Double Exponential Time Research Articles

Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics, including credulous reasoning. When asking for projected counts, we are interested in counting the number of extensions of a given argumentation framework, while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by the treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming. Our algorithms run in double or triple exponential time in the treewidth, depending on the semantics under consideration. Finally, we establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension under the exponential time hypothesis (ETH).

Fundamental Logic is Decidable

It is shown that Holliday’s propositional Fundamental Logic is decidable in polynomial time and that first-order Fundamental Logic is decidable in double-exponential time. The proof also yields a double-exponential–time decision procedure for first-order orthologic.

Crystal structure, luminescence properties and thermal stability of novel Sr2CaLa(VO4)3: Sm3+ phosphor synthesized by the combustion method

Sr3-xCaxLa(VO4)3: Sm3+ (x = 0, 0.5, 1.0, 1.5, 2.0, 2.5) and novel Sr2CaLa(VO4)3: ySm3+ (SCLVO: ySm3+, y = 0.01, 0.03, 0.05, 0.07, 0.09) phosphors were synthesized by the citric acid-assisted sol-combustion method. The crystal structure, morphology, concentration-dependent emission spectrum, decay time, and thermal stability of phosphors were studied. These synthesized phosphors have a hexagonal structure with a space group of R-3 m (166). High-purity Sr3-xCaxLa(VO4)3: Sm3+ phosphors were obtained when x < 2.5. The change in particle shape and unit cell parameters of the Sr3-xCaxLa(VO4)3: Sm3+ phosphors led to reduced luminescence performance when the Ca2+ concentration was x > 1. The optimal doping concentration of Sm3+ in the SCLVO host is 3 mol%. Under the excitation of 402 nm near-ultraviolet (NUV) light, all phosphors in this series emit orange–red light at 603 nm, which is attributed to the 4G5/2→6H7/2 transition of Sm3+. The SCLVO: 0.03Sm3+ phosphor exhibits a double exponential decay time of 0.872 ms, excellent thermal stability with an activation energy Ea of 0.232 eV, a chromaticity coordinate of (0.5817, 0.4136), and a high color purity of 98.48%. These experimental results demonstrate the potential application of phosphor SCLVO: 0.03Sm3+ in the red component of white light-emitting diodes (WLEDs).

Immersed cycles and the JSJ decomposition

We present an algorithm to construct the JSJ decomposition of one-ended hyperbolic groups which are fundamental groups of graphs of free groups with cyclic edge groups. Our algorithm runs in double exponential time, and is the first algorithm on JSJ decompositions to have an explicit time bound. Our methods are combinatorial/geometric and rely on analysing properties of immersed cycles in certain CAT(0) square complexes.

Influence of Longer Range Transfer of Vapor Interface Modified Caging Constraints on the Spatially Heterogeneous Dynamics of Glass-Forming Liquids

Based on the Elastically Collective Nonlinear Langevin Equation theory of bulk relaxation in glass-forming liquids, and our recent ideas of how interface-nucleated modification of caging constraints are spatially transferred into the interior of a thick film, we present a force-based theory for dynamical gradients in thick films with one vapor interface that includes collective elasticity effects. Quantitative applications to the foundational hard sphere fluid and polymer melts of diverse fragilities are presented. We predict a roughly exponential spatial variation of total activation barrier which is non-perturbatively modified to ~10 particle diameters into film. This leads to prediction of a reduced alpha relaxation time gradient of a double exponential form characterized by a nearly constant spatial decay length, in qualitative accord with simulations. These spatially heterogeneous changes of the temperature variation of the alpha time result in a large and long-range gradient of the local glass transition temperature. An average interfacial layer thickness relevant to ensemble-averaged dielectric and other experiments is also computed. It is predicted to be rather large at the bulk glass transition temperature, decreasing roughly linearly with heating. Spatially inhomogeneous power law decoupling of the alpha relaxation time from its bulk value is predicted, with an effective exponent that decays to zero with distance from the free surface in a nearly exponential manner, trends which are in qualitative accord with recent simulations. This behavior and the double exponential alpha time gradient are related and can be viewed as consequences of an effective quasi-universal factorization of the total barrier in films into the product of its bulk temperature dependent value times a function solely of location in the film.

