Factors of low individual degree polynomials

Kaltofen (Randomness in computation, vol 5, pp 375–412, 1989) proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen’s work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in Kopparty et al. (2014), and the question of bounding the depth of the circuits computing the factors of a polynomial. We are able to answer these questions in the affirmative for the interesting class of polynomials of bounded individual degrees, which contains polynomials such as the determinant and the permanent. We show that if \({P(x_{1},\ldots,x_{n})}\) is a polynomial with individual degrees bounded by r that can be computed by a formula of size s and depth d, then any factor \({f(x_{1},\ldots, x_{n})}\) of \({P(x_{1},\ldots,x_{n})}\) can be computed by a formula of size \({\textsf{poly}((rn)^{r},s)}\) and depth d + 5. This partially answers the question above posed in Kopparty et al. (2014), who asked if this result holds without the dependence on r. Our work generalizes the main factorization theorem from Dvir et al. (SIAM J Comput 39(4):1279–1293, 2009), who proved it for the special case when the factors are of the form \({f(x_{1}, \ldots, x_{n}) \equiv x_{n} - g(x_{1}, \ldots, x_{n-1})}\). Along the way, we introduce several new technical ideas that could be of independent interest when studying arithmetic circuits (or formulas).     

Evaluation Method,Dataset Size or Dataset Content: How to Evaluate Algorithms for Image Matching?

Flutter shutter (coded exposure) cameras allow to extend indefinitely the exposure time for uniform motion blurs. Recently, Tendero et al. (SIAM J Imaging Sci 6(2):813–847, 2013) proved that for a fixed known velocity v the gain of a flutter shutter in terms of root means square error (RMSE) cannot exceeds a 1.1717 factor compared to an optimal snapshot. The aforementioned bound is optimal in the sense that this 1.1717 factor can be attained. However, this disheartening bound is in direct contradiction with the recent results by Cossairt et al. (IEEE Trans Image Process 22(2), 447–458, 2013). Our first goal in this paper is to resolve mathematically this discrepancy. An interesting question was raised by the authors of reference (IEEE Trans Image Process 22(2), 447–458, 2013). They state that the “gain for computational imaging is significant only when the average signal level J is considerably smaller than the read noise variance \(\sigma _r^2\)” (Cossairt et al., IEEE Trans Image Process 22(2), 447–458, 2013, p. 5). In other words, according to Cossairt et al. (IEEE Trans Image Process 22(2), 447–458, 2013) a flutter shutter would be more efficient when used in low light conditions e.g., indoor scenes or at night. Our second goal is to prove that this statement is based on an incomplete camera model and that a complete mathematical model disproves it. To do so we propose a general flutter shutter camera model that includes photonic, thermal (The amplifier noise may also be mentioned as another source of background noise, which can be included w.l.o.g. in the thermal noise) and additive [The additive (sensor readout) noise may contain other components such as reset noise and quantization noise. We include them w.l.o.g. in the readout.] (sensor readout, quantification) noises of finite variances. Our analysis provides exact formulae for the mean square error of the final deconvolved image. It also allows us to confirm that the gain in terms of RMSE of any flutter shutter camera is bounded from above by 1.1776 when compared to an optimal snapshot. The bound is uniform with respect to the observation conditions and applies for any fixed known velocity. Incidentally, the proposed formalism and its consequences also apply to e.g., the Levin et al. motion-invariant photography (ACM Trans Graphics (TOG), 27(3):71:1–71:9, 2008; Method and apparatus for motion invariant imag- ing, October 1 2009. US Patent 20,090,244,300, 2009) and variant (Cho et al. Motion blur removal with orthogonal parabolic exposures, 2010). In short, we bring mathematical proofs to the effect of contradicting the claims of Cossairt et al. (IEEE Trans Image Process 22(2), 447–458, 2013). Lastly, this paper permits to point out the kind of optimization needed if one wants to turn the flutter shutter into a useful imaging tool.     

