Minimal Circuit Size Problem (MCSP)

• MCSP is defined as deciding whether a Boolean function, presented by its truth table, has a Boolean circuit of size at most s.
• It serves as a central compression problem that connects key areas like cryptography, learning theory, and pseudorandomness within computational complexity.
• Recent approaches using convex optimization and proof complexity reveal essential insights into circuit stability, NP-intermediate challenges, and quantum extensions.

The Minimum Circuit Size Problem (MCSP) concerns determining, for a given Boolean function represented by its truth table and a threshold $s$, whether there exists a Boolean circuit of size at most $s$ that computes the function. MCSP occupies a central position in computational complexity theory—serving as a canonical "compression" problem and as a natural barrier between key complexity classes—yet its precise computational hardness remains unresolved. MCSP is tightly connected to pseudorandomness, learning theory, cryptography, proof complexity, and is increasingly studied in quantum and convex-optimization contexts.

1. Formal Definition and Basic Properties

MCSP is formally defined as follows. Given a function $f: \{0,1\}^n \to \{0,1\}$ (represented by its $2^n$-bit truth table $tt(f)$) and an integer $s$:

$$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$

For any polynomially constructible size function $s(n)\geq n$, denote by $\mathrm{MCSP}[s]$ the variant that asks, on input $tt(f)$, whether $s$0 has a circuit of size at most $s$1 (Modanese, 2020). MCSP is evidently in NP since a small circuit serves as a witness, verified by checking $s$2 on all $s$3 inputs.

It is unknown whether MCSP is NP-complete under standard reductions; evidence exists only for NP-hardness under more general reductions (randomized, non-uniform, or oracle-aided) (Hirsch et al., 25 Dec 2025).

MCSP has a Kolmogorov complexity-inspired variant, MKTP, where the threshold is expressed in terms of KT-complexity rather than circuit size. MCSP and MKTP are polynomially related in the sense that small circuit size gives short algorithms and vice versa (Allender et al., 2017).

2. Circuit Complexity Landscape and Lipschitz Stability

Let $s$4 denote the minimum size of any circuit over a finite complete basis $s$5 computing $s$6. A foundational result establishes a Lipschitz-type stability of optimal circuit size under local perturbations:

Theorem: (Single-Bit Perturbation Bound)

For $s$7 differing on exactly one input ($s$8), and for any finite complete basis $s$9 with unit-cost gates, we have:

$f: \{0,1\}^n \to \{0,1\}$0

where $f: \{0,1\}^n \to \{0,1\}$1 depends only on $f: \{0,1\}^n \to \{0,1\}$2. In the AIG basis ($f: \{0,1\}^n \to \{0,1\}$3, AND gates counted, free inversion), $f: \{0,1\}^n \to \{0,1\}$4, so the variation is at most $f: \{0,1\}^n \to \{0,1\}$5 (Krinkin, 10 Mar 2026).

Global Hamming Ball Bound:

For any $f: \{0,1\}^n \to \{0,1\}$6 with Hamming distance $f: \{0,1\}^n \to \{0,1\}$7:

$f: \{0,1\}^n \to \{0,1\}$8

For AIG, this is tight at $f: \{0,1\}^n \to \{0,1\}$9 based on exhaustive SAT-based optimal circuit size computations; the maximum observed size change for one-bit mutation is exactly $2^n$0 (Krinkin, 10 Mar 2026).

Implications for MCSP:

• This stability under perturbation undergirds average-case lower bounds and concentration results for MCSP.
• In reductions (e.g., from graph problems), it prevents circuit-size "jumps" and enables fine control of reductions' approximation parameters.
• It constrains the search landscape for practical and heuristic MCSP solvers, allowing local improvements to be bounded in effect.

3. Complexity-Theoretic Status and Hardness Magnification

MCSP remains a prominent NP-intermediate candidate:

• Contained in NP, but not known to be NP-complete (Modanese, 2020, Hirsch et al., 25 Dec 2025, Allender et al., 2017).
• Magnification paradigms show even mild progress—such as ruling out poly$2^n$1-space streaming algorithms for MCSP$2^n$2—would separate P from NP. For example:
  • If MCSP$2^n$3 requires superpolynomial space or time for streaming algorithms, then $2^n$4 (Modanese, 2020).
  • Equivalent results hold for sublinear-time Shrinking Cellular Automata: a language outside SCA$2^n$5 (even slightly) yields $2^n$6 (Modanese, 2020).
• Hardness against extremely sparse tally sets would force $2^n$7; in particular, MCSP cannot be hard for all sparse sets under polynomial-time truth-table reductions unless the exponential hierarchy separates (Fu, 2020).

Recent work links MCSP to the range avoidance problem, showing that any language reducible in randomized polynomial time to MCSP is in $2^n$8. This containment is established via randomized inversion routines for one-way functions, simulated via MCSP oracles (Hirsch et al., 25 Dec 2025).