Instantaneous and dynamical decoherence

Two manifestations of decoherence, called instantaneous and dynamical, are investigated. The former reflects the suppression of the interference between the components of the current state while the latter reflects that within the initial state. These types of decoherence are computed in the case of the Brownian motion and the harmonic and anharmonic oscillators within the semiclassical approximation. A remarkable phenomenon, namely the opposite orientation of the time arrow of the dynamical variables compared to that of the quantum fluctuations generates a double exponential time dependence of the dynamical decoherence in the presence of a harmonic force. For the weakly anharmonic oscillator the dynamical decoherence is found to depend in a singular way on the amount of the anharmonicity.

String constraints with concatenation and transducers solved efficiently

String analysis is the problem of reasoning about how strings are manipulated by a program. It has numerous applications including automatic detection of cross-site scripting, and automatic test-case generation. A popular string analysis technique includes symbolic executions, which at their core use constraint solvers over the string domain, a.k.a. string solvers. Such solvers typically reason about constraints expressed in theories over strings with the concatenation operator as an atomic constraint. In recent years, researchers started to recognise the importance of incorporating the replace-all operator (i.e. replace all occurrences of a string by another string) and, more generally, finite-state transductions in the theories of strings with concatenation. Such string operations are typically crucial for reasoning about XSS vulnerabilities in web applications, especially for modelling sanitisation functions and implicit browser transductions (e.g. innerHTML). Although this results in an undecidable theory in general, it was recently shown that the straight-line fragment of the theory is decidable, and is sufficiently expressive in practice. In this paper, we provide the first string solver that can reason about constraints involving both concatenation and finite-state transductions. Moreover, it has a completeness and termination guarantee for several important fragments (e.g. straight-line fragment). The main challenge addressed in the paper is the prohibitive worst-case complexity of the theory (double-exponential time), which is exponentially harder than the case without finite-state transductions. To this end, we propose a method that exploits succinct alternating finite-state automata as concise symbolic representations of string constraints. In contrast to previous approaches using nondeterministic automata, alternation offers not only exponential savings in space when representing Boolean combinations of transducers, but also a possibility of succinct representation of otherwise costly combinations of transducers and concatenation. Reasoning about the emptiness of the AFA language requires a state-space exploration in an exponential-sized graph, for which we use model checking algorithms (e.g. IC3). We have implemented our algorithm and demonstrated its efficacy on benchmarks that are derived from cross-site scripting analysis and other examples in the literature.

Nicotinic acetylcholine receptors (nAChRs) at zebrafish red and white muscle show different properties during development.

Nicotinic acetylcholine receptors (nAChRs) are highly expressed at the vertebrate neuromuscular junction (NMJ) where they are required for muscle activation. Understanding the factors that underlie NMJ development is critical for a full understanding of muscle function. In this study we performed whole cell and outside-out patch clamp recordings, and single-cell RT-qPCR from zebrafish red and white muscle to examine the properties of nAChRs during the first 5 days of development. In red fibers miniature endplate currents (mEPCs) exhibit single exponential time courses at 1.5 days postfertilization (dpf) and double exponential time courses from 2 dpf onwards. In white fibers, mEPCs decay relatively slowly, with a single exponential component at 1.5 dpf. By 2 and 3 dpf, mEPC kinetics speed up, and decay with a double exponential component, and by 4 dpf the exponential decay reverts back to a single component. Single channel recordings confirm the presence of two main conductance classes of nAChRs (∼45 pS and ∼65 pS) in red fibers with multiple time courses. Two main conductance classes are also present in white fibers (∼55 pS and ∼73 pS), but they exhibit shorter mean open times by 5 dpf compared with red muscle. RT-qPCR of mRNA for nicotinic receptor subunits supports a switch from γ to ε subunits in white fibers but not in red. Our findings provide a developmental profile of mEPC properties from red and white fibers in embryonic and larval zebrafish, and reveal previously unknown differences between the NMJs of these muscle fibers.© 2015 Wiley Periodicals, Inc. Develop Neurobiol 76: 916-936, 2016.