Fast Image Inpainting Based on Coherence Transport

High-quality image inpainting methods based on nonlinear higher-order partial differential equations have been developed in the last few years. These methods are iterative by nature, with a time variable serving as iteration parameter. For reasons of stability a large number of iterations can be needed which results in a computational complexity that is often too large for interactive image manipulation.Based on a detailed analysis of stationary first order transport equations the current paper develops a fast noniterative method for image inpainting. It traverses the inpainting domain by the fast marching method just once while transporting, along the way, image values in a coherence direction robustly estimated by means of the structure tensor. Depending on a measure of coherence strength the method switches continuously between diffusion and directional transport. It satisfies a comparison principle. Experiments with the inpainting of gray tone and color images show that the novel algorithm meets the high level of quality of the methods of Bertalmio et al. (SIG-GRAPH ’00: Proc. 27th Conf. on Computer Graphics and Interactive Techniques, New Orleans, ACM Press/Addison-Wesley, New York, pp. 417–424, 2000), Masnou (IEEE Trans. Image Process. 11(2):68–76, 2002), and Tschumperlé (Int. J. Comput. Vis. 68(1):65–82, 2006), while being faster by at least an order of magnitude.     

The complexity of estimating min-entropy

Goldreich et al. (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZK-complete. We consider the analogous problem for estimating the min-entropy and prove that it is SBP-complete, where SBP is the class of promise problems that correspond to approximate counting of NP witnesses. The result holds even when the sampling circuits are restricted to be 3-local. For logarithmic-space samplers, we observe that this problem is NP-complete by a result of Lyngsø and Pedersen on hidden Markov models (JCSS 65(3):545–569, 2002).     

<Emphasis Type="Italic">δ</Emphasis>-equality of intuitionistic fuzzy sets: a new proximity measure and applications in medical diagnosis

Intuitionistic fuzzy set is capable of handling uncertainty with counterpart falsities which exist in nature. Proximity measure is a convenient way to demonstrate impractical significance of values of memberships in the intuitionistic fuzzy set. However, the related works of Pappis (Fuzzy Sets Syst 39(1):111–115, 1991), Hong and Hwang (Fuzzy Sets Syst 66(3):383–386, 1994), Virant (2000) and Cai (IEEE Trans Fuzzy Syst 9(5):738–750, 2001) did not model the measure in the context of the intuitionistic fuzzy set but in the Zadeh’s fuzzy set instead. In this paper, we examine this problem and propose new notions of δ-equalities for the intuitionistic fuzzy set and δ-equalities for intuitionistic fuzzy relations. Two fuzzy sets are said to be δ-equal if they are equal to an extent of δ. The applications of δ-equalities are important to fuzzy statistics and fuzzy reasoning. Several characteristics of δ-equalities that were not discussed in the previous works are also investigated. We apply the δ-equalities to the application of medical diagnosis to investigate a patient’s diseases from symptoms. The idea is using δ-equalities for intuitionistic fuzzy relations to find groups of intuitionistic fuzzified set with certain equality or similar degrees then combining them. Numerical examples are given to illustrate validity of the proposed algorithm. Further, we conduct experiments on real medical datasets to check the efficiency and applicability on real-world problems. The results obtained are also better in comparison with 10 existing diagnosis methods namely De et al. (Fuzzy Sets Syst 117:209–213, 2001), Samuel and Balamurugan (Appl Math Sci 6(35):1741–1746, 2012), Szmidt and Kacprzyk (2004), Zhang et al. (Procedia Eng 29:4336–4342, 2012), Hung and Yang (Pattern Recogn Lett 25:1603–1611, 2004), Wang and Xin (Pattern Recogn Lett 26:2063–2069, 2005), Vlachos and Sergiadis (Pattern Recogn Lett 28(2):197–206, 2007), Zhang and Jiang (Inf Sci 178(6):4184–4191, 2008), Maheshwari and Srivastava (J Appl Anal Comput 6(3):772–789, 2016) and Support Vector Machine (SVM).     

