Minimal Circuit Size Problem Overview

• MCSP is defined as determining if a Boolean function has a circuit of size at most s, serving as a fundamental query in computational complexity.
• Recent SAT-based local search methods have improved practical minimization bounds, as seen in tighter implementations of majority and threshold functions.
• The problem's links to hardness magnification, quantum circuit generalizations, and continuous relaxations via deep learning underscore its broad impact on complexity theory.

The Minimal Circuit Size Problem (MCSP) is a central question in computational complexity and circuit theory, asking whether, for a given Boolean function (specified by its truth table) and a size parameter $s$, there exists a Boolean circuit of size at most $s$ that computes the function exactly. MCSP has deep connections to fundamental questions in complexity theory, derandomization, cryptographic hardness amplification, and proof complexity, and continues to motivate both algorithmic innovations and lower-bound methodology. The problem also admits quantum extensions, variants for restricted circuit classes, and continuous relaxations involving deep learning models such as ResNets.

1. Formal Definition and Classical Complexity Position

Given a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ described via its $2^n$-bit truth-table, the decision version of MCSP asks: Does there exist a circuit $C$ (using standard fan-in-2 gates: AND, OR, NOT) of size at most $s$ such that $C(x) = f(x)$ for all $x \in \{0,1\}^n$? The function $s$ is typically parameterized as $s(n)$ (most commonly $n^{O(1)}$).

MCSP lies in NP: a candidate circuit serves as a witness verifiable in poly$(2^n, s)$ time. However, MCSP is not known to be NP-complete, nor in P, and is a chief candidate for natural NP-intermediate status. There is no known reduction from SAT or other NP-complete problems placing MCSP in the NP-complete class, and in fact, there are results ruling out (under plausible assumptions) NP-completeness for MCSP under various reductions (Allender et al., 2017, Fu, 2020).

Several parameterizations exist, including restrictions to specific gate types, monotone circuits, multi-output variants, or function classes with small variable dependencies.

2. Exact Minimization Algorithms and Local Search Heuristics

The space of size-$s$ Boolean circuits is $s^{\Theta(s)}$, making MCSP intractable for even moderate $s$ by naive enumeration. Recent practical advances utilize algorithmic local search, with the most notable being the SAT-based circuit local improvement paradigm (Kulikov et al., 2021). The methodology is as follows:

• Given an initial circuit $C$ (not necessarily optimal), enumerate all connected subcircuits of a fixed moderate size $s' \leq s$.
• For each such subcircuit, use a propositional encoding (SAT) to check for substitutable strictly smaller subcircuits (functionally identical on inherited inputs).
• Replace improvable subcircuits and iterate until no further improvements exist.

The SAT encoding uses gate-type, wire-selection, and evaluation variables with constraints enforcing structure, semantics (Tseitin style), output consistency, and full functional equivalence on the truth table.

Empirical results include automatic sharpening of upper bounds for standard symmetric functions: e.g., demonstrating that the MAJ$_7$ function can be implemented with 10 gates (improving over the hand-derived 11) and yielding new bounds for small threshold and majority functions. The local improvement approach scales to subcircuits of size up to 12 and, while not solving MCSP in the worst case, provides a robust framework for experimental circuit optimization (Kulikov et al., 2021).

3. Lower Bounds and Proof Complexity Barriers

MCSP is a natural meta-complexity problem central to contemporary lower-bound frameworks. Austrin and Risse (Austrin et al., 2023) establish the first unconditional superconstant lower bounds for MCSP in strong algebraic proof systems—specifically, the Sum-of-Squares (SoS) hierarchy.

Key results include:

• Any SoS refutation that a Boolean function $f$ does not have a size-$s$ circuit must have degree $\Omega(s^{1-\epsilon})$ for $s > \mathrm{poly}(n)$, for every $\epsilon>0$.
• There are no low-degree SoS proofs separating NP from P/poly, which significantly constrains the applicability of SDP-based algorithmic paradigms to MCSP and related meta-complexity statements.
• For explicit functions with circuit complexity $>2^{n^\alpha}$, any SoS refutation certifying unsatisfiability under size bound $s=2^{n^\alpha}$ must have syntactic size at least $2^{2^{\Omega(n^\alpha)}}$.
• The framework generalizes to monotone circuit lower bounds for monotone Boolean slice functions.

The core methodology is to transfer degree lower bounds for SoS refutations of XOR-CSPs on explicit expander graphs (via reductions using partial assignments and substitutability in constraint–variable incidence graphs) to the MCSP setting. This approach robustly rules out a large class of SDP-based techniques for proving strong statements about MCSP (Austrin et al., 2023).

