What Is Encubation What Is Encubation . . . How to Explain . . . Why Encubation? How to Explain . . . How to Explain . . . What About Human . . . Vladik Kreinovich, Rohan Baingolkar, Human Computations . . . Swapnil S. Chauhan, and Human Computations . . . Ishtjot S. Kamboj Home Page Department of Computer Science Title Page University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA vladik@utep.edu ◭ ◮ rubaingolkar@miners.utep.edu sschauhan@miners.utep.edu Page 1 of 9 iskambo@miners.utep.edu Go Back Full Screen Close Quit
What Is Encubation 1. What Is Encubation What Is Encubation . . . How to Explain . . . • It is known that: How to Explain . . . – some algorithms are feasible, and How to Explain . . . – some take too long to be practical. What About Human . . . Human Computations . . . • For example: Human Computations . . . – if the running time of an algorithm is 2 n , where Home Page n = len( x ) is the bit size of the input x , Title Page – then already for n = 500, the computation time ◭◭ ◮◮ exceeds the lifetime of the Universe. ◭ ◮ • In computer science, it is usually assumed that an algo- Page 2 of 9 rithm A is feasible if and only if A is polynomial-time . Go Back Full Screen Close Quit
What Is Encubation 2. What Is Encubation (cont-d) What Is Encubation . . . How to Explain . . . • In other words, an algorithm is feasible if: How to Explain . . . – its number of computational steps t A ( x ) on any in- How to Explain . . . put x What About Human . . . – is bounded by a polynomial P ( n ) of the input Human Computations . . . length n = len( x ). Human Computations . . . Home Page • An interesting encubation phenomenon is that: Title Page – once we succeed in finding a polynomial-time algo- ◭◭ ◮◮ rithm for solving a problem, ◭ ◮ – eventually it turns out to be possible to further decrease its computation time Page 3 of 9 – until we either reach the cubic time t A ( x ) ≈ n 3 or Go Back reach some even faster time n α for α < 3. Full Screen Close Quit
What Is Encubation 3. How to Explain Encubation? What Is Encubation . . . How to Explain . . . • According to modern physics, the Universe has ≈ 10 90 How to Explain . . . particles. How to Explain . . . • There are ≈ 10 42 moments of time. What About Human . . . • The number of moments of time can be obtained if we Human Computations . . . divide: Human Computations . . . Home Page – the lifetime of the Universe ( T ≈ 20 billion years) Title Page – by the smallest possible time ∆ t . ◭◭ ◮◮ • ∆ t is the time that light passes through the size-wise smallest possible stable particle – a proton. ◭ ◮ • This means that overall: Page 4 of 9 – even if each elementary particle is a processor that Go Back operates as fast as physically possible, Full Screen – the largest possible number of computational steps that we can perform is 10 90 · 10 42 = 10 132 . Close Quit
What Is Encubation 4. How to Explain Encubation (cont-d) What Is Encubation . . . How to Explain . . . • This is the largest possible number of computational How to Explain . . . steps t ( n ). How to Explain . . . • The largest possible input size comes if you input 1 bit What About Human . . . per unit time. Human Computations . . . • Thus, during the lifetime of the Universe, the largest Human Computations . . . possible length of the input is n ≈ 10 42 bits. Home Page • If an algorithm is feasible, then: Title Page ◭◭ ◮◮ – for the largest possible length n of the input – it should still perform the physically possible num- ◭ ◮ ber of steps. Page 5 of 9 • For t ( n ) ≈ n α and n ≈ 10 42 this means that Go Back t ( n ) ≈ n α ≤ 10 132 . Full Screen • Thus, we get α ≤ 132 42 = 22 7 ≈ 3. Close Quit
What Is Encubation 5. How to Explain Encubation (cont-d) What Is Encubation . . . How to Explain . . . • For t A ( n ) = n α , we got α ≤ 22 7 ≈ 3. How to Explain . . . How to Explain . . . • This is exactly what we want to explain. What About Human . . . • Comment. Since 22 7 ≈ π , maybe π and not 3 is the Human Computations . . . actual upper bound? Human Computations . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 9 Go Back Full Screen Close Quit
What Is Encubation 6. What About Human Computations? What Is Encubation . . . How to Explain . . . • What if instead computability in a computer we con- How to Explain . . . sider computability in a human brain? How to Explain . . . • Let us repeat similar computations for such human What About Human . . . computing. Human Computations . . . Human Computations . . . • A human life lasts for ≈ 80 years. Home Page • Each year has ≈ 30 million second, so overall, we get ≈ 2 . 4 · 10 9 seconds. Title Page ◭◭ ◮◮ • Brain processing is performed by neurons. ◭ ◮ • Typical neurons involved in thinking and processing Page 7 of 9 data have an operation time about 100 milliseconds. Go Back • This is about 0.1 seconds. Full Screen • Thus, during the lifetime, we have ≈ 2 . 4 · 10 10 moments of time. Close Quit
What Is Encubation 7. Human Computations (cont-d) What Is Encubation . . . How to Explain . . . • There are about 10 10 neuron in a brain. How to Explain . . . • Thus, overall: How to Explain . . . What About Human . . . – if all the neurons are active all the time, Human Computations . . . – we can perform t ( n ) ≈ (2 . 4 · 10 10 ) · 10 10 ≈ 10 20 Human Computations . . . computational steps. Home Page • Similarly to the physical case: Title Page – we can gauge the largest possible size ◭◭ ◮◮ – by assuming that enter 1 bit every single moment ◭ ◮ of time. Page 8 of 9 • Thus, the largest input size is n ≈ 10 10 . Go Back Full Screen Close Quit
8. Human Computations (cont-d) What Is Encubation What Is Encubation . . . How to Explain . . . • Similarly to the physical case, let us check for which α : How to Explain . . . How to Explain . . . – the number of computational steps t ( n ) needed to What About Human . . . process the largest possible input n ≈ 10 10 Human Computations . . . Human Computations . . . – does not exceed the largest possible number of com- putation al steps: t ( n ) = n α ≤ 10 20 . Home Page • In this case, we conclude that α ≤ 2. Title Page • So, only quadratic-time (and faster) algorithms are fea- ◭◭ ◮◮ sible in terms of human computations. ◭ ◮ • This makes sense; for example: Page 9 of 9 – sorting algorithms that describe how we sort by Go Back hand (such as insertion sort), Full Screen – are indeed quadratic-time. Close Quit
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