We survey recent developments related to the Minimum Circuit Size Problem.

Language and Automata Theory and Applications, 14th International Conference LATA 2020, Milan, Italy, March 4–6, 2020, Proceedings

Programs over semigroups are a well-studied model of computation for boolean functions. It has been used successfully to characterize, in algebraic terms, classes of problems that can, or cannot, be solved within certain resources. The interest of the approach is that the algebraic complexity of the semigroups required to capture a class should be a good indication of its expressive (or computing) power. In this paper we derive algebraic characterizations for some small classes of boolean functions, all of which have depth-3 AC° circuits, namely k-term DNF, k-DNF, k-decision lists, decision trees of bounded rank, and DNF. The interest of such classes, and the reason for this investigation, is that they have been intensely studied in computational learning theory.

Algebraic characterizations of small classes of boolean functions

Garbling schemes, also known as decomposable randomized encodings (DRE), have found many applications in cryptography. However, despite a large body of work on constructing such schemes, very little is known about their limitations.
We initiate a systematic study of the DRE complexity of Boolean functions, obtaining the following main results: 
- Near-quadratic lower bounds. We use a classical lower bound technique of Neciporuk [Dokl. Akad. Nauk SSSR '66] to show an Ω(n²/log n) lower bound on the size of any DRE for many explicit Boolean functions. For some natural functions, we obtain a corresponding upper bound, thus settling their DRE complexity up to polylogarithmic factors. Prior to our work, no superlinear lower bounds were known, even for non-explicit functions. 
- Garbling-friendly PRFs. We show that any exponentially secure PRF has Ω(n²/log n) DRE size, and present a plausible candidate for a "garbling-optimal" PRF that nearly meets this bound. This candidate establishes a barrier for super-quadratic DRE lower bounds via natural proof techniques. In contrast, we show a candidate for a weak PRF with near-exponential security and linear DRE size. Our results establish several qualitative separations, including near-quadratic separations between computational and information-theoretic DRE size of Boolean functions, and between DRE size of weak vs. strong PRFs.

https://drops.dagstuhl.de/opus/volltexte/2020/11771/pdf/LIPIcs-ITCS-2020-86.pdf/

On the Complexity of Decomposable Randomized Encodings, Or: How Friendly Can a Garbling-Friendly PRF Be?

Cette these porte sur des minorants pour des mesures de complexite liees a des sous-classes de la classe P de langages pouvant etre decides en temps polynomial par des machines de Turing. Nous considerons des modeles de calcul non uniformes tels que les programmes sur monoides et les programmes de branchement. Notre premiere contribution est un traitement abstrait de la methode de Neciporuk pour prouver des minorants, independamment de toute mesure de complexite specifique. Cette methode donne toujours les meilleurs minorants connus pour des mesures telles que la taille des programmes de branchements deterministes et non deterministes ou des formules avec des operateurs booleens binaires arbitraires ; nous donnons une formulation abstraite de la methode et utilisons ce cadre pour demontrer des limites au meilleur minorant obtenable en utilisant cette methode pour plusieurs mesures de complexite. Par la, nous confirmons, dans ce cadre legerement plus general, des resultats de limitation precedemment connus et exhibons de nouveaux resultats de limitation pour des mesures de complexite auxquelles la methode de Neciporuk n'avait jamais ete appliquee. Notre seconde contribution est une meilleure comprehension de la puissance calculatoire des programmes sur monoides issus de petites varietes de monoides finis. Les programmes sur monoides furent introduits a la fin des annees 1980 par Barrington et Therien pour generaliser la reconnaissance par morphismes et ainsi obtenir une caracterisation en termes de semi-groupes finis de NC^1 et de ses sous-classes. Etant donne une variete V de monoides finis, on considere la classe P(V) de langages reconnus par une suite de programmes de longueur polynomiale sur un monoide de V : lorsque l'on fait varier V parmi toutes les varietes de monoides finis, on obtient differentes sous-classes de NC^1, par exemple AC^0, ACC^0 et NC^1 quand V est respectivement la variete de tous les monoides aperiodiques finis, resolubles finis et finis. Nous introduisons une nouvelle notion de docilite pour les varietes de monoides finis, renforcant une notion de Peladeau. L'interet principal de cette notion est que quand une variete V de monoides finis est docile, nous avons que P(V) contient seulement des langages reguliers qui sont quasi reconnus par morphisme par des monoides de V. De nombreuses questions ouvertes a propos de la structure interne de NC^1 seraient reglees en montrant qu'une variete de monoides finis appropriee est docile, et, dans cette these, nous debutons modestement une etude exhaustive de quelles varietes de monoides finis sont dociles. Plus precisement, nous portons notre attention sur deux petites varietes de monoides aperiodiques finis bien connues : DA et J. D'une part, nous montrons que DA est docile en utilisant des arguments de theorie des semi-groupes finis. Cela nous permet de deriver une caracterisation algebrique exacte de la classe des langages reguliers dans P(DA). D'autre part, nous montrons que J n'est pas docile. Pour faire cela, nous presentons une astuce par laquelle des programmes sur monoides de J peuvent reconnaitre beaucoup plus de langages reguliers que seulement ceux qui sont quasi reconnus par morphisme par des monoides de J. Cela nous amene a conjecturer une caracterisation algebrique exacte de la classe de langages reguliers dans P(J), et nous exposons quelques resultats partiels appuyant cette conjecture. Pour chacune des varietes DA et J, nous exhibons egalement une hierarchie basee sur la longueur des programmes a l'interieur de la classe des langages reconnus par programmes sur monoides de la variete, ameliorant par la les resultats de Tesson et Therien sur la propriete de longueur polynomiale pour les monoides de ces varietes.