Quantified Constraints and Containment Problems

The quantified constraint satisfaction problem $\mathrm{QCSP}(\mathcal{A})$ is the problem to decide whether a positive Horn sentence, involving nothing more than the two quantifiers and conjunction, is true on some fixed structure $\mathcal{A}$. We study two containment problems related to the QCSP. Firstly, we give a combinatorial condition on finite structures $\mathcal{A}$ and $\mathcal{B}$ that is necessary and sufficient to render $\mathrm{QCSP}(\mathcal{A}) \subseteq \mathrm{QCSP}(\mathcal{B})$. We prove that $\mathrm{QCSP}(\mathcal{A}) \subseteq \mathrm{QCSP}(\mathcal{B})$, that is all sentences of positive Horn logic true on $\mathcal{A}$ are true on $\mathcal{B}$, iff there is a surjective homomorphism from $\mathcal{A}^{|A|^{|B|}}$ to $\mathcal{B}$. This can be seen as improving an old result of Keisler that shows the former equivalent to there being a surjective homomorphism from $\mathcal{A}^\omega$ to $\mathcal{B}$. We note that this condition is already necessary to guarantee containment of the $\Pi_2$ restriction of the QCSP, that is $\Pi_2$-$\mathrm{CSP}(\mathcal{A}) \subseteq \Pi_2$-$\mathrm{CSP}(\mathcal{B})$. The exponent's bound of ${|A|^{|B|}}$ places the decision procedure for the model containment problem in non-deterministic double-exponential time complexity. We further show the exponent's bound $|A|^{|B|}$ to be close to tight by giving a sequence of structures $\mathcal{A}$ together with a fixed $\mathcal{B}$, $|B|=2$, such that there is a surjective homomorphism from $\mathcal{A}^r$ to $\mathcal{B}$ only when $r \geq |A|$. Secondly, we prove that the entailment problem for positive Horn fragment of first-order logic is decidable. That is, given two sentences $\varphi$ and $\psi$ of positive Horn, we give an algorithm that determines whether $\varphi \rightarrow \psi$ is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive first-order logic (i.e. positive Horn plus disjunction) is undecidable. In the final part of the paper we ponder a notion of Q-core that is some canonical representative among the class of templates that engender the same QCSP. Although the Q-core is not as well-behaved as its better known cousin the core, we demonstrate that it is still a useful notion in the realm of QCSP complexity classifications.

Temperature and Voltage Coupling to TRPM8 Channel Opening

Expressed in somatosensory neurons of the dorsal root and trigeminal ganglion, the transient receptor potential melastatin 8 (TRPM8) channel is a Ca2+-permeable cation channel activated by cold, voltage, PIP2 and menthol. Although TRPM8 channel gating has been characterized at the single channel and macroscopic current levels, there is currently no consensus regarding the extent to which temperature and voltage sensors couple to the conduction gate. In the present study we extended the range of voltages at which the TRPM8-induced ionic currents were measured and made careful measurements of the maximum open probability the channel can attain at different temperatures by means of fluctuation analysis. The first direct measurements of TRPM8 channel temperature-driven conformational rearrangements provided here suggest that temperature alone is able to open the channel and that the opening reaction is voltage-independent. Voltage is a partial activator of TRPM8 channels, since absolute open probability values measured with fully activated voltage sensors are less than 1 and they decrease as temperature rises. By unveiling the fast temperature-dependent deactivation process, we show that TRPM8 channel deactivation is well described by a double exponential time course. The fast and slow deactivation processes are temperature-dependent with enthalpy changes of 27.2 kcalmol-1 and 30.8 kcalmol-1. The overall Q10 for the closing reaction is about 33. A three-tiered allosteric model containing four voltage sensors and four temperature sensors can account for the complex deactivation kinetics and coupling between voltage and temperature sensor activation and channel opening.

