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Stochastic computing is a collection of techniques that represent continuous values by streams of random bits. Complex computations can then be computed by simple bit-wise operations on the streams.

Despite the similarity in their names, stochastic computing is distinct from the study of randomized algorithms.

Motivation and a simple example

Suppose that $p,q\in [0,1]$ is given, and we wish to compute $p\times q$ . Stochastic computing performs this operation using probability instead of arithmetic.

Specifically, suppose that there are two random, independent bit streams called stochastic numbers (i.e. Bernoulli processes), where the probability of a one in the first stream is $p$ , and the probability in the second stream is $q$ . We can take the logical AND of the two streams.

$a_{i}$	1	0	1	1	0	1	...
$b_{i}$	1	1	0	1	1	0	...
$a_{i}\land b_{i}$	1	0	0	1	0	0	...

The probability of a one in the output stream is $pq$ . By observing enough output bits and measuring the frequency of ones, it is possible to estimate $pq$ to arbitrary accuracy.

The operation above converts a fairly complicated computation (multiplication of $p$ and $q$ ) into a series of very simple operations (evaluation of $a_{i}\land b_{i}$ ) on random bits.

More generally speaking, stochastic computing represents numbers as streams of random bits and reconstructs numbers by calculating frequencies. The computations are performed on the streams and translate complicated operations on $p$ and $q$ into simple operations on their stream representations. (Because of the method of reconstruction, devices that perform these operations are sometimes called stochastic averaging processors.) In modern terms, stochastic computing can be viewed as an interpretation of calculations in probabilistic terms, which are then evaluated with a Gibbs sampler. It can also be interpreted as a hybrid analog/digital computer.

History

A photograph of the RASCEL stochastic computer.

The RASCEL stochastic computer, circa 1969

Stochastic computing was first introduced in a pioneering paper by John von Neumann in 1953.^[1] However, the theory could not be fully developed until advances in computing of the 1960s,^[2] ^[3] mostly through a series of simultaneous and parallel efforts in the US^[4] and the UK.^[5] By the late 1960s, attention turned to the design of special-purpose hardware to perform stochastic computation. A host^[6] of these machines were constructed between 1969 and 1974; RASCEL^[7] is pictured in this article.

Despite the intense interest in the 1960s and 1970s, stochastic computing ultimately failed to compete with more traditional digital logic, for reasons outlined below. The first (and last) International Symposium on Stochastic Computing^[8] took place in 1978; active research in the area dwindled over the next few years.

Although stochastic computing declined as a general method of computing, it has shown promise in several applications. Research has traditionally focused on certain tasks in machine learning and control.^[9] ^[10] Somewhat recently, interest has turned towards stochastic decoding, which applies stochastic computing to the decoding of error correcting codes.^[11] More recently, stochastic circuits have been successfully used in image processing tasks such as edge detection. ^[12]

Strengths and weaknesses

Although stochastic computing was a historical failure, it may still remain relevant for solving certain problems. To understand when it remains relevant, it is useful to compare stochastic computing with more traditional methods of digital computing.

Strengths

Suppose we wish to multiply two numbers each with $n$ bits of precision. Using the typical long multiplication method, we need to perform $n^{2}$ operations. With stochastic computing, we can AND together any number of bits and the expected value will always be correct. (However, with a small number of samples the variance will render the actual result highly inaccurate).

Moreover, the underlying operations in a digital multiplier are full adders, whereas a stochastic computer only requires an AND gate. Additionally, a digital multiplier would naively require $2n$ input wires, whereas a stochastic multiplier would only require 2 input wires. (If the digital multiplier serialized its output, however, it would also require only 2 input wires.)

Additionally, stochastic computing is robust against noise; if a few bits in a stream are flipped, those errors will have no significant impact on the solution.

Finally, stochastic computing provides an estimate of the solution that grows more accurate as we extend the bit stream. In particular, it provides a rough estimate very rapidly. This property is usually referred to as progressive precision, which suggests that the precision of stochastic numbers (bit streams) increases as computation proceeds. ^[13] It is as if the most significant bits of the number arrive before its least significant bits; unlike the conventional arithmetic circuits where the most significant bits usually arrive last. In some iterative systems the partial solutions obtained through progressive precision can provide faster feedback than through traditional computing methods, leading to faster convergence.

