Abstract: This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many **valued** propositional **logic**. The generalization of Kalmar’s proof of deducibility for **two** **valued** tautologies inside **classical** propositional **logic** gives us a possibility to suggest some method for defining of **two** types axiomatic systems for any version of 3-**valued** **logic**, completeness of which is easy proved direct, without of loading into **two** **valued** **logic**. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-**valued** (k≥3) logics and may be for fuzzy **logic** also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.

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0 1 1 1 1 Suppose we treat 1 as the only designated semantic value (hence the bold font). Intu- itively, this is to say that some premises entail a conclusion iff, whenever all the premises have value 1, the conclusion also has value 1. We have now defined a **logic** using our four- **valued** truth-tables. (Throughout, I take a **logic** just to be any **two**-place relation which relates sets of sentences—think ‘premises’—to individual sentences—think ‘conclu- sion’.) This four-**valued** definition may be unfamiliar, but the **logic** it defines is very familiar indeed: it is our old friend, **classical** sentential **logic**. Otherwise put: the **two** different systems of truth-tables yield the very same **logic** (see Theorem 1 ).

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The main motivation for making the transition from binary to multi-**valued** quantum **logic** is to avail of the greater information capacity of multi-level atomic systems. Using more than **two** levels in each ion in the linear ion trap scheme for example, we could reduce the number of ions needed to be stored in the trap. This is an advantage because the bottleneck for implementing this scheme, and in many others, is the maintenance of a macroscopically coherent state of all the ions for a sufficiently long time before, subject to environmental noise, the coherence vanishes and the computation is corrupted. Another difference between binary and multi-**valued** **logic** that applies equally well to the **classical** and quantum domains is the trade-off in processing time – executing a large number of small (2 or 4-dim) binary gates versus a small number of large (d or d 2 -dim) multi-**valued** gates. If large single-qubit operations are more viable than doing many small ones in sequence, then the multi-**valued** case will have an advantage.

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Fuzzy reasoning is an important tool of fuzzy control. The most influential approach is CRI based on fuzzy set theory presented by Zadeh [18]. As an improvement for the CRI method, Wang [14] proposed a new method, called the full implication triple I method, for fuzzy reasoning. As for fuzzy reasoning, the results of inference completely depend on the choice of fuzzy sets of fuzzy antecedents and fuzzy consequences as well as fuzzy connectives. However, fuzzy sets are chosen by people subjectively [8]. Naturally, we hope this fuzzy reasoning system has good robustness. It is important to measure the degree of deviation between fuzzy sets when we discuss robustness of fuzzy reasoning. Therefore, the largest perturbation and the average perturbation [16], δ-equalities [2], δ-sensitivity, largest δ-sensitivity and average δ-sensitivity [9,10] had been proposed. Based on these notions, [2,7,10,16] discussed robustness of various kinds fuzzy reasoning systems from different views. Because the behavior of a fuzzy **logic** system is mainly determined by the fuzzy connectives and fuzzy implication operators. Therefore, based on this point, Dai et al. [4] and Jin et al. [7] investigated the robustness of fuzzy reasoning via **logic** similarity degree. As an alternative for the **logic** similarity degree, a new metric was proposed and **two** kinds of suitable fuzzy metric spaces for fuzzy reasoning were found by Wang and Duan [15]. Based on the **logic** similarity degree, Duan and Li [5] compared the structures of four special **logic** metric spaces and analyzed the results from the standpoint of robustness of fuzzy reasoning.

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Abstract—One of the bottlenecks in interprocedural analysis is the difficulty in handling complex parameters. This paper proposes a novel approach to solve this problem: symbolic procedure summary, which is constructed using region- based symbolic three-**valued** **logic** (RSTVL). RSTVL is a memory model that can describe memory state of variables and all kinds of associations among them. Based on the re- sult of intraprocedural analysis, we construct symbolic pro- cedure summary described by RSTVL, and instantiate it at call site using the calling context. Our approach improves analysis precision as it achieves context-sensitive and field- sensitive interprocedural analysis. We apply this approach in defect detection, and experimental results show that it can effectively reduce both false negatives and false positives of defect detecting, and improve test accuracy at the same time.

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There are several features of type theory that are of especial benefit for proof assistants: each object carries a type which gives information about that object, and the type theory itself has a primitive notion of computation. We contend that the intuitions behind type theory apply outside of intuitionistic mathematics, and that these advantages would prove beneficial when applied to other forms of proof. It is equally natural in **classical** mathematics to divide mathematical objects into types, and it would be of as much benefit to take advantage of the information provided by an object’s type in a **classical** proof. The notion of computation is an important part of **classical** mathematics. When formally proving a property of a program, we may be perfectly satisfied with a **classical** proof, which could well be shorter or easier to find.