Temperature and Voltage Coupling to Channel Opening in Transient Receptor Potential Melastatin 8 (TRPM8)

Expressed in somatosensory neurons of the dorsal root and trigeminal ganglion, the transient receptor potential melastatin 8 (TRPM8) channel is a Ca(2+)-permeable cation channel activated by cold, voltage, phosphatidylinositol 4,5-bisphosphate, and menthol. Although TRPM8 channel gating has been characterized at the single channel and macroscopic current levels, there is currently no consensus regarding the extent to which temperature and voltage sensors couple to the conduction gate. In this study, we extended the range of voltages where TRPM8-induced ionic currents were measured and made careful measurements of the maximum open probability the channel can attain at different temperatures by means of fluctuation analysis. The first direct measurements of TRPM8 channel temperature-driven conformational rearrangements provided here suggest that temperature alone is able to open the channel and that the opening reaction is voltage-independent. Voltage is a partial activator of TRPM8 channels, because absolute open probability values measured with fully activated voltage sensors are less than 1, and they decrease as temperature rises. By unveiling the fast temperature-dependent deactivation process, we show that TRPM8 channel deactivation is well described by a double exponential time course. The fast and slow deactivation processes are temperature-dependent with enthalpy changes of 27.2 and 30.8 kcal mol(-1). The overall Q10 for the closing reaction is about 33. A three-tiered allosteric model containing four voltage sensors and four temperature sensors can account for the complex deactivation kinetics and coupling between voltage and temperature sensor activation and channel opening.

Automata-theoretic decision of timed games

The solution of games is a key decision problem in the context of verification of open systems and program synthesis. Given a game graph and a specification, we wish to determine if there exists a strategy of the protagonist that allows to select only behaviors fulfilling the specification. In this paper, we consider timed games, where the game graph is a timed automaton and the specification is given by formulas of the temporal logics Ltl and Ctl. We present an automata-theoretic approach to solve the addressed games, extending to the timed framework a successful approach to solve discrete games. The main idea of this approach is to translate the timed automaton A, modeling the game graph, into a tree automaton AT accepting all trees that correspond to a strategy of the protagonist. Then, given an automaton corresponding to the specification, we intersect it with the tree automaton AT and check for the nonemptiness of the resulting automaton. Our approach yields a decision algorithm running in exponential time for Ctl and in double exponential time for Ltl. The obtained algorithms are optimal in the sense that their computational complexity matches the known lower bounds.

Succinctness of the Complement and Intersection of Regular Expressions

We study the succinctness of the complement and intersection of regular expressions. In particular, we show that when constructing a regular expression defining the complement of a given regular expression, a double exponential size increase cannot be avoided. Similarly, when constructing a regular expression defining the intersection of a fixed and an arbitrary number of regular expressions, an exponential and double exponential size increase, respectively, cannot be avoided. All mentioned lower bounds improve the existing ones by one exponential and are tight in the sense that the target expression can be constructed in the corresponding time class, that is, exponential or double exponential time. As a by-product, we generalize a theorem by Ehrenfeucht and Zeiger stating that there is a class of DFAs which are exponentially more succinct than regular expressions, to a fixed alphabet. When the given regular expressions are one-unambiguous, as for instance required by the XML Schema specification, the complement can be computed in polynomial time whereas the bounds concerning intersection continue to hold. For the subclass of single-occurrence regular expressions, we prove a tight exponential lower bound for intersection.

Effects of KCNQ channel modulators on the M-type potassium current in primate retinal pigment epithelium

Recently, we demonstrated the expression of KCNQ1, KCNQ4, and KCNQ5 transcripts in monkey retinal pigment epithelium (RPE) and showed that the M-type current in RPE cells is blocked by the specific KCNQ channel blocker XE991. Using patch-clamp electrophysiology, we investigated the pharmacological sensitivity of the M-type current in isolated monkey RPE cells to elucidate the subunit composition of the channel. Most RPE cells exhibited an M-type current with a voltage for half-maximal activation of approximately -35 mV. The M-type current activation followed a double-exponential time course and was essentially complete within 1 s. The M-type current was inhibited by micromolar concentrations of the nonselective KCNQ channel blockers linopirdine and XE991 but was relatively insensitive to block by 10 μM chromanol 293B or 135 mM tetraethylammonium (TEA), two KCNQ1 channel blockers. The M-type current was activated by 1) 10 μM retigabine, an opener of all KCNQ channels except KCNQ1, 2) 10 μM zinc pyrithione, which augments all KCNQ channels except KCNQ3, and 3) 50 μM N-ethylmaleimide, which activates KCNQ2, KCNQ4, and KCNQ5, but not KCNQ1 or KCNQ3, channels. Application of cAMP, which activates KCNQ1 and KCNQ4 channels, had no significant effect on the M-type current. Finally, diclofenac, which activates KCNQ2/3 and KCNQ4 channels but inhibits KCNQ5 channels, inhibited the M-type current in the majority of RPE cells but activated it in others. The results indicate that the M-type current in monkey RPE is likely mediated by channels encoded by KCNQ4 and KCNQ5 subunits.