Pupil detection for head-mounted eye tracking in the wild: an evaluation of the state of the art

Robust and accurate detection of the pupil position is a key building block for head-mounted eye tracking and prerequisite for applications on top, such as gaze-based human–computer interaction or attention analysis. Despite a large body of work, detecting the pupil in images recorded under real-world conditions is challenging given significant variability in the eye appearance (e.g., illumination, reflections, occlusions, etc.), individual differences in eye physiology, as well as other sources of noise, such as contact lenses or make-up. In this paper we review six state-of-the-art pupil detection methods, namely ElSe (Fuhl et al. in Proceedings of the ninth biennial ACM symposium on eye tracking research & applications, ACM. New York, NY, USA, pp 123–130, 2016), ExCuSe (Fuhl et al. in Computer analysis of images and patterns. Springer, New York, pp 39–51, 2015), Pupil Labs (Kassner et al. in Adjunct proceedings of the 2014 ACM international joint conference on pervasive and ubiquitous computing (UbiComp), pp 1151–1160, 2014. doi: 10.1145/2638728.2641695), SET (Javadi et al. in Front Neuroeng 8, 2015), Starburst (Li et al. in Computer vision and pattern recognition-workshops, 2005. IEEE Computer society conference on CVPR workshops. IEEE, pp 79–79, 2005), and ?wirski (?wirski et al. in Proceedings of the symposium on eye tracking research and applications (ETRA). ACM, pp 173–176, 2012. doi: 10.1145/2168556.2168585). We compare their performance on a large-scale data set consisting of 225,569 annotated eye images taken from four publicly available data sets. Our experimental results show that the algorithm ElSe (Fuhl et al. 2016) outperforms other pupil detection methods by a large margin, offering thus robust and accurate pupil positions on challenging everyday eye images.     

Information Lower Bounds via Self-Reducibility

We use self-reduction methods to prove strong information lower bounds on two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product (IP). In our first result we affirm the conjecture that the information cost of GHD is linear even under the uniform distribution, which strengthens the Ω(n) bound recently shown by Kerenidis et al. (2012), and answers an open problem from Chakrabarti et al. (2012). In our second result we prove that the information cost of IP_n is arbitrarily close to the trivial upper bound n as the permitted error tends to zero, again strengthening the Ω(n) lower bound recently proved by Braverman and Weinstein (Electronic Colloquium on Computational Complexity (ECCC) 18, 164 2011). Our proofs demonstrate that self-reducibility makes the connection between information complexity and communication complexity lower bounds a two-way connection. Whereas numerous results in the past (Chakrabarti et al. 2001; Bar-Yossef et al. J. Comput. Syst. Sci. 68(4), 702–732 2004; Barak et al. 2010) used information complexity techniques to derive new communication complexity lower bounds, we explore a generic way in which communication complexity lower bounds imply information complexity lower bounds in a black-box manner.     

Content-Based Image Retrieval Using Iterative Search

The aim of Content-based Image Retrieval (CBIR) is to find a set of images that best match the query based on visual features. Most existing CBIR systems find similar images in low level features, while Text-based Image Retrieval (TBIR) systems find images with relevant tags regardless of contents in the images. Generally, people are more interested in images with similarity both in contours and high-level concepts. Therefore, we propose a new strategy called Iterative Search to meet this requirement. It mines knowledge from the similar images of original queries, in order to compensate for the missing information in feature extraction process. To evaluate the performance of Iterative Search approach, we apply this method to four different CBIR systems (HOF Zhou et al. in ACM international conference on multimedia, 2012; Zhou and Zhang in Neural information processing—international conference, ICONIP 2011, Shanghai, 2011, HOG Dalal and Triggs in IEEE computer society conference on computer vision pattern recognition, 2005, GIST Oliva and Torralba in Int J Comput Vision 42:145–175, 2001 and CNN Krizhevsky et al. in Adv Neural Inf Process Syst 25:2012, 2012) in our experiments. The results show that Iterative Search improves the performance of original CBIR features by about \(20\%\) on both the Oxford Buildings dataset and the Object Sketches dataset. Meanwhile, it is not restricted to any particular visual features.     