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In order to have the n-**valued** ÃLukasiewicz-Moisil **logic** (LML) algebra repre- sent correctly the basic behavior of quantum systems [7], [17] –which is usually observed through measurements that involve a quantum system interactions with a macroscopic measuring instrument– several of these axioms have to be significantly changed so that the resulting lattice becomes non-distributive and also (possibly) non-associative. Several encouraging results in this direction were recently obtained by Dala Chiara and coworkers. With an appropriately defined quantum **logic** of events one can proceed to define Hilbert, or ‘nu- clear’/Frechet, spaces in order to be able to utilize the ‘standard’ procedures of quantum theories [17].

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Further, first-order logic can be used to axiomatize other types of feature structures in addition to attribute-value structures such as "set-valued" elements and express a wider variety[r]

A full set of logical functions can be created from a maximum function so long as the output can be complemented. The minimum function is used with the maximum function, but this can be created by inverting the inputs and output of a maximum function. The grounded output encoding can be complemented as shown in figure 2b. The inverters may be omitted if n-type transistors can be used as the current switches. In this case, the complemented encoding would be smaller than the un-complemented encoding. This is because the grounding transistor must be an n-type transistor and hence the inverter in the un-complemented encoding shown in figure 2a cannot be removed. Even using p-type switch transistors, the complemented encoder uses only one more inverter (i.e. **two** transistors) than the un-complemented encoder.

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The split circuitry is used for converting the four **logic** levels i.e 0,1,2,3 into the series of individual pulses for 4 outputs which can be used to ON and OFF the switch matrix as shown below with its simulation waveform. The split circuit is basically an encoder. To avoid conversion we require split circuitry which can be used to switch the switch network according to truth table. The output of Quaternary split is used to switch the particular transmission gate in switch matrix. As told earlier it consists of differential amplifier, using this we create level converter which is nothing but the comparative ladder having some reference values and series of op-amp. After that there is a XOR gate in series which converts thermometer code into the required output.

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By considering the measurements of the x and y components of spin of a 4-geon with spin-half it is possible to construct a modular lattice of propositions. It has been reported by Friedman and Sorkin[13] that manifolds with the trans- formation properties of a spinor can be constructed. For the construction which follows, we require the 4-geon to have more than one possible outcome from a Stern-Gerlach apparatus. We will consider **two** possible outcomes (> 0, ≤ 0); the exact spectrum, whether it is finite or infinite, continuous or discrete is not important. The choice of x and y-spin and the restriction to **two** outcomes is made to give a simple model of the spin for a spin-half particle; momentum and position could equally well have been used.

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In Fig 4.4 (a), when both bit line and word line are activated, the input data is written to the memory node. Once the NMOS switch is OFF, the memory node latches to one of three **logic** voltage levels, depending on the written input. Fig 4.4 (b) shows the simulation results for the write operation. By adjusting parameters and/or supply voltage, each **logic** state can be designed to take place near the valley of the NDC’s I-V characteristics in order to minimize the current required to hold the states. Fig 4.5 shows three **logic** states when NDC parameters are: C j1~j4 = C j =0.176aF, C g1 = 0.86aF, and C g2 =

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Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would r[r]

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by PLC controller where the last has a wide range in in- dustrial application. The experiments that are imple- mented in this lab can be divided into three main groups according to application field in **classical** control ap- proach: **Logic** process, power system and interactive process control, where by using this trainer six experi- ments can be done in **logic** process, ten experiments in power system and eight experiments in the interactive process control. Every experiment has a procedure for connection on the board and program for writing with PLC memory and then run the connection. For example, the implementation of traffic light control can be dis- cussed by the procedure as follows:

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From all the key features discussed above, the design of multiplier for RLNS based applications using TVL is proposed in this work. This method is more effectual in reducing the computational complexity of the multiplier design as logarithmic property is involved. To further reduce the number of input output terminals, overall area occupied and the delay values of the circuit, TVL is incorporated. The challenging task for the design of pro- posed idea lie in the conversion of ternary inputs to its corresponding logarithmic values and vice versa. This forms the major contribution of this work, along with the error correction circuit of ternary logarithmic and an- tilogarithmic conversion process. As far as dealing with logarithmic numbers with base value 2 (binary **logic**), there are several procedures followed to ensure correction process for the logarithmic and antilogarithmic con- version circuits. They can summarized as Look Up Table (LUT) based approach [35] [36], improving the accu- racy of Mitchell’s approach [37] using correction term based, Divided approximation [38]-[41], Operand de- composition [42] [43] and so on. But in the case of TVL, Mitchell’s approach cannot be applied, as the ap- proaches do not provide even approximate results for all values in the conversion process. Also use of LUT is not appreciated as it involves in increasing the area utilization of the design. Hence different approach is fol- lowed for reducing the error value of the ternary logarithmic and antilogarithmic conversion process (discussed in Section 4)

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SKIN METABOLISM IN VITRO IN Two CHILDREN WITH PROGERIA COMPARED WITH ONE NORMAL GIRL. Patient Age (tir)[r]

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In the first time, we gave full classification of the group. Number of classes of the classification is k-1. We named them self-m-al at 2≤m≤k. We built the theory of self-k-al functions and proved 10 important theorems. We get numerical results for 3-**valued** **logic** and found that inclusion graphs of clones of self-dual and of self-3-al functions are not lattices.