4. Connections to Other Central Problems

MCSP interfaces with numerous domains:

• Isomorphism Problems: Via entropy-gap reductions, $2^n$9 and other group-theoretic isomorphism problems reduce (ZPP Turing reductions) to MKTP and substantially, to MCSP. This leverages information-theoretic encodings and interactive proof techniques rather than hardness-based PRG constructions (Allender et al., 2017).
• Learning and Cryptography: MCSP in BPP implies efficient PAC learning and breaks one-way functions; conversely, hardness of MCSP underlies the security of cryptographic primitives and learning-theoretic separation results (Chia et al., 2021).
• Proof Complexity: Degree-$tt(f)$0 lower bounds hold for SoS refutations of MCSP-falsity (i.e., for any $tt(f)$1 with no small circuit), and similar in the monotone and "almost-easily-approximable" settings (Austrin et al., 2023).

The quantum generalization of MCSP introduces variants for quantum Boolean functions, unitaries, and quantum states, with protocols in QCMA and reductions not possible for classical MCSP. These quantum MCSPs relate to quantum pseudorandomness, quantum learning, and quantum gravity conjectures (Chia et al., 2021).

5. Approximability, Convexity, and Algorithmic Approaches

Recent developments recast circuit minimization as a (nearly) convex optimization problem—especially in the "Harder Than Monte Carlo" (HTMC) regime, where functions are efficiently $tt(f)$2-approximable only with superquadratic blowup in $tt(f)$3 (Jacot, 25 Nov 2025). Main elements:

• The class of functions $tt(f)$4 $tt(f)$5-approximable by circuits of size at most $tt(f)$6 is convex for $tt(f)$7.
• An explicit norm (HTMC norm) can be defined on functions, tightly sandwiching the minimal circuit size in terms of a ResNet parameter norm (weighted $tt(f)$8).
• Minimizing the ResNet norm over the function class yields (up to power-of-two factors) a nearly minimum Boolean circuit—deep learning thus conducts a convexification of MCSP in the HTMC regime.
• Frank–Wolfe style methods, while not polynomial time for worst-case MCSP, conceptually link convex optimization and circuit-size minimization.

Notable limitations:

• The convexification only holds for $tt(f)$9; many real-world functions have smaller $s$0.
• The ResNet parameter norm has an exponent gap in the sandwich bound, still open.

6. Proof Complexity and Lower Bounds

Proof-complexity approaches to MCSP have yielded unconditional lower bounds in static proof systems:

• Sum-of-Squares Lower Bounds: Any degree-$s$1 SoS refutation of MCSP requires $s$2 for $s$3 having no circuit of size $s$4 (for $s$5). For "almost easily approximable" functions—close to analytic or slice functions—SoS proof size grows doubly-exponentially (Austrin et al., 2023).
• These bounds hold both in the general and monotone (slice-function) cases, preventing any low-degree SDP-based algorithmic resolution of MCSP.
• The reduction from MCSP to XOR-CSPs on expanders underpins these lower bounds using substitution and restriction techniques.

A plausible implication is that any attempt to resolve MCSP in polynomial time within strong SDP hierarchies, or via monotone circuit methods, is obstructed at a fundamental level by these lower bounds.

7. Open Problems and Future Directions

Numerous foundational questions surrounding MCSP remain open or only partially resolved:

• Is MCSP NP-complete under natural reductions? This remains a longstanding open problem (Hirsch et al., 25 Dec 2025, Modanese, 2020, Fu, 2020).
• Can the exponent gap in the convex-analytic embedding (from ResNet minimization to circuit size) be closed, and are there polynomial-time algorithms for MCSP in any regime?
• What is the quantum complexity of MCSP variants? Are they in NP, QCMA, or complete for these classes (Chia et al., 2021)?
• Are there explicit, infinite families attaining tightness in the $s$6 or $s$7 circuit-size change bounds for single- or few-bit truth table perturbations (Krinkin, 10 Mar 2026)?
• Can uniform fine-grained hardness (in streaming or uniform automata models) of MCSP be established, thereby resolving central class separations in complexity theory (Modanese, 2020)?
• Further connections, e.g., to quantum tomography, efficiency of learning in BQP/poly, and general explicit encodings, remain to be fully explored (Allender et al., 2017, Chia et al., 2021).

Summary Table: Selected Key Results

Area	Key Result / Theorem	Reference
Perturbation stability	$s$8	(Krinkin, 10 Mar 2026)
Streaming lower bound magnification	Poly$s$9 streaming for MCSP$$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$0 $$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$1 P=NP	(Modanese, 2020)
Proof complexity (SoS)	SoS degree $$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$2 for MCSP-falsity	(Austrin et al., 2023)
Quantum setting	MCSP variants in QCMA; search-to-decision for unitaries	(Chia et al., 2021)
Convex approximation (HTMC/ResNet norm)	Shallow ResNet norm $$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$3 minimal circuit size for $$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$4	(Jacot, 25 Nov 2025)
Isomorphism reductions	GI, CodeEquiv, PermGroupConj $$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$5 ZPP$$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$6	(Allender et al., 2017)
Range avoidance connection	$$\mathrm{MCSP} = \{ (tt(f), s) \mid \exists\, \text{Boolean circuit } C \text{ of size} \leq s \text{ such that } C(x) = f(x) \, \forall x \}$$7	(Hirsch et al., 25 Dec 2025)

MCSP thus persists as a focal point for several foundational challenges across classical and quantum complexity, bridging barriers in learning, derandomization, cryptography, and proof-theoretic lower bounds.