CS3102 Theory of Computation Warm up: Is this function computable - - PowerPoint PPT Presentation

CS3102 Theory of Computation

Warm up: Is this function computable with a NAND-Circuit? 𝑔 𝑏 = 1 if UVA wins against ND tonight 0 otherwise

Logistics

• Quiz 3 out

– Due Thursday

• Exercise 3 is out

– Last exercise before midterm

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Last Time

• Showing that NAND/AON can compute any

finite function

• Started showing how we could have a

function that simulates programs

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How are programs run?

• Have a table of variables
• Execute code in sequence
• Update values in table
• Return a value from the table

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Programs as Bits

• To evaluate a program with another program, we need to

convert the first program into bits 1. Number each variable (first n go to input, last m to outputs) 2. Represent each line as 3 numbers (outvar, in1, in2) 3. Represent program as (n,m,[Lines])

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Variable Number a b 1 temp1 2 temp2 3 return 4 (2,0,0) (3,1,1) (4,2,3) (2,1,[(2,0,0),(3,1,1),(4,2,3)])

XOR to bits

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Variable Number 𝑜 = 𝑛 = 𝑡 = Total bits =

XOR to bits

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Variable Number a b 1 u 2 v 3 w 4 return 5 𝑜 = 2 𝑛 = 1 𝑡 = 4 Total bits = 3 [numbers per line] ⋅ 3 [bits per number] ⋅ 4[lines] + 6 [length of 𝑜 + 𝑛]

How big is this?

1. Number each variable 2. Represent each line as 3 numbers (outvar, in1, in2) 3. Represent program as (n,m,[Lines])

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⌈log2 3𝑡⌉ bits each

𝑇 𝑡 ≤ 4𝑡⌈log2 3𝑡⌉ ℓ = ⌈log2 3𝑡⌉

3𝑡 ⋅ ⌈log2 3𝑡⌉ bits 2⌈log2 𝑡⌉ bits

Defining EVAL

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Input: bit string representing a program (first 𝑇(𝑡) bits) plus input values (remaining 𝑜 bits) Output: the result of running the represented program

• n the provided input, or 𝑛 0's if there's a "compile

error"

𝐹𝑊𝐵𝑀𝑡,𝑜,𝑛: 0,1 𝑇 𝑡 +𝑜 → 0,1 𝑛

Defining the EVAL function

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Variable Value 1 2 3 4 5 Representation: (2, 0, 1), (3, 0, 2), (4, 1, 2), (5, 3, 4) Input: 0, 1

𝑜 = 2 𝑛 = 1

Psuedocode for EVAL

• Table 𝑈:

– holds variables and their values

• 𝐻𝐹𝑈(𝑈, 𝑗)

– Returns the bit of 𝑈 associated with variable 𝑗

• 𝑉𝑄𝐸𝐵𝑈𝐹(𝑈, 𝑗, 𝑐)

– Returns a new table such that variable 𝑗's value has been changed to 𝑐

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𝑈

Psuedocode for EVAL

• Input:

– Numbers 𝑜, 𝑛, 𝑡, 𝑢 representing the number of inputs, outputs, variables, and lines respectively – 𝑀, a list of triples representing the program – A string 𝑦 to be given as input to the program

• Output:

– Evaluation of the program represented by 𝑀 when run

• n input 𝑦

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Let 𝑈 be table of size 𝑢 For 𝑗 in range(𝑜): 𝑈 = UPDATE(𝑈, 𝑗, 𝑦[𝑗]) For (𝑗,𝑘,𝑙) in 𝑀: 𝑏 = GET(𝑈, 𝑘) 𝑐 = GET(𝑈, 𝑙) 𝑈 = UPDATE(𝑈, 𝑗, NAND(𝑏,𝑐)) For 𝑗 in range(𝑛): 𝑍[𝑗] = GET(𝑈, 𝑢 − 𝑛 + 𝑗) Return 𝑍

EVAL in NAND

• Next we implement 𝐹𝑊𝐵𝑀𝑡,𝑜,𝑛 using NAND

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𝐻𝐹𝑈(𝑈, 𝑗)

• Get the bit at “row” 𝑗 of 𝑈
• Look familiar?
• How many gates to implement?

