Multi-modal Logic Research Articles

Fluorescence-Based Multimodal DNA Logic Gates.

The use of DNA structures in creating multimodal logic gates bears high potential for building molecular devices and computation systems. However, due to the complex designs or complicated working principles, the implementation of DNA logic gates within molecular devices and circuits is still quite limited. Here, we designed simple four-way DNA logic gates that can serve as multimodal platforms for simple to complex operations. Using the proximity quenching of the fluorophore-quencher pair in combination with the toehold-mediated strand displacement (TMSD) strategy, we have successfully demonstrated that the fluorescence output, which is a result of gate opening, solely relies on the oligonucleotide(s) input. We further demonstrated that this strategy can be used to create multimodal (tunable displacement initiation sites on the four-way platform) logic gates including YES, AND, OR, and the combinations thereof. The four-way DNA logic gates developed here bear high promise for building biological computers and next-generation smart molecular circuits with biosensing capabilities.

Algorithmic properties of modal and superintuitionistic logics of monadic predicates over finite Kripke frames

Abstract We show that the monadic fragment of the modal predicate logic of a single Kripke frame with finitely many possible worlds, but possibly infinite domains, is decidable. This holds true even for multimodal logics with equality, regardless of whether equality is interpreted as identity or as congruence. By the Gödel–Tarski translation, similar results follow for superintuitionistic predicate logics, with or without equality. Using these observations, we establish upper algorithmic bounds, which match the known lower bounds, for monadic fragments of some modal predicate logics. In particular, we prove that, if $L$ is a propositional modal logic contained in $\textbf{S5}$, $\textbf{GL.3}$ or $\textbf{Grz.3}$ and the class of finite Kripke frames validating $L$ is recursively enumerable, then the monadic fragment with equality of the predicate logic of finite Kripke frames validating $L$ is $\varPi ^{0}_{1}$-complete; this, in particular, holds if $L$ is one of the following propositional logics: $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{KTB}$, $\textbf{K4}$, $\textbf{K4.3}$, $\textbf{S4}$, $\textbf{S4.3}$, $\textbf{GL}$, $\textbf{Grz}$, $\textbf{K5}$, $\textbf{K45}$ and $\textbf{S5}$. We also prove that monadic fragments with equality of logics $\textbf{QAlt}^=_{n}$ and $\textbf{QTAlt}^=_{n}$ are decidable. The obtained results are easily extendable to the multimodal versions of the predicate logics we consider and to logics with the Barcan formula.

Depth-Bounded Fuzzy Bisimulation for Fuzzy Modal Logic

We introduce depth-bounded fuzzy bisimulation between fuzzy Kripke models. Roughly speaking, a depth-bounded fuzzy bisimulation is a decreasing sequence of fuzzy binary relations whose infimum is a fuzzy bisimulation. We provide logical characterizations of depth-bounded fuzzy bisimulations between fuzzy Kripke models w.r.t. a fuzzy multimodal logic over complete residuated lattices, including fuzzy invariance of formulas of with a modal depth bounded by under the th component of a depth-bounded fuzzy bisimulation, as well as the Hennessy-Milner property of depth-bounded fuzzy bisimulations. We also provide a polynomial-time algorithm for computing the th component of the greatest depth-bounded fuzzy bisimulation between two finite fuzzy Kripke models when the underlying complete residuated lattice is linear.

Using n-ary multi-modal logics in argumentation frameworks to reason about ethics

Autonomous behaviors may raise ethical issues that agents must consider in their reasoning. Some approaches use deontic logics, while others consider a value-based argumentation framework. However, no work combines both modal logic and argumentation to reason about ethics. Hence, we propose a new argumentation framework where arguments are built from a n-ary multi-modal logic. It allows to express different kinds of operators, e.g. nullary choice or moral worth operators, dyadic deontic operators or mental states. However the standard attacks based on logical contradictions are no longer sufficient to catch an intuitive meaning for attacks. Hence, we enrich standard attacks by characterizing how oppositions between modal operators arise. Furthermore we show the standard logic-based attacks have a quasi-symmetry property, i.e. when an argument attacks another, this argument is necessarily attacked by another one. Our modal attacks do not have this property, which is highly relevant to decide a dilemma.

1,2,3-Triazoles: Controlled Switches in Logic Gate Applications.