Weaknesses

Stochastic computing is, by its very nature, random. When we examine a random bit stream and try to reconstruct the underlying value, the effective precision can be measured by the variance of our sample. In the example above, the digital multiplier computes a number to $2n$ bits of accuracy, so the precision is $2^{-2n}$ . If we are using a random bit stream to estimate a number and want the standard deviation of our estimate of the solution to be at least $2^{-2n}$ , we would need $O(2^{4n})$ samples. This represents an exponential increase in work. In certain applications, however, the progressive precision property of stochastic computing can be exploited to compensate this exponential loss.

Second, stochastic computing requires a method of generating random biased bit streams. In practice, these streams are generated with pseudo-random number generators. Unfortunately, generating (pseudo-)random bits is fairly costly (compared to the expense of, e.g., a full adder). Therefore, the gate-level advantage of stochastic computing is typically lost.

Third, the analysis of stochastic computing assumes that the bit streams are independent (uncorrelated). If this assumption does not hold, stochastic computing can fail dramatically. For instance, if we try to compute $p^{2}$ by multiplying a bit stream for $p$ by itself, the process fails: since $a_{i}\land a_{i}=a_{i}$ , the stochastic computation would yield $p\times p=p$ , which is not generally true (unless $p=$ 0 or 1). In systems with feedback, the problem of decorrelation can manifest in more complicated ways. Systems of stochastic processors are prone to latching, where feedback between different components can achieve a deadlocked state.^[14] A great deal of effort must be spent decorrelating the system to attempt to remediate latching.

Fourth, although some digital functions have very simple stochastic counterparts (such as the translation between multiplication and the AND gate), many do not. Trying to express these functions stochastically may cause various pathologies. For instance, stochastic decoding requires the computation of the function $f(p,q)\rightarrow pq/(pq+(1-p)(1-q))$ . There is no single bit operation that can compute this function; the usual solution involves producing correlated output bits, which, as we have seen above, can cause a host of problems. Still other functions (such as the averaging operator $f(p,q)\rightarrow (p+q)/2$ ) require stream decimation. Since decimation discards information, it leads to the problem of attenuation.

Stochastic decoding

Although stochastic computing has a number of defects when considered as a method of general computation, there are certain applications that highlight its strengths. One notable case occurs in the decoding of certain error correcting codes.

In developments unrelated to stochastic computing, highly effective methods of decoding LDPC codes using the belief propagation algorithm were developed. Belief propagation in this context involves iteratively reestimating certain parameters using two basic operations (essentially, a probabilistic XOR operation and an averaging operation).

In 2003, researchers realized that these two operations could be modeled very simply with stochastic computing.^[15] Moreover, since the belief propagation algorithm is iterative, stochastic computing provides partial solutions that may lead to faster convergence. Hardware implementations of stochastic decoders have been built on FPGAs. ^[16] The proponents of these methods argue that the performance of stochastic decoding is competitive with digital alternatives.

Variants of stochastic computing

There are a number of variants of the basic stochastic computing paradigm. Further information can be found in the referenced book by Mars and Poppelbaum.

Bundle Processing involves sending a fixed number of bits instead of a stream. One of the advantages of this approach is that the precision is improved. To see why, suppose we transmit $s$ bits. In regular stochastic computing, we can represent a precision of roughly $O(1/{\sqrt {s}})$ different values, because of the variance of the estimate. In bundle processing, we can represent a precision of $1/s$ . However, bundle processing retains the same robustness to error of regular stochastic processing.

Ergodic Processing involves sending a stream of bundles, which captures the benefits of regular stochastic and bundle processing.

