Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on Theoretical Computer Science (ICTCS'2007

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}. For a class F of boolean functions and a class G of graphs, an (F, G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #Pcomplete to compute the number of fixed points in an (F, G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas

Holographic Algorithm with Matchgates Is Universal for Planar #\#CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Comment: 94 page

Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}? ? {0,1}, SoS requires degree ?(s^{1-?}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ? P/poly. We also show that for any 0 < ? < 1 there are Boolean functions with circuit complexity larger than 2^{n^?} but SoS requires size 2^{2^?(n^?)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q

Sum-of-Squares Lower Bounds for the Minimum Circuit Size Problem

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $\Omega(s^{1-\epsilon})$ to prove that $f$ does not have circuits of size $s$ (for any $s > \mathrm{poly}(n)$). As a corollary we obtain that there are no low degree SoS proofs of the statement NP $\not \subseteq$ P/poly. We also show that for any $0 < \alpha < 1$ there are Boolean functions with circuit complexity larger than $2^{n^{\alpha}}$ but SoS requires size $2^{2^{\Omega(n^{\alpha})}}$ to prove this. In addition we prove analogous results on the minimum \emph{monotone} circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system $Q$ has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, $Q$ is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for $Q$.Comment: A conference version appeared previously in CCC'2