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Shannon counting gives an efficiently explicit circuit-hard function
Statement
The Shannon counting argument itself gives a polynomial-time computable family of Boolean functions having exponential circuit complexity.
Facts & Assumptions
Given: the Shannon counting argument.
Counting circuits gives only an upper bound on how many truth tables they represent, by Counting bounded-size Boolean circuits.
The resulting theorem says that almost all truth tables are hard, by Almost all Boolean functions require exponential circuit size.
Refutation
The argument in [L1] and [L2] compares two cardinalities and concludes that the complement of the set of small-circuit truth tables is nonempty (and large). One can make an ineffective-for-complexity selection computable by enumerating every bounded-size circuit, forming all of their truth tables, and choosing the lexicographically first missing table. That exhaustive procedure, however, takes time exponential (indeed much larger) in the truth-table length and supplies no polynomial-time algorithm for evaluating the selected function on an input.
Consequently the counting proof establishes existential hardness and even permits a brute-force computable choice, but it does not by itself produce an efficiently explicit hard family. Such a family requires an additional efficient construction and lower-bound argument absent from the count.
This distinction refutes the claim that the Shannon counting argument itself supplies a polynomial-time computable hard family.
Depends on
Used by
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Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Arora and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
- Lance Fortnow, Counting Complexity (standard reference, not scraped)