Burst Processing encodes a number by a higher base increasing stream. For instance, we would encode 4.3 with ten decimal digits as

4444444555

since the average value of the preceding stream is 4.3. This representation offers various advantages: there is no randomization since the numbers appear in increasing order, so the PRNG issues are avoided, but many of the advantages of stochastic computing are retained (such as partial estimates of the solution). Additionally, it retains the linear precision of bundle and ergodic processing.

References

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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↑ 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Template:Cite doi
↑ Template:Cite doi
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[1] 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[2] 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[3] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[4] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[5] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[6] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[7] Template:Cite thesis

[8] 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[9] 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[10] 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[11] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[12] Template:Cite doi

[13] Template:Cite doi

[14] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

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Schoenflies problem: Difference between revisions

Revision as of 21:15, 7 December 2013

Contents

Motivation and a simple example

History

Strengths and weaknesses

Strengths

Weaknesses

Stochastic decoding

Variants of stochastic computing

References

Further reading

Navigation menu

@@ Line 1: / Line 1: @@
+'''Stochastic computing''' is a collection of techniques that represent continuous values by streams of random bits. Complex computations can then be computed by simple bit-wise operations on the streams.
+Despite the similarity in their names, stochastic computing is distinct from the study of [[randomized algorithm]]s.
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+== Motivation and a simple example ==
+Suppose that <math>p,q \in [0,1]</math> is given, and we wish to compute <math>p \times q</math>. Stochastic computing performs this operation using probability instead of arithmetic.
+Specifically, suppose that there are two random, independent bit streams called ''stochastic number''s (i.e. [[Bernoulli process]]es), where the probability of a one in the first stream is <math>p</math>, and the probability in the second stream is <math>q</math>. We can take the [[Logical_and#Applications_in_computer_programming|logical AND]] of the two streams.
+{| class="wikitable" style="text-align:left;"
+! <math>a_i</math>
+| 1 || 0 || 1 || 1 || 0 || 1 || ...
+|-
+! <math>b_i</math>
+| 1 || 1 || 0 || 1 || 1 || 0 || ...
+|-
+! <math>a_i \land b_i</math>
+| 1 || 0 || 0 || 1 || 0 || 0 || ...
+|}
+The probability of a one in the output stream is <math>pq</math>.  By observing enough output bits and measuring the frequency of ones, it is possible to estimate <math>pq</math> to arbitrary accuracy.
+The operation above converts a fairly complicated computation (multiplication of <math>p</math> and <math>q</math>) into a series of very simple operations (evaluation of <math>a_i \land b_i</math>) on random bits.
+More generally speaking, stochastic computing represents numbers as streams of random bits and reconstructs numbers by calculating frequencies.  The computations are performed on the streams and translate complicated operations on <math>p</math> and <math>q</math> into simple operations on their stream representations.  (Because of the method of reconstruction, devices that perform these operations are sometimes called stochastic averaging processors.)  In modern terms, stochastic computing can be viewed as an interpretation of calculations in probabilistic terms, which are then evaluated with a [[Gibbs sampling|Gibbs sampler]]. It can also be interpreted as a hybrid [[Analog computer|analog]]/[[Computer|digital]] computer.
+== History ==
+[[Image:RASCEL stochastic computer 1969.png|thumb|alt=A photograph of the RASCEL stochastic computer.|The RASCEL stochastic computer, circa 1969]]
