Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size ≤ s(n). We show: Learning Speedups. If C[poly(n)] admits a randomized weak learning algorithm under the uniform distribution with membership queries that runs in time 2n/nω(1), then for every k ≥ 1 and ε > 0 the class C[n k ] can be learned to high accuracy in time O(2n ε ). There is ε > 0 such that C[2n ε ] can be learned in time 2n/nω(1) if and only if C[poly(n)] can be learned in time 2(log n) O(1) . Equivalences between Learning Models. We use learning speedups to obtain equivalences between various randomized learning and compression models, including sub-exponential time learning with membership queries, sub-exponential time learning with membership and equivalence queries, probabilistic function compression and probabilistic average-case function compression. A Dichotomy between Learnability and Pseudorandomness. In the non-uniform setting, there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure pseudorandom functions computable in C[poly(n)]. Lower Bounds from Nontrivial Learning. If for each k ≥ 1, (depth-d)-C[n k ] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2n/nω(1), then for each k ≥ 1, BPE * (depth-d)-C[n k ]. If for some ε > 0 there are P-natural proofs useful against C[2n ε ], then ZPEXP * C[poly(n)]. Karp-Lipton Theorems for Probabilistic Classes. If there is a k > 0 such that BPE ⊆ i.o.Circuit[n k ], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2n/3 ], then ZPEXP ⊆ i.o.ESUBEXP. Hardness Results for MCSP. All functions in non-uniform NC1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC0 circuits. In particular, if MCSP ∈ TC0 then NC1 = TC0

Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size <= s(n). We show: Learning Speedups: If C[s(n)] admits a randomized weak learning algorithm under the uniform distribution with membership queries that runs in time 2^n/n^{omega(1)}, then for every k >= 1 and epsilon > 0 the class C[n^k] can be learned to high accuracy in time O(2^{n^epsilon}). There is epsilon > 0 such that C[2^{n^{epsilon}}] can be learned in time 2^n/n^{omega(1)} if and only if C[poly(n)] can be learned in time 2^{(log(n))^{O(1)}}. Equivalences between Learning Models: We use learning speedups to obtain equivalences between various randomized learning and compression models, including sub-exponential time learning with membership queries, sub-exponential time learning with membership and equivalence queries, probabilistic function compression and probabilistic average-case function compression. A Dichotomy between Learnability and Pseudorandomness: In the non-uniform setting, there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure pseudorandom functions computable in C[poly(n)]. Lower Bounds from Nontrivial Learning: If for each k >= 1, (depth-d)-C[n^k] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2^n/n^{omega(1)}, then for each k >= 1, BPE is not contained in (depth-d)-C[n^k]. If for some epsilon > 0 there are P-natural proofs useful against C[2^{n^{epsilon}}], then ZPEXP is not contained in C[poly(n)]. Karp-Lipton Theorems for Probabilistic Classes: If there is a k > 0 such that BPE is contained in i.o.Circuit[n^k], then BPEXP is contained in i.o.EXP/O(log(n)). If ZPEXP is contained in i.o.Circuit[2^{n/3}], then ZPEXP is contained in i.o.ESUBEXP. Hardness Results for MCSP: All functions in non-uniform NC^1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC^0 circuits. In particular, if MCSP is in TC^0 then NC^1 = TC^0

Unconditional Lower Bounds in Complexity Theory

This work investigates the hardness of solving natural computational problems according to different complexity measures. Our results and techniques span several areas in theoretical computer science and discrete mathematics. They have in common the following aspects: (i) the results are unconditional, i.e., they rely on no unproven hardness assumption from complexity theory; (ii) the corresponding lower bounds are essentially optimal. Among our contributions, we highlight the following results. Constraint Satisfaction Problems and Monotone Complexity. We introduce a natural formulation of the satisfiability problem as a monotone function, and prove a near-optimal 2^{Ω (n/log n)} lower bound on the size of monotone formulas solving k-SAT on n-variable instances (for a large enough k ∈ ℕ). More generally, we investigate constraint satisfaction problems according to the geometry of their constraints, i.e., as a function of the hypergraph describing which variables appear in each constraint. Our results show in a certain technical sense that the monotone circuit depth complexity of the satisfiability problem is polynomially related to the tree-width of the corresponding graphs. Interactive Protocols and Communication Complexity. We investigate interactive compression protocols, a hybrid model between computational complexity and communication complexity. We prove that the communication complexity of the Majority function on n-bit inputs with respect to Boolean circuits of size s and depth d extended with modulo p gates is precisely n/log^{ϴ(d)} s, where p is a fixed prime number, and d ∈ ℕ. Further, we establish a strong round-separation theorem for bounded-depth circuits, showing that (r+1)-round protocols can be substantially more efficient than r-round protocols, for every r ∈ ℕ. Negations in Computational Learning Theory. We study the learnability of circuits containing a given number of negation gates, a measure that interpolates between monotone functions, and the class of all functions. Let C^t_n be the class of Boolean functions on n input variables that can be computed by Boolean circuits with at most t negations. We prove that any algorithm that learns every f ∈ C^t_n with membership queries according to the uniform distribution to accuracy ε has query complexity 2^{Ω (2^t sqrt(n)/ε)} (for a large range of these parameters). Moreover, we give an algorithm that learns C^t_n from random examples only, and with a running time that essentially matches this information-theoretic lower bound. Negations in Theory of Cryptography. We investigate the power of negation gates in cryptography and related areas, and prove that many basic cryptographic primitives require essentially the maximum number of negations among all Boolean functions. In other words, cryptography is highly non-monotone. Our results rely on a variety of techniques, and give near-optimal lower bounds for pseudorandom functions, error-correcting codes, hardcore predicates, randomness extractors, and small-bias generators. Algorithms versus Circuit Lower Bounds. We strengthen a few connections between algorithms and circuit lower bounds. We show that the design of faster algorithms in some widely investigated learning models would imply new unconditional lower bounds in complexity theory. In addition, we prove that the existence of non-trivial satisfiability algorithms for certain classes of Boolean circuits of depth d+2 leads to lower bounds for the corresponding class of circuits of depth d. These results show that either there are no faster algorithms for some computational tasks, or certain circuit lower bounds hold

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum