completion task. To overcome the problem, we propose an association rules enhanced knowledge graph attention network (AR-KGAN). In this paper, triplets and logical rules are jointly modeled in the proposed unified framework to achieve more predictive entity and relation embeddings. Association rules and corresponding correlation degrees between them can be automatically obtained according to our designed mining algorithm. The major component of AR-KGAN is an encoder of an effective neighborhood aggregator, which addresses the problems by aggregating neighbors with both association rules based and graph based attention weights. The decoder enables AR-KGAN to be translational between entities and relations while keeping the superior link prediction performance. Then, the global loss is minimized over both atomic and complex formulas to achieve the embedding task. In this manner, we learn embeddings compatible with triplets and association rules, which are certainly more predictive for knowledge acquisition and inference. The results show that the proposed AR-KGAN model achieves significant and consistent improvements over state-of-the-art methods on three benchmark datasets.

Akiba's logic of indeterminacy

ABSTRACT A standard approach to indeterminacy treats ‘determinately’ and ‘indeterminately’ as modal operators. Determinacy behaves like necessity; indeterminacy like contingency. This raises two questions. What is the appropriate modal system for these operators? And how should we interpret that system? Ken Akiba has developed an account of ontic indeterminacy that interprets possible worlds as worldly precisifications. He argues that this account vindicates S4 as the logic of indeterminacy. In this paper I explore one significant and surprising consequence of this view. I do this in two stages. First, I prove a technical result, which I call the Infinite Indeterminacy Theorem. Roughly put, this theorem states that, at any given precisification w, either every instance of indeterminacy admits of never-ending higher-order indeterminacy, or else there's indeterminacy in an infinite amount of distinct atomic formulas. In slogan form: either all indeterminacy is infinitely ascending or else there's infinitely wide indeterminacy. Second, I unpack the metaphysical consequences of this technical result (under its intended interpretation) and assess their plausibility. I conclude that these consequences commit Akiba's theory to highly contentious – and arguably untenable – views about the metaphysics of indeterminacy and related matters.

Adelfa: A System for Reasoning about LF Specifications

We present a system called Adelfa that provides mechanized support for reasoning about specifications developed in the Edinburgh Logical Framework or LF. Underlying Adelfa is a new logic named L_LF. Typing judgements in LF are represented by atomic formulas in L_LF and quantification is permitted over contexts and terms that appear in such formulas. Contexts, which constitute type assignments to uniquely named variables that are modelled using the technical device of nominal constants, are characterized in L_LF by context schemas that describe their inductive structure. We present these formulas and an associated semantics before sketching a proof system for constructing arguments that are sound with respect to the semantics. We then outline the realization of this proof system in Adelfa and illustrate its use through a few example proof developments. We conclude the paper by relating Adelfa to existing systems for reasoning about LF specifications.

A framework for the specification and validation of dynamic reconfigurable systems

We study a framework for the specification and validation of dynamic reconfigurable systems. The framework is based on configuration logic for the description of architecture styles which are families of architectures sharing common connectivity features. We express specifications in the Temporal Configuration Logic (TCL), a linear time temporal logic built from atomic formulas characterizing system configurations and temporal modalities. Two non-trivial benchmarks are introduced to show the adequacy of TCL for the specification of dynamic reconfigurable systems. We study an effective model-checking procedure based on SMT techniques for a non-trivial fragment of TCL which has been implemented in a prototype runtime verification tool. We provide preliminary experimental results illustrating the capabilities of the tool on the considered benchmark systems.