4. Hardness Magnification, Reductions, and Derandomization Barriers

Hardness results for MCSP are tightly intertwined with well-known complexity separations. It is shown that any nontrivial lower bound for MCSP in extremely weak uniform computational models, such as sublinear-time shrinking cellular automata (SCAs), would suffice to separate P from NP—demonstrating "hardness magnification" (Modanese, 2020). Specifically, MCSP is so robust that proving even modest lower bounds in weak models already implies major complexity theory breakthroughs.

On the landscape of reductions, MCSP has resisted NP-hardness proofs under all standard reductions. It is shown that MCSP cannot be ZPP-hard under polynomial-time truth-table reductions unless EXP $\ne$ ZPP (Fu, 2020). This creates a significant barrier to direct hardness-of-MCSP results and connects the problem’s complexity status to major open derandomization questions.

Reductions to and from MCSP (and its Kolmogorov complexity analog MKTP) include randomized reductions from natural isomorphism problems (e.g., Graph Isomorphism, Linear Code Equivalence), establishing that GI $\in$ ZPP$^{\mathrm{MKTP}}$ via entropy-based interactive proof translations. For MCSP itself, such reductions face hurdles due to a logarithmic factor in known circuit compression bounds, currently precluding direct non-trivial reductions of GI or similar problems to MCSP (Allender et al., 2017).

5. Extensions, Parameterized Variants, and Quantum Generalizations

MCSP admits various interesting generalizations:

• Multi-output AND-function minimization: For functions of the form $f_j(x) = \bigwedge_{i \in T_j} x_i$ (collections of ANDs over overlapping variable subsets), the problem of jointly minimizing the total number of AND gates is NP-hard already for $|T_j| \leq 3$, but unlike general MCSP, it is in NP and admits constant-factor approximations for small $k$ (Armbruster, 2024).
• CNOT-circuit minimization: In the context of linear reversible/quantum circuits (CNOT-only), the MCSP becomes distance computation in the Cayley graph of $GL(n,2)$; efficient synthesis algorithms exploiting symmetries enable optimal decompositions up to $n=7$ qubits. Group theory and combinatorics supply structural insights and asymptotic bounds (Christensen et al., 3 Mar 2025).
• Quantum MCSP: For quantum circuits, MCSP variants (for functions, unitaries, states) have been introduced, formalized as MQCSP, UMCSP, and SMCSP. These lie in QCMA (classical-witness quantum-verifiable), and admit search-to-decision and self-reduction results for certain cases. Important connections exist to quantum learning theory, quantum cryptography, complexity of quantum pseudorandomness, state tomography, and quantum gravity (via volume=complexity in AdS/CFT) (Chia et al., 2021).

6. Continuous and Approximate Relaxations: Deep Learning Perspective

A recent development relates MCSP to deep learning through the lens of convex function-approximation (Jacot, 25 Nov 2025). In sufficiently hard regimes—i.e., when the minimal circuit size required for $\epsilon$-approximation scales as $C(f,\epsilon) \leq c \epsilon^{-\gamma}$ with $\gamma > 2$ (the "harder than Monte Carlo" or HTMC regime)—the set of $\gamma$-computable functions admits (up to scaling) a convex norm structure.

Deep residual networks (ResNets) equipped with a weighted $\ell_1$-norm on parameters induce a continuous ResNet norm, which can be polynormally sandwiched between the HTMC circuit-norm and the ResNet-norm. Minimizing this ResNet norm when fitting data yields a solution that is within a constant multiplicative factor (in terms of gates) of MCSP optimality for the function approximation task. This perspective recasts deep learning as an implicit Occam’s razor, providing a geometric and complexity-theoretic rationale for the empirical tendency of neural networks to find minimal-complexity interpolators (Jacot, 25 Nov 2025).

7. Open Problems and Future Directions

MCSP remains open on numerous fronts:

• It is not known whether MCSP is NP-complete, and completeness under standard reductions is widely conjectured to be false under plausible complexity assumptions.
• Proving nontrivial circuit lower bounds for MCSP in even restricted uniform models would constitute a breakthrough for P vs NP via hardness magnification (Modanese, 2020).
• Reductions from natural computational problems (e.g., Graph Isomorphism, SZK) directly to MCSP remain open, due in part to technical hurdles in circuit compression and entropy arguments (Allender et al., 2017).
• For monotone and quantum circuit-size analogs, sharp lower bounds and completeness classifications remain active research directions.
• Bridging continuous complexity (as in ResNet-based norms or convex relaxations) with traditional combinatorial circuit complexity is a promising route for both theoretical advances and practical minimization algorithms (Jacot, 25 Nov 2025).

MCSP sits at the intersection of computational complexity, circuit optimization, proof complexity, learning theory, and quantum computing, continuing to motivate foundational work across these domains.