A 1,2,3-triazole-based chemosensor is used for selective switching in logic gate operations through colorimetric and fluorometric response mechanisms. The molecular probe synthesized via "click chemistry" resulted in a non-fluorescent 1,4-diaryl-1,2,3-triazole with a phenol moiety (PTP). However, upon sensing fluoride, it TURNS ON the molecule's fluorescence. The TURN-OFF order occurs through fluorescence quenching of the sensor when metal ions, e.g., Cu2+, and Zn2+, are added to the PTP-fluoride ensemble. A detailed characterization using Nuclear Magnetic Resonance (NMR) spectroscopy in a sequential titration study substantiated the photophysical characteristics of PTP through UV-Vis absorption and fluorescence profiles. A combination of fluorescence OFF-ON-OFF sequences provides evidence of 1,2,3-triazoles being controlled switches applicable to multimodal logic operations. The "INH" gate was constructed based on the fluorescence output of PTP when the inputs are F- and Zn2+. The "IMP" and "OR" gates were created on the colorimetric output responses using the probe's absorption with multiple inputs (F- and Zn2+ or Cu2+). The PTP sensor is the best example of the "Write-Read-Erase-Read" mimic.

UNDER LOCK AND KEY: A PROOF SYSTEM FOR A MULTIMODAL LOGIC

AbstractWe present a proof system for a multimode and multimodal logic, which is based on our previous work on modal Martin-Löf type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e., a small 2-category. The logic is extended to a lambda calculus, establishing a Curry–Howard correspondence.

CANONICITY IN POWER AND MODAL LOGICS OF FINITE ACHRONAL WIDTH

AbstractWe develop a method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones. This is applied to many systems, including the logics of finite width, and a broader class of multimodal logics of ‘finite achronal width’ that are introduced here.

Simulations and bisimulations for fuzzy multimodal logics over Heyting algebras

In the present paper, we study fuzzy multimodal logics over complete Heyting algebras and Kripke models for these logics. We introduce two types of simulations (forward and backward) and five types of bisimulations (forward, backward, forward-backward, backward-forward and regular) between Kripke models, as well as the corresponding presimulations and prebisimulations, which are simulations and bisimulations with relaxed conditions. For each type of simulations and bisimulations an efficient algorithm has been provided that works as follows: it computes the greatest presimulation/prebisimulation of that type, and then checks whether it meets the additional condition: if it does, then it is also the greatest simulation/ bisimulation of that type, otherwise, there is not any simulation/bisimulation of that type. The algorithms are inspired by algorithms for checking the existence and computing the greatest simulations and bisimulations between fuzzy automata. We also demonstrate the application of these algorithms in the state reduction of Kripke models. We show that forward bisimulation fuzzy equivalences on the Kripke model provide reduced models equivalent to the original model concerning plus-formulas, backward bisimulation fuzzy equivalences provide reduced models equivalent concerning minus-formulas, while regular bisimulation fuzzy equivalences provide reduced models equivalent concerning all modal formulas.

A reduction-based cut-free Gentzen calculus for dynamic epistemic logic

Abstract Dynamic epistemic logic (DEL) is a multi-modal logic for reasoning about the change of knowledge in multi-agent systems. It extends epistemic logic by a modal operator for actions which announce logical formulas to other agents. In Hilbert-style proof calculi for DEL, modal action formulas are reduced to epistemic logic, whereas current sequent calculi for DEL are labelled systems which internalize the semantic accessibility relation of the modal operators, as well as the accessibility relation underlying the semantics of the actions. We present a novel cut-free ordinary sequent calculus, called $ \textbf{G4}_{P,A}[] $, for propositional DEL. In contrast to the known sequent calculi, our calculus does not internalize the accessibility relations, but—similar to Hilbert style proof calculi—action formulas are reduced to epistemic formulas. Since no ordinary sequent calculus for full S5 modal logic is known, the proof rules for the knowledge operator and the Boolean operators are those of an underlying S4 modal calculus. We show the soundness and completeness of $ \textbf{G4}_{P,A}[] $ and prove also the admissibility of the cut-rule and of several other rules for introducing the action modality.

A linear logic framework for multimodal logics

AbstractOne of the most fundamental properties of a proof system is analyticity, expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $\mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning “if H follows from G and F follows from H, then F follows from G” can be eliminated. The proof of cut admissibility is usually a tedious, error-prone process through several proof transformations, thus requiring the assistance of (semi-)automatic procedures. In a previous work by Miller and Pimentel, linear logic ( $\mathsf{LL}$ ) was used as a logical framework for establishing sufficient conditions for cut admissibility of object logical systems (OL). The OL’s inference rules are specified as an $\mathsf{LL}$ theory and an easy-to-verify criterion sufficed to establish the cut-admissibility theorem for the OL at hand. However, there are many logical systems that cannot be adequately encoded in $\mathsf{LL}$ , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( $\mathsf{LNS}$ ) presentation of $\mathsf{MMLL}$ (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for $\mathsf{LNS}$ systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the $\mathsf{LNS}$ system for intuitionistic logic.

Interpreting Reo Circuits as PDL models

Reo is a coordination-based language with the proposal of connecting different systems and interfaces. It aims to develop communication between different systems with a high level of abstraction and without many restrictions. PDL is a multimodal logic tailored to reason about programs. It is proved to be sound, complete, decidable and has a simple Kripke semantics. This work intends to provide an interpretation of Reo circuits as PDL models. These are the first steps towards providing a dynamic logic tailored to reason directly about Reo circuits.

