# Questions tagged [boolean-complexity]

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### Shannon's result that some Boolean functions require exponential circuits

In 1949 Shannon proved, using a non-constructive counting argument, that some boolean functions have exponential circuit complexity, see [1] and many texts on computational complexity. This result has ...
• 8,328
67 views

### Is $f(X)f^d(X) = 0$ for a Boolean function $f$?

I'm currently trying to understand a step in the proof for in the Crama and Hammer book on Boolean Functions. The proof is Proposition 4.12, which claims that the self-dualization of Boolean $f$ is ...
• 429
29 views

### sum of Boolean characters larger degree

I was curious if someone knew the answer/reference for the following. So it is well-known that if $S\in \{0,1\}^n$, then $$\frac{1}{2^n}\sum_{x\in \{0,1\}^n} (-1)^{\langle S, x\rangle}=1$$ if and ...
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### Approximate the parity function in L1-norm

Consider the parity function $MOD_2(x) = x_1 \oplus \cdots \oplus x_n$ for $x \in \mathbb{F}_2^n$. I am concerned about the degree bounds for a real polynomial $f$ which approximates $MOD_2$ well in ...
• 21
77 views

### Lower Bound on Parity of Boolean Functions

Let's say we have boolean functions $f_1, \cdots, f_n$, each of which operates on pairwise disjoint variables (i.e. the variables for each function are unique to that function). Then, how can we show ...
• 23
1 vote
72 views

### Can Boolean circuits of polylog depth represent all Boolean functions?

Consider a Boolean circuit using (2-input) logical-and, (2-input) logical-or and logical-not as basic components. The depth of the Boolean circuit is the length of the longest path from the input to ...
• 296
1 vote
120 views

### How to construct a carry-lookahead adder of the optimal $O(n)$ size

Problem (TL;DR): I'd like to know how to construct a CLA adder that has $O(n)$ size and $O(\log n)$ depth using only fan-in 2 AND gates and XOR gates, as suggested in this answer and this answer. ...
• 31
1 vote
83 views

### How many different boolean functions exist up to permutation of its $n$ variables

i am relatively new here, so if this was asked before, feel free to redirect me. I am searching for an answer in form of a (iterative or recursive) Formula or even better, an algorithm to list them ...
1 vote
35 views

### Boolean function represented as a column vector: easy way to see if it has full degree?

If you have a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ on $n$ variables, you can represent this function as a $2^n$ vector indexed by the input space such that $f_x = f(x)$. For ...
• 991
1 vote
44 views

### Algorithm design: Model redundancy in tests

I've run across an interesting problem at work that I'm not quite sure how to grapple. Broadly, there is a suite of of $n$ tests to ensure the quality of a product. However, the tests are both time-...
• 11
1 vote
20 views

### Why is End-Of-The-Line defined in terms of "Arithmetic circuits" instead of "Boolean circuits"

The definition of PPAD (Polynomial parity arguments on directed graphs) revolves around the definition of "End-Of-The-Line" An exponentially large polynomial-depth arithmetic circuit, $f$, ...
• 327
1 vote
67 views

### Acyclic boolean circuit (DAG)

If a function f has a while loop or for loop, can I compile this function into an ...
1 vote
56 views

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