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Born into an aristocratic family, Borumand was raised in a household frequented by musicians, poets, and artists, and, at the age of twelve, he began studying tār (long-necked plucked lute) with the renowned Qolam Hossein Darvish Khan (1872– 1926). In 1922, he was sent to Germany, where he attended secondary school and also became familiar with European music, taking piano lessons for **two** years. While studying medicine, Borumand continued to pursue his musical interests, but failing eyesight and eventual blindness led to his return to Iran in 1938, from which time he devoted himself entirely to Iranian music. Borumand spent the next thirty years liv- ing a rather secluded life and continuing to study tār and setār. It wasn’t until the 1960s that he emerged into public life, after being invited to advise on and participate in a number of government-sponsored activities and—from the mid-1960s—to teach radif at the University of Tehran’s Fine Arts Faculty (before the formal establishment of the Music Department in 1969) and later at the Markaz. Borumand was not primarily a per- former and was therefore little known to the general public; however, he became highly regarded as a teacher, largely through his detailed knowledge of the radif, accumulated through many years of study with several prominent masters. Borumand’s significance lies largely in the fact that he studied with masters who were in direct receipt of Mirza Abdollah’s radif (see During et al. 1991: 62; Nettl 1987: 142–143), his main teachers— Esma’il Khan Qahremani and Haji Aqa Mohammad—being among the most highly regarded of Mirza Abdollah’s pupils. In particular, as noted, many consider the radif that Borumand studied intensively with Qahremani over a period of about twelve years (During et al. 1991: 63) to be the version closest to Mirza Abdollah’s original, particularly since the route of transmission through Qahremani and Borumand was not affected by Darvish Khan’s attempts to popularize the tradition. As Fayaz observes:

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2 Abstract
A recent survey has shown that there are approximately seventeen collections of **classical** antiquities in South African museums which fall into **two** main categories: museums attached to higher education institutions and public state museums. While these collections were once on display all but four of these collections are currently boxed-up and in storage. Furthermore the survey shows that the information pertaining to these collections is often lacking, unreliable or simply lost. It has been hypothesised that the disinterest in **classical** antiquities may have been as a result of the heritage transformation that took place in South Africa post-1994 and that, as the promotion of South African heritage has become the primary focus, more of these “Eurocentric” collections are being placed in storage. Samantha Masters‟ NRF Thuthuka-funded project has as its primary goal to research, collect data and digitise all seventeen **classical** antiquities collections in an electronic database. Another goal is to ascertain whether the shift in heritage policy post-1994 may have had an effect on the display of these collections. As a component of the broader Thuthuka project, this thesis examines **two** collections of **classical** antiquities held at the Durban University of Technology (DUT) and the University of KwaZulu-Natal (UKZN), Museum of **Classical** Archaeology in Durban, in KwaZulu-Natal. It provides a digital solution for artefact conservation, preserving the data related to the artefacts and making this data accessible for future research. In addition, this thesis ascertains how these collections were acquired, and as a result, determines how and to what extent the journeys and histories of these **two** collections have been influenced by the shift in heritage policies. Contrary to the initial hypothesis, the examination of the history of these **two** Durban collections reveals that though other collections were affected by changes in heritage policy, neither of these collections was greatly influenced by heritage transformation in post-apartheid South Africa.

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In this research work we have presented a low power design for static memory cells using NDC characteristics of SETs. The circuit parameters have been chosen to make the best trade-off between power and reliability for a particular NDC structure with about 4 times power improvement over the previous work. A new read/write circuit for the memory cell has been proposed. It has also been shown that both temperature and background charges affect the standby power and/or reliability of the memory cell in a negative way. A novel static memory cell with multiple **logic** states based on SET devices has been presented. The parameters of the SET blocks with NDC are optimized for proper memory operations. The standby power consumption is 0.98nW for 3 **logic** SET memory cell and 5.06 nW for four **logic** SET memory cell at 77K with Cg/Ct = 3 and Co=2Cg (Ct =0.1aF). A static memory using the bistable latch comprised of SET blocks has been designed and simulated. It should also be pointed out that the size of the MVL SET static memory cell is much smaller than the conventional CMOS static memory cell.

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