Small Molecule Binding to Proteins: Affinity and Binding/Unbinding Dynamics from Atomistic Simulations

A dynamic situation! Molecular dynamics simulations at equilibrium are shown to correctly identify the binding mode of dimethyl sulfoxide in the rotamase FKBP without using any experimental information or bias. Furthermore, both the binding and unbinding kinetics were found to have a double-exponential time dependence. Detailed facts of importance to specialist readers are published as ”Supporting Information”. Such documents are peer-reviewed, but not copy-edited or typeset. They are made available as submitted by the authors. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Recovery from Short-Term Depression and Facilitation Is Ultrafast and Ca2+Dependent at Auditory Hair Cell Synapses

Short-term facilitation and depression coexist at many CNS synapses. Facilitation, however, has not been fully characterized at hair cell synapses. Using paired recordings and membrane capacitance measurements we find that paired-pulse plasticity at an adult frog auditory hair cell synapse depends on pulse duration and interpulse intervals. For short 20 ms depolarizing pulses, and interpulse intervals between 15 and 50 ms, facilitation occurred when hair cells were held at -90 mV. However, hair cells held at -60 mV displayed only paired-pulse depression. Facilitation was dependent on residual free Ca2+ levels because it was greatly reduced by the Ca2+ buffers EGTA and BAPTA. Furthermore, low external Ca2+ augmented facilitation, whereas depression was augmented by high external Ca2+, consistent with depletion of a small pool of fast releasing synaptic vesicles. Recovery from depression had a double-exponential time course with a fast component that may reflect the rapid replenishment of a depleted vesicle pool. We suggest that hair cells held at more depolarized in vivo-like resting membrane potentials have a tonic influx of Ca2+; they are thus in a dynamic state of continuous vesicle release, pool depletion and replenishment. Further Ca2+ influx during paired-pulse stimuli then leads to depression. However, at membrane potentials of -90 mV, ongoing release and pool depletion are minimized, so facilitation is revealed at time intervals when rapid vesicle pool replenishment occurs. Finally, we propose that vesicle pool replenishment kinetics is not rate limited by vesicle endocytosis, which is too slow to influence the rapid pool replenishment process.

Double-exponential inseparability of Robinson subsystem Q+

AbstractIn this work a double exponential time inseparability result is proven for a finitely axiomatizable first order theory Q+. The theory, subset of Presburger theory of addition S+, is the additive fragment of Robinson system Q. We prove that every set that separates Q+ from the logically false sentences of addition is not recognizable by any Turing machine working in double exponential time. The lower bound is given both in the non-deterministic and in the linear alternating time models.The result implies also that any theory of addition that is consistent with Q+—in particular any theory contained in S+—is at least double exponential time difficult. Our inseparability result is an improvement on the known lower bounds for arithmetic theories.Our proof uses a refinement and adaptation of the technique that Fischer and Rabin used to prove the difficulty of S+. Our version of the technique can be applied to any incomplete finitely axiomatizable system in which all of the necessary properties of addition are provable.

Metric propositional neighborhood logics on natural numbers

Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is "met by" the current one and to an interval that "meets" the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the length of intervals. We argue that MPNL can be successfully used in different areas of computer science to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alternative to well-established logical frameworks such as Duration Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined sub-fragment of the two-variable fragment $${{\rm FO}^2[\mathbb{N},=,<,s]}$$ of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full $${{\rm FO}^2[\mathbb{N},=,<,s]}$$ , but, unexpectedly, the latter (and hence the former) turns out to be undecidable.