https://tel.archives-ouvertes.fr/tel-01935719/document

The limits of Nečiporuk's method and the power of programs over monoids taken from small varieties of finite monoids

For a size parameter s: ℕ → ℕ, the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}ⁿ → {0,1} (represented by a string of length N : = 2ⁿ) is at most a threshold s(n). A recent line of work exhibited "hardness magnification" phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant μ₁ > 0, if MCSP[2^{μ₁⋅ n}] cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^{1.01}, then P≠NP.
In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: 
1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute MCSP[2^{μ₂⋅n}] in time N^{1.99}, for some constant μ₂ > μ₁. 
2) A non-deterministic (or parity) branching program of size o(N^{1.5}/log N) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Neciporuk method to MKTP, which previously appeared to be difficult. 
3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least N^{1.5-o(1)}. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models.
The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola 2019). En route, we obtain several related results: 
1) There exists a (local) hitting set generator with seed length O(√N) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs. 
2) Any read-once co-non-deterministic branching program computing MCSP must have size at least 2^Ω(N).

One-Tape Turing Machine and Branching Program Lower Bounds for MCSP.

The program-over-monoid model of computation originates with Barrington's proof that it captures the complexity class NC^1. Here we make progress in understanding the subtleties of the model. First, we identify a new tameness condition on a class of monoids that entails a natural characterization of the regular languages recognizable by programs over monoids from the class. Second, we prove that the class known as DA satisfies tameness and hence that the regular languages recognized by programs over monoids in DA are precisely those recognizable in the classical sense by morphisms from QDA. Third, we show by contrast that the well studied class of monoids called J is not tame and we exhibit a regular language, recognized by a program over a monoid from J, yet not recognizable classically by morphisms from the class QJ. Finally, we exhibit a program-length-based hierarchy within the class of languages recognized by programs over monoids from DA.

https://hal.archives-ouvertes.fr/hal-02303526/document

The Power of Programs over Monoids in DA.

A formulation of Neciporuk’s lower bound method slightly more inclusive than the usual complexity-measure-specific formulation is presented. Using this general formulation, limitations to lower bounds achievable by the method are obtained for several computation models, such as branching programs and Boolean formulas having access to a sublinear number of nondeterministic bits. In particular, it is shown that any lower bound achievable by the method of Neciporuk for the size of nondeterministic and parity branching programs is at most O(n3/2/logn).

https://dl.acm.org/doi/pdf/10.1145/3013516

Nondeterminism and An Abstract Formulation of Neciporuk’s Lower Bound Method

The model of programs over (finite) monoids, introduced by Barrington and Therien, gives an interesting way to characterise the circuit complexity class Open image in new window and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in Open image in new window, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from Open image in new window, based on the length of programs but also some parametrisation of Open image in new window. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in Open image in new window. We show that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in Open image in new window.

/pdf/the-power-of-programs-over-monoids-in-j-4s3i2m8rr4.pdf

The Power of Programs over Monoids in J

A formulation of "Neciporuk's lower bound method" slightly more inclusive than the usual complexity-measure-specific formulation is presented. Using this general formulation, limitations to lower bounds achievable by the method are obtained for several computation models, such as branching programs and Boolean formulas having access to a sublinear number of nondeterministic bits. In particular, it is shown that any lower bound achievable by the method of Neciporuk for the size of nondeterministic and parity branching programs is at most $O(n^{3/2}/\log n)$.

Nondeterminism and an abstract formulation of Ne\v{c}iporuk's lower bound method

The model of programs over (finite) monoids, introduced by Barrington and Th{e}rien, gives an interesting way to characterise the circuit complexity class $\mathsf{NC^1}$ and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in $\mathbf{J}$, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from $\mathbf{J}$, based on the length of programs but also some parametrisation of $\mathbf{J}$. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in $\mathbf{J}$. We show that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in $\mathbf{J}$.

Nathan Grosshans

Papers

The Power of Programs over Monoids in DA.

Nondeterminism and An Abstract Formulation of Neciporuk’s Lower Bound Method

The Power of Programs over Monoids in J

Nondeterminism and an abstract formulation of Ne\v{c}iporuk's lower bound method

The Power of Programs over Monoids in J