Correctness and concurrent complexity of the Black-White Bakery Algorithm

Lamport’s Bakery Algorithm (Commun ACM 17:453–455, 1974) implements mutual exclusion for a fixed number of threads with the first-come first-served property. It has the disadvantage, however, that it uses integer communication variables that can become arbitrarily large. Taubenfeld’s Black-White Bakery Algorithm (Proceedings of the DISC. LNCS, vol 3274, pp 56–70, 2004) keeps the integers bounded, and is adaptive in the sense that the time complexity only depends on the number of competing threads, say N. The present paper offers an assertional proof of correctness and shows that the concurrent complexity for throughput is linear in N, and for individual progress is quadratic in N. This is proved with a bounded version of UNITY, i.e., by assertional means.     

What Would Edmonds Do? Augmenting Paths and Witnesses for Degree-Bounded MSTs

Given a graph and degree upper bounds on vertices, the BDMST problem requires us to find a minimum cost spanning tree respecting the given degree bounds. This problem generalizes the Travelling Salesman Path Problem (TSPP), even in unweighted graphs, and so we expect that it is necessary to relax the degree constraints to get efficient algorithms. Könemann and Ravi (Proceedings of the Thirty Second Annual ACM Symposium on Theory of Computing, pp. 537–546, 2000; Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, pp. 389–395, 2003) give bicriteria approximation algorithms for the problem using local search techniques of Fischer (Technical Report 14853, Cornell University, 1993). Their algorithms find solutions which make a tradeoff of the approximation factor for the cost of the resulting tree against the factor by which degree constraints are violated. In particular, they give an algorithm which, for a graph with a spanning tree of cost C and degree B, and for parameters b,w>1, produces a tree whose cost is at most wC and whose degree is at most \(\frac{w}{w-1}bB+\log_{b}n.\) A primary contribution of Könemann and Ravi is to use a Lagrangean relaxation to formally relate the BDMST problem to what we call the MDMST problem, which is the problem of finding an MST of minimum degree in a graph. In their solution to the MDMST problem, they make central use of a local-search approximation algorithm of Fischer.In this paper, we give the first approximation algorithms for the BDMST problem—both our algorithms find trees of optimal cost. We achieve this improvement using a primal-dual cost bounding methodology from Edmonds’ weighted matching algorithms which was not previously used in this context. In order to follow Edmonds’ approach, we develop algorithms for a variant of the MDMST problem in which there are degree lower bound requirements. This variant may be of independent interest; in particular, our results extend to a generalized version of the BDMST problem in which both upper and lower degree bounds are given.First we give a polynomial-time algorithm that finds a tree of optimal cost and with maximum degree at most \(\frac{b}{2-b}B+O(\log_{b}n)\) for any b∈(1,2). We also give a quasi-polynomial-time approximation algorithm which produces a tree of optimal cost C and maximum degree at most B+O(log?n/log?log?n). That is, the error is additive as well as restricted just to the degree. This further improvement in degree is obtained by using augmenting-path techniques that search over a larger solution space than Fischer’s local-search algorithm.     