KEY AGREEMENT PROTOCOL OVER THE RING OF MULTIVARIATE POLYNOMIALS

The key agreement protocol (KAP) using matrices over the ring of multivariate polynomials is presented. The compromisation of proposed KAP relies on the solution of multivariate quadratic (MQ) system of equations problem – the problem, which is reckoned as being NP-complete. The general method of solving MQ problem is Grobner basis algorithm, which is of exponential or even double exponential time in general case. For special cases such as overdefined and sparse systems, there are some special solution methods, i.e. XL and XSL algorithms. By choosing suitable security parameters for the compromisation of the proposed KAP, we obtained a random not overdefined and not sparse system of MQ equations and hence we recon that our KAP compromasation relies on the hard MQ problem.

Open channel block and beyond

After settling down from the excitement of seeing single channels open and close in the membrane of a living cell for the first time (Neher & Sakmann, 1976), the question that arose was what could these extraordinary recordings tell us? At the time we had no idea how many different applications we would soon find for the patch clamp technique. Most of these applications followed from the subsequent improvement of the gigaseal, which dramatically increased the power of the patch clamp. But as a ‘first application’ for single channel recording, in an article in The Journal of Physiology, Neher & Steinbach (1978) turned to a question about the mechanism of action of local anaesthetics. It was known that local anaesthetics reduce the current through the membranes of excitable cells. The endplate current at the neuromuscular synapse becomes smaller in the presence of a local anaesthetic, and its decay follows a double-exponential time course. These results could be accounted for by a model in which a channel first opens, and a local anaesthetic molecule then binds to a site within the open channel to block it (Ruff, 1977). These drugs could thus be pictured as acting like a plug in the drain of a bathtub. This model fitted the data well, and most researchers regarded it as very appealing. Neher and Steinbach noted a number of studies that had invoked a sequential mechanism of channel blockade, but this mechanism was not generally accepted as rigorously proven. The key result of two exponentials in the decay of the synaptic current indicated that the drug must induce the appearance of a new kinetic state of the channel. From an analysis of macroscopic currents alone, it was very difficult to decide whether the new state was an open or closed channel, or whether entry into the new state occurs from an open or closed channel. Single channel recordings from a frog skeletal muscle fibre Patch electrode solution containing 0.1 μm suberyldicholine alone in A, together with 5 μm (B), 10 μm (C), and 50 μm (D) QX-222. (Reproduced from Fig. 2 of Neher & Steinbach, 1978.) QX-222 and QX314 are both charged, and if their binding site resides within the channel, then the transmembrane potential should influence binding. The single channel data permitted a direct determination of the blocking and unblocking rates (F and G), and these rates did indeed depend on voltage. The magnitude of this voltage dependence suggested that the drug traverses 78% of the membrane potential upon binding. This study established open channel block as an important mechanism of drug action. But the study had much greater significance in providing the first illustration of the remarkable power of single channel recording in the elucidation of kinetic mechanism. By observing the closed state, the open state and the blocked state directly, some very strong statements could be made about how these states interconvert. For any ion channel mechanism, the single-molecule transitions contain information not readily extracted from the macroscopic currents that report the activity of populations of channels. This work thus initiated a new era of mechanistic studies of ion channels. A rigorous mathematical formulation of single-channel kinetics provided the essential theoretical tools for interpreting complex lifetime distributions (Colquhoun & Hawkes, 1982). Single channel recordings revealed correlations in the lifetimes of adjacent gating intervals that added to the information content uniquely available in this form of data (Jackson et al. 1983), and again a theoretical framework served to guide these studies (Fredkin et al. 1985). The enduring contribution of this paper was its illustration of the extraordinary power of single-channel recording in model discrimination and the elucidation of kinetic mechanism. The many applications of single-channel kinetics that followed this work brought our understanding of ion channels to a new level of detail. These studies offer valuable lessons that should prove useful in the study of other forms of single-molecule kinetics, most notably single-molecule fluorescence.