P vs NP
P vs NP es uno de los problemas abiertos más importantes de la informática teórica y de las matemáticas modernas. Pregunta, en esencia, qué tan difícil es resolver problemas frente a qué tan difícil es verificar una solución.
A Topological Obstruction to Shallow Boolean Circuits
Collapsibility, Compatibility Complexes, and Depth Lower Bounds
Abstract
This paper introduces a topological framework for studying lower bounds in Boolean circuit complexity. To each Boolean language, a canonical simplicial complex is associated, encoding literal compatibility across accepting assignments. It is shown that any family of Boolean circuits of constant depth and polynomial size induces simplicial complexes that are collapsible to dimension one, and therefore have trivial second homology. Explicit CNF formulas are then exhibited whose associated complexes have nontrivial second homology, yielding a topological obstruction to constant-depth circuit computation. Finally, a conjectural extension to logarithmic-depth circuits (NC¹) is formulated as an independent open problem, and the intrinsic limitations of the method are explicitly delineated.
Significance for the P vs NP Problem
The P vs NP problem has resisted resolution not merely because of technical difficulty, but because many of the classical tools of complexity theory—diagonalization, relativization, algebrization, and natural proofs—fail to capture truly global structural constraints on efficient computation. Most unsuccessful approaches attempt to reason directly about running time or circuit size, without access to invariants that remain meaningful across representations.
The contribution of this work is to introduce precisely such an invariant.
The central result is not a full separation between P and NP, but something conceptually prior: it demonstrates that restricted efficient computation enforces unavoidable global topological constraints on the space of solutions. Specifically, it is shown that constant-depth Boolean circuits can only induce compatibility complexes that are topologically flat—collapsible to dimension one—whereas explicit CNF formulas in NP necessarily give rise to nontrivial second homology.
This is relevant to P vs NP for three fundamental reasons:
The limitation does not arise from counting arguments, random restrictions, or ad hoc constructions, but from the impossibility of generating certain global topological