Pressure drop of three-phase liquid–liquid–gas slug flow in round microchannels

In this paper we present a model for the calculation of pressure drop of three-phase liquid–liquid–gas slug flow in microcapillaries of a circular cross section. Introduced models consist of terms attributing for frictional and interfacial pressure drop, incorporating the presence of a stagnant thin film at the wall of the channel. Different formulations of the interfacial pressure drop equation were employed, using expressions developed by Bretherton (J Fluid Mech 10:166–188, 1961), Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989). Models were validated experimentally using oleic acid–water–nitrogen and heptane–water–nitrogen three-phase flows in round Teflon or Radel R microchannels of 254- and 508-µm nominal inner diameter, for capillary numbers Ca _b between 10^?4 and 4.9 × 10^?1 and Reynolds numbers Re between 0.095 and 300. Best agreement between measured and calculated values of pressure drop, with relative error between ?22 and 19 % or ?20 and 16 %, is reached for Warnier’s or Ratulowski and Chang’s interfacial pressure drop equation, respectively. The results prove that three-phase slug flow pressure drop can be successfully predicted by extending existing two-phase slug flow correlations. Good agreement of Bretherton’s equation was reached only at lower Ca numbers, indicating that an extension of the interfacial pressure drop equation as performed by Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989) for higher capillary numbers is necessary. Additionally it was demonstrated that pressure drop increases substantially if dry slug flow occurs or if microchannels with significant surface roughness are employed. Those influences were not accounted for in the models presented.     

On variable-weighted exact satisfiability problems

We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(||C||2^0.2441n ) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n ²·||C||?+?2^0.40567n ) time. In recent years only the unweighted counterparts of these problems have been studied (Dahllöf and Jonsson, An algorithm for counting maximum weighted independent sets and its applications. In: Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms, pp. 292–298, 2002; Dahllöf et al., Theor Comp Sci 320: 373–394, 2004; Porschen, On some weighted satisfiability and graph problems. In: Proceedings of the 31st Conference on Current Trends in Theory and Practice of Informatics (SOFSEM 2005). Lecture Notes in Comp. Science, vol. 3381, pp. 278–287. Springer, 2005).     

Mesh generation for confined fusion plasma simulation

XGC1 and M3D-C ¹ are two fusion plasma simulation codes being developed at Princeton Plasma Physics Laboratory. XGC1 uses the particle-in-cell method to simulate gyrokinetic neoclassical physics and turbulence (Chang et al. Phys Plasmas 16(5):056108, 2009; Ku et al. Nucl Fusion 49:115021, 2009; Admas et al. J Phys 180(1):012036, 2009). M3D-\(C^1\) solves the two-fluid resistive magnetohydrodynamic equations with the \(C^1\) finite elements (Jardin J comput phys 200(1):133–152, 2004; Jardin et al. J comput Phys 226(2):2146–2174, 2007; Ferraro and Jardin J comput Phys 228(20):7742–7770, 2009; Jardin J comput Phys 231(3):832–838, 2012; Jardin et al. Comput Sci Discov 5(1):014002, 2012; Ferraro et al. Sci Discov Adv Comput, 2012; Ferraro et al. International sherwood fusion theory conference, 2014). This paper presents the software tools and libraries that were combined to form the geometry and automatic meshing procedures for these codes. Specific consideration has been given to satisfy the mesh configuration and element shape quality constraints of XGC1 and M3D-\(C^1\).     

A polynomial-time algorithm for a flow-shop batching problem with equal-length operations

A flow-shop batching problem with consistent batches is considered in which the processing times of all jobs on each machine are equal to p and all batch set-up times are equal to s. In such a problem, one has to partition the set of jobs into batches and to schedule the batches on each machine. The processing time of a batch B _i is the sum of processing times of operations in B _i and the earliest start of B _i on a machine is the finishing time of B _i on the previous machine plus the set-up time s. Cheng et al. (Naval Research Logistics 47:128–144, 2000) provided an O(n) pseudopolynomial-time algorithm for solving the special case of the problem with two machines. Mosheiov and Oron (European Journal of Operational Research 161:285–291, 2005) developed an algorithm of the same time complexity for the general case with more than two machines. Ng and Kovalyov (Journal of Scheduling 10:353–364, 2007) improved the pseudopolynomial complexity to \(O(\sqrt{n})\). In this paper, we provide a polynomial-time algorithm of time complexity O(log?³ n).     

Performance of One-Round Walks in Linear Congestion Games

We investigate the approximation ratio of the solutions achieved after a one-round walk in linear congestion games. We consider the social functions Sum, defined as the sum of the players’ costs, and Max, defined as the maximum cost per player, as a measure of the quality of a given solution. For the social function Sum and one-round walks starting from the empty strategy profile, we close the gap between the upper bound of \(2+\sqrt{5}\approx 4.24\) given in Christodoulou et al. (Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS), LNCS, vol. 3884, pp. 349–360, Springer, Berlin, 2006) and the lower bound of 4 derived in Caragiannis et al. (Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP), LNCS, vol. 4051, pp. 311–322, Springer, Berlin, 2006) by providing a matching lower bound whose construction and analysis require non-trivial arguments. For the social function Max, for which, to the best of our knowledge, no results were known prior to this work, we show an approximation ratio of \(\Theta(\sqrt[4]{n^{3}})\) (resp. \(\Theta(n\sqrt{n})\)), where n is the number of players, for one-round walks starting from the empty (resp. an arbitrary) strategy profile.     

Connectivity Preserving Multivalued Functions in Digital Topology

We study connectivity preserving multivalued functions (Kovalevsky in A new concept for digital geometry, shape in picture, 1994) between digital images. This notion generalizes that of continuous multivalued functions (Escribano et al. in Discrete geometry for computer imagery, lecture notes in computer science, 2008; Escribano et al. in J Math Imaging Vis 42:76–91, 2012) studied mostly in the setting of the digital plane \({\mathbb {Z}}^2\). We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morphological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.     

Quantum and Classical Query Complexities of Local Search Are Polynomially Related

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).     

Grounds for Trust: Essential Epistemic Opacity and Computational Reliabilism

Several philosophical issues in connection with computer simulations rely on the assumption that results of simulations are trustworthy. Examples of these include the debate on the experimental role of computer simulations (Parker in Synthese 169(3):483–496, 2009; Morrison in Philos Stud 143(1):33–57, 2009), the nature of computer data (Barberousse and Vorms, in: Durán, Arnold (eds) Computer simulations and the changing face of scientific experimentation, Cambridge Scholars Publishing, Barcelona, 2013; Humphreys, in: Durán, Arnold (eds) Computer simulations and the changing face of scientific experimentation, Cambridge Scholars Publishing, Barcelona, 2013), and the explanatory power of computer simulations (Krohs in Int Stud Philos Sci 22(3):277–292, 2008; Durán in Int Stud Philos Sci 31(1):27–45, 2017). The aim of this article is to show that these authors are right in assuming that results of computer simulations are to be trusted when computer simulations are reliable processes. After a short reconstruction of the problem of epistemic opacity, the article elaborates extensively on computational reliabilism, a specified form of process reliabilism with computer simulations located at the center. The article ends with a discussion of four sources for computational reliabilism, namely, verification and validation, robustness analysis for computer simulations, a history of (un)successful implementations, and the role of expert knowledge in simulations.     

The computational capability of chemical reaction automata

We propose a new computing model called chemical reaction automata (CRAs) as a simplified variant of reaction automata (RAs) studied in recent literature (Okubo in RAIRO Theor Inform Appl 48:23–38 2014; Okubo et al. in Theor Comput Sci 429:247–257 2012a, Theor Comput Sci 454:206–221 2012b). We show that CRAs in maximally parallel manner are computationally equivalent to Turing machines, while the computational power of CRAs in sequential manner coincides with that of the class of Petri nets, which is in marked contrast to the result that RAs (in both maximally parallel and sequential manners) have the computing power of Turing universality (Okubo 2014; Okubo et al. 2012a). Intuitively, CRAs are defined as RAs without inhibitor functioning in each reaction, providing an offline model of computing by chemical reaction networks (CRNs). Thus, the main results in this paper not only strengthen the previous result on Turing computability of RAs but also clarify the computing powers of inhibitors in RA computation.