+Stochastic computing was first introduced in a pioneering paper by [[John von Neumann]] in 1953.<ref>{{cite conference |
+last = von Neumann | first = J.  | title = Probabilistic logics and the synthesis of reliable organisms from unreliable components |
+booktitle = The Collected Works of John von Neumann | publisher =
+Macmillan | year = 1963 | isbn = 978-0-393-05169-8}}</ref> However, the
+theory could not be fully developed until advances in computing of the 1960s,<ref>{{cite conference
+| last1 = Petrovic | first1= R. | last2=Siljak | first2=D. | title=
+Multiplication by means of coincidence | year = 1962 | booktitle =
+ACTES Proc. of 3rd Int. Analog Comp. Meeting}}</ref>
+<ref>
+{{citation
+| last=Afuso
+| first=C.
+| title=Quart. Tech. Prog. Rept.
+| location=Department of Computer Science, University of Illinois, Urbana, Illinois
+| year=1964
+}}</ref>
+mostly through a series of simultaneous and parallel efforts in the US<ref>
+{{cite journal
+| last1=Poppelbaum
+| first1=W.
+| last2=Afuso
+| first2=C.
+| last3=Esch
+| first3=J.
+| title=Stochastic computing elements and systems
+| journal=AFIPS FJCC
+| volume=31
+| pages=635–644
+| year=1967
+}}</ref>
+and the UK.<ref>
+{{cite journal
+| last=Gaines
+| first=B.
+| title=Stochastic Computing
+| journal=AFIPS SJCC
+| year=1967
+| volume=30
+| pages=149–156
+}}</ref>
+By the late 1960s, attention turned to the design of
+special-purpose hardware to perform stochastic computation.  A host<ref>
+{{cite book
+| last1=Mars
+| first1=P.
+| last2=Poppelbaum
+| first2=W.
+| title=Stochastic and deterministic averaging processors
+| year=1981
+}}
+</ref>
+of these machines were constructed between 1969 and 1974; RASCEL<ref>
+{{cite thesis
+| last=Esch
+| first=J.
+| title=RASCEL, a programmable analog computer based on a regular array of stochastic computing element logic
+| year=1969
+| location=University of Illinois, Urbana, Illinois
+}}
+</ref>
+is pictured in this article.
+Despite the intense interest in the 1960s and 1970s, stochastic
+computing ultimately failed to compete with more traditional digital
+logic, for reasons outlined below.  The first (and last)
+International Symposium on Stochastic Computing<ref>
+{{cite conference
+| title=Proceedings of the first International Symposium on Stochastic Computing and its Applications
+| location= Toulouse, France
+| year=1978
+}}
+</ref>
+took place in 1978; active research in the area dwindled over the next
+few years.
+Although stochastic computing declined as a general method of
+computing, it has shown promise in several applications.  Research has
+traditionally focused on certain tasks in machine learning and
+control.<ref>
+{{cite conference
+| booktitle=Advances in Information Systems Science
+| title=Stochastic Computing Systems
+| last=Gaines
+| first= B. R.
+| editor-last=Tou
+| editor-first=Julius
+| publisher=Plenum Press
+| volume=2
+| year=1969
+}}
+</ref>
+<ref>
+{{cite conference
+| booktitle=FPGAs for Custom Computing Machines, Proceedings IEEE, NAPA
+| title=A stochastic neural architecture that exploits dynamically reconfigurable FPGAs
+| last=van Daalen, M. R. et al
+| year=1993
+}}
+</ref>
+Somewhat recently, interest has turned towards stochastic
+decoding, which applies stochastic computing to the decoding of error
+correcting codes.<ref>
+{{cite journal
+| title=Iterative decoding  using stochastic computation
+| last1=Gaudet
+| first1=Vincent
+| last2=Rapley
+| first2=Anthony
+| journal=Electronic Letters
+| volume=39
+| number=3
+| pages=299–301
+|date=February 2003
+}}
+</ref> More recently, stochastic circuits have been successfully used in [[image processing]] tasks such as [[edge detection]].
+<ref>{{Cite doi|10.1145/2463209.2488901}}</ref>
+== Strengths and weaknesses ==
+Although stochastic computing was a historical failure, it may still remain relevant for
+solving certain problems.  To understand when it remains relevant, it is useful to
+compare stochastic computing with more traditional methods of digital computing.
+=== Strengths ===
+Suppose we wish to multiply
+two numbers each with <math>n</math> bits of precision.
+Using the typical [[Multiplication_algorithm#Long_multiplication|long
+multiplication]] method, we need to perform
+<math>n^2</math> operations.  With stochastic computing, we can
+AND together any number of bits and the expected value will always
+be correct.  (However, with a small number of samples the variance will
