We investigate the complexity of the syntactic isomorphism problem of CNF Boolean Formulas (CSFI): given two CNF Boolean formulas $\varphi(a_{1},\ldots,a_{n})$ and $\varphi(b_{1},\ldots,b_{n})$ decide whether there exists a permutation of clauses, a permutation of literals and a bijection between their variables such that $\varphi(a_{1},\ldots,a_{n})$ and $\varphi(b_{1},\ldots,b_{n})$ become syntactically identical. We first ... more >>>

Koiran's real $\tau$-conjecture asserts that if a non-zero real polynomial can be written as $f=\sum_{i=1}^{p}\prod_{j=1}^{q}f_{ij},$
where each $f_{ij}$ contains at most $k$ monomials, then the number of distinct real roots of $f$ is polynomial in $pqk$. We show that the conjecture implies quite a strong property of the ... more >>>

In this survey, I discuss the general question of what evidence can we use to predict the answer for open questions in computational complexity, as well as the concrete evidence currently known for two conjectures: Khot's Unique Games Conjecture and Feige's Random 3SAT Hypothesis.

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