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UPDATE

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• To change index 𝑗 of table 𝑈 to bit 𝑐
• For every index except 𝑗, return the same value
• For index 𝑗, return b instead
• Define 𝐹𝑅𝑉𝐵𝑀𝑘: 0,1 ℓ → {0,1} which returns 1 if

the input binary number is equal to j

Note: 𝐹𝑅𝑉𝐵𝑀𝑘 can be done in 𝑑 ⋅ ℓ gates 𝑉𝑄𝐸𝐵𝑈𝐹ℓ: 0,1 𝑡ℓ+ℓ+1 → 0,1 2ℓ

UPDATE pseudocode

For 𝑘 in range(2ℓ): 𝑏 = 𝐹𝑅𝑉𝐵𝑀𝑇

𝑘(𝑗)

𝑜𝑓𝑥𝑈[𝑘] = 𝐽𝐺(𝑏, 𝑐, 𝑈[𝑘]) Return 𝑜𝑓𝑥𝑈

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Runs 2ℓ times

Conclusion

• What we know:

– We can compute any finite function with circuits – We can compute a function to evaluate programs of a certain size

• Big question:

– How expensive are functions? – Some are more expensive than others, how big could they get? – If I wanted to be able to evaluate a program for any function 0,1 𝑜 → {0,1}, how big would the eval circuit need to be?

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Complexity

• The "complexity" of a function:

– Measure of the resources required to compute that function

• Complexity Class:

– A set of functions defined by a complexity measure

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Categorizing Functions by Circuit Size

• No functions require more than 𝑑𝑛2𝑜 gates

– Proved last class

• Some functions require much less

– E.g. IF

• Observation: some functions are more "complicated"

than others!

• Idea: categorize functions by resources required to

implement them using a particular computing model

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SIZE

• 𝑇𝐽𝑎𝐹(𝑡): The set of all functions that can be

implemented by a circuit of at most s NAND gates

• TCS also uses:

– 𝑇𝐽𝑎𝐹𝑜,𝑛 𝑡 :The set of all 𝑜-input, 𝑛-output functions that can be implemented with at most 𝑡 NAND gates – 𝑇𝐽𝑎𝐹𝑜 𝑡 : The set of all 𝑜-input, 1-output functions that can be implemented with at most 𝑡 NAND gates

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𝑇𝐽𝑎𝐹(1000𝑛2𝑜) Contains all functions 𝑔: 0,1 𝑜 → 0,1 𝑛

Comparing Classes

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If 𝑦 ≤ 𝑧, then 𝑇𝐽𝑎𝐹 𝑡 ⊆ 𝑇𝐽𝑎𝐹(𝑧)

Theorem

• Let 𝑇𝐽𝑎𝐹𝐵𝑃𝑂 𝑡 represent the set of all

functions that can be computed using at most 𝑡 AND/OR/NOT gates

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𝑇𝐽𝑎𝐹 𝑡 2 ⊆ 𝑇𝐽𝑎𝐹𝐵𝑃𝑂 𝑡 ⊆ 𝑇𝐽𝑎𝐹 3𝑡

Proof

𝑇𝐽𝑎𝐹 𝑡 2 ⊆ 𝑇𝐽𝑎𝐹𝐵𝑃𝑂 𝑡 ⊆ 𝑇𝐽𝑎𝐹 3𝑡

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O, Ω, Θ

• Groups functions together
• Each uses a function as a bound for other functions
• O (Big-Oh):

– O(f(n)) = the set of all functions "asymptotically upper-bounded" by f

• Ω (Big-Omega):

– Ω(f(n)) = the set of all functions "asymptotically lower-bounded" by f

• Θ (Big-Theta):

– Θ(f(n)) = the set of all functions "asymptotically tight-bounded" by f

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Definitions

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• 𝑃(𝑕 𝑜 )
• At most within constant of 𝑕 for large 𝑜
• {𝑔: ℝ → ℝ|∃ constants 𝑑, 𝑜0 > 0 s.t. ∀𝑜 > 𝑜0, 𝑔 𝑜 ≤ 𝑑 ⋅ 𝑕(𝑜)}
• Ω(𝑕 𝑜 )
• At least within constant of 𝑕 for large 𝑜
• {𝑔: ℝ → ℝ|∃ constants 𝑑, 𝑜0 > 0 s.t. ∀𝑜 > 𝑜0, 𝑔 𝑜 ≥ 𝑑 ⋅ 𝑕(𝑜)}
• Θ 𝑕 𝑜
• “Tightly” within constant of 𝑕 for large 𝑜
• Ω 𝑕 𝑜

∩ 𝑃(𝑕 𝑜 )

Showing Big-Oh

• To show: 𝑜 log 𝑜 ∈ 𝑃 𝑜2

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Showing Big-Omega

• To Show: 2𝑜 ∈ Ω 𝑜2

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Showing Big-Theta

• To Show: log𝑦 𝑜 = Θ log𝑧 𝑜

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