+render the actual result highly inaccurate).
+Moreover, the underlying operations in a digital multiplier are
+[[Adder_(electronics)#Full_adder|full adders]], whereas a stochastic
+computer only requires an [[And gate|AND gate]].  Additionally,
+a digital multiplier would naively require <math>2n</math> input wires,
+whereas a stochastic multiplier would only require 2 input wires.
+(If the digital multiplier serialized its output, however, it would also
+require only 2 input wires.)
+Additionally, stochastic computing is robust against noise; if a few
+bits in a stream are flipped, those errors will have no significant impact
+on the solution.
+Finally, stochastic computing provides an estimate of the solution
+that grows more accurate as we extend the bit stream.  In particular,
+it provides a rough estimate very rapidly.  This property is usually
+referred to as ''progressive precision'', which suggests that the precision
+of stochastic numbers (bit streams) increases as computation proceeds.
+<ref>{{Cite doi|10.1145/2465787.2465794}}</ref>
+It is as if the [[most significant bit]]s of the number
+arrive before its [[least significant bit]]s; unlike the
+conventional [[Arithmetic logic unit|arithmetic circuits]] where the most
+significant bits usually arrive last. In some
+iterative systems the partial solutions obtained through progressive precision can provide faster feedback
+than through traditional computing methods, leading to faster
+convergence.
+=== Weaknesses ===
+Stochastic computing is, by its very nature, random.  When we examine
+a random bit stream and try to reconstruct the underlying value, the effective precision
+can be measured by the variance of our sample.  In the example above, the digital multiplier
+computes a number to <math> 2n </math> bits of accuracy, so the
+precision is <math> 2^{-2n}</math>.  If we are using a random bit
+stream to estimate a number and want the standard deviation of our
+estimate of the solution to be at least <math> 2^{-2n}</math>, we
+would need <math> O(2^{4n}) </math> samples.  This represents an
+exponential increase in work. In certain applications, however, the
+progressive precision property of stochastic computing can be exploited
+to compensate this exponential loss.
+Second, stochastic computing requires a method of generating random
+biased bit streams.  In practice, these streams are generated with
+[[PRNG|pseudo-random number generators]].  Unfortunately, generating
+(pseudo-)random bits is fairly costly (compared to the expense of,
+e.g., a full adder).  Therefore, the gate-level advantage of
+stochastic computing is typically lost.
+Third, the analysis of stochastic computing assumes that the bit
+streams are independent (uncorrelated).  If this assumption does not
+hold, stochastic computing can fail dramatically.  For instance, if we
+try to compute <math>p^2</math> by multiplying a bit stream for
+<math>p</math> by itself, the process fails: since <math>a_i \land
+a_i=a_i</math>, the stochastic computation would yield <math> p \times p
+= p </math>, which is not generally true (unless <math>p=</math>0 or 1).
+In systems with feedback, the problem of decorrelation can manifest in
+more complicated ways.  Systems of stochastic processors are prone to
+''latching'', where feedback between different components can achieve
+a deadlocked state.<ref>
+{{cite journal
+| last1=Winstead
+| first1=C.
+| last2=Rapley
+| first2=A.
+| last3=Gaudet
+| first3=V.
+| last4=Schlegel
+| first4=C.
+| title=Stochastic iterative decoders
+| journal=IEEE International Symposium on Information Theory
+| location=Adelaide Australia
+|date=September 2005
+}}
+</ref>
+A great deal of effort must be spent decorrelating the system to
+attempt to remediate latching.
+Fourth, although some digital functions have very simple stochastic
+counterparts (such as the translation between multiplication and the
+AND gate), many do not.  Trying to express these functions stochastically
+may cause various pathologies.  For instance, stochastic decoding requires
+the computation of the function <math>f(p,q)\rightarrow pq/(pq + (1-p)(1-q))</math>.
+There is no single bit operation that can compute this function; the usual solution
+involves producing correlated output bits, which, as we have seen above, can cause
+a host of problems.  Still other functions (such as the