Alternative Semantics for Normative Reasoning with an Application to Regret and Responsibility

We provide a fine-grained analysis of notions of regret and responsibility (such as agent-regret and individual responsibility) in terms of a language of multimodal logic. This language undergoes a detailed semantic analysis via two sorts of models: (i) relating models, which are equipped with a relation of propositional pertinence, and (ii) synonymy models, which are equipped with a relation of propositional synonymy. We specify a class of strictly relating models and show that each synonymy model can be transformed into an equivalent strictly relating model. Moreover, we define an axiomatic system that captures the notion of validity in the class of all strictly relating models.

Adding abductive reasoning to a propositional logic

AbstractWe present a technique for obtaining a logic with abductive reasoning extending a given propositional logic. Abduction, along with deduction and induction, is recognized as important for machine learning, namely in identifying possible causes that may lead to the occurrence of an event and in providing new ways for a computational device to achieve a certain objective. Each rule in the original calculus induces a set of multiple-conclusion abductive rules. Moreover, rules stating generic properties of abduction have to be added. In the induced logic, the deductive mechanism of the base logic coexists with this abductive component. A new notion of a multiple-conclusion derivation had to be developed. Due to the canonical nature of obtaining such a logic, we prove the preservation of soundness, completeness, decidability and computational complexity. These concepts and results are illustrated in a robot navigation problem using a multimodal logic.

Towards a Formalisation of Justification and Justifiability

We introduce the logic QKSD which is a normal multi-modal logic over finitely many modalities that additionally supports bounded quantification of modalities. An important feature of this logic is that it allows to quantify over the information components of systems and, hence, can be used to derive justifications. We compare the proposed logic with Artemov's justification logic and also report on a prototypical implementation of a satisfiability solver of this logic and show some examples.

Modal context restriction for multiagent BDI logics

We present and discuss a novel language restriction for modal logics for multiagent systems, called modal context restriction, that reduces the complexity of the satisfiability problem from EXPTIME complete to NPTIME complete. We focus on BDI multimodal logics that contain fix-point modalities like common beliefs and mutual intentions together with realism and introspection axioms. We show how this combination of modalities and axioms affects complexity of the satisfiability problem and how it can be reduced by restricting the modal context of formulas.

Dynamic Epistemic Logic in Game Design

Dynamic Epistemic Logics are a set of multimodal logics that deal with knowledge and change. We argue that the theory for this formalism is mature enough to start a practical implementation, while at the same time having a sizable amount of theoretical expansions. We also claim that these two factors make it an attractive formalism for new features in games featuring an agent’s internal model, like planning for gathering information, acting without revealing one’s goals or feeding possibly false information to a set of agents to influence their beliefs, augmenting already existing mechanisms.

Efficient loop-check for multimodal KD45n logic

We introduce sequent calculus for multi-modal logic KD45n which uses efficient loop-check. Efficiency of the used loop-check is obtained by using marked modal operator squarei which is used as an alternative to sequent with histories ([2,3]).We use inference rules with or branches to make all rules invertible or semi-invertible. We showthe maximum height of the constructed derivation tree. Also polynomial space complexity is proved.

Cut elimination for knowledge logic with interaction

In the article the multimodal logic Tn with central agent interaction axiom is analysed. The Hilbert type calculi is presented, then Gentzen type calculi with cut is derived and the proof of cutelimination theorem is outlined. The work shows that it is possible to construct a Gentzen type calculi without cut for this logic.

Duality for $$\kappa $$-additive complete atomic modal algebras

In this paper, we give a duality theorem between the category of $\kappa$-additive complete atomic modal algebras and the category of $\kappa$-downward directed multi-relational Kripke frames, for any cardinal number $\kappa$. Multi-relational Kripke frames are not Kripke frames for multi-modal logic, but frames for monomodal logics in which the modal operator $\Diamond$ does not distribute over (possibly infinite) disjunction, in general. We first define homomorphisms of multi-relational Kripke frames, and then show the equivalence between the category of $\kappa$-downward directed multi-relational Kripke frames and the category $\kappa$-complete neighborhood frames, from which the duality theorem follows. We also present another direct proof of this duality based on the technique given by Minari.

Formalne związki między powinnością a dobrem. Rozważania na marginesie prac Henryka Elzenberga

The paper presents a relative preference semantics for multimodal logic of good and ought inspired by the axiological writings of Henryk Elzenberg. Its central concept is the act of preference between alternative possibilities performed by a metaempirical will, guided only by pure reasons. In semantics, the act of a metaempirical will is modeled using an orderly relation between possible worlds. Using this logic, I formalize some relationships between good and ought posited by Elzenberg.