+averaging operator <math>f(p,q)\rightarrow (p+q)/2</math>) require
+stream decimation.  Since decimation discards information, it leads to the problem
+of ''attenuation''.
+== Stochastic decoding ==
+Although stochastic computing has a number of defects when considered
+as a method of general computation, there are certain applications
+that highlight its strengths.  One notable case occurs in the
+decoding of certain error correcting codes.
+In developments unrelated to stochastic computing, highly effective
+methods of decoding [[Low-density parity-check code|LDPC codes]] using
+the [[belief propagation]] algorithm were
+developed.  Belief propagation in this context involves iteratively
+reestimating certain parameters using two basic operations
+(essentially, a probabilistic XOR operation and an averaging
+operation).
+In 2003, researchers realized that these two operations could be
+modeled very simply with stochastic computing.<ref>
+{{cite journal
+| title=Iterative decoding  using stochastic computation
+| last1=Gaudet
+| first1=Vincent
+| last2=Rapley
+| first2=Anthony
+| journal=Electronic Letters
+| volume=39
+| number=3
+| pages=299–301
+|date=February 2003
+}}
+</ref>
+Moreover, since the
+belief propagation algorithm is iterative, stochastic computing provides partial
+solutions that may lead to faster convergence.
+Hardware implementations of stochastic decoders have been built on [[Field-programmable gate array|FPGAs]].
+<ref>
+{{cite conference
+| title=Stochastic implementation of LDPC decoders
+| last1=Gross
+| first1=W.
+| last2=Gaudet
+| first2=V.
+| last3=Milner
+| first3=A.
+| booktitle=Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers
+| year=2006
+}}
+</ref>
+The proponents of these methods argue that the performance of stochastic decoding is
+competitive with digital alternatives.
+== Variants of stochastic computing ==
+There are a number of variants of the basic stochastic computing
+paradigm.  Further information can be found in the referenced book by
+Mars and Poppelbaum.
+''Bundle Processing'' involves sending a fixed number of
+bits instead of a stream.  One of the advantages of this approach is
+that the precision is improved.  To see why, suppose we transmit
+<math>s</math> bits.  In regular stochastic computing, we can
+represent a precision of roughly <math>O(1/\sqrt{s})</math> different
+values, because of the variance of the estimate.  In bundle
+processing, we can represent a precision of <math>1/s</math>.
+However, bundle processing retains the same robustness to error of
+regular stochastic processing.
+''Ergodic Processing'' involves sending a stream of bundles, which
+captures the benefits of regular stochastic and bundle processing.
+''Burst Processing'' encodes a number by a higher base increasing
+stream.  For instance, we would encode 4.3 with ten decimal digits as
+::: 4444444555
+since the average value of the preceding stream is 4.3.  This
+representation offers various advantages: there is no randomization
+since the numbers appear in increasing order,
+so the PRNG issues are avoided, but many of the advantages of
+stochastic computing are retained (such as partial estimates of the
+solution).  Additionally, it retains the linear precision of bundle
+and ergodic processing.
+==References==
+{{reflist}}
+==Further reading==
+* {{cite journal|url=http://pages.cpsc.ucalgary.ca/~gaines/reports/COMP/IdentSC/IdentSC.pdf|title=Techniques of Identification with the Stochastic Computer|last=Gaines|first=Brian R. |journal=Proceedings IFAC Symposium on "The Problems of Identification in Automatic Control Systems", Section 6 Special Identification Instruments, Prague June 12–19, 1967|year=1967|accessdate=2013-11-11}}
+* {{cite journal|url=http://web.eecs.umich.edu/~alaghi/ACM_TECS_2013.pdf|title=Survey of Stochastic Computing|last1=Alaghi|first1=Armin|last2=Hayes|first2=John P.|journal=ACM Transactions on Embedded Computing Systems|year=2013|accessdate=2013-11-11}}
+[[Category:History of computing hardware]]
+[[Category:Models of computation]]
+[[Category:Stochastic algorithms]]

Schoenflies problem: Difference between revisions

Revision as of 21:15, 7 December 2013

Motivation and a simple example

History

Strengths and weaknesses

Strengths

Weaknesses

Stochastic decoding

Variants of stochastic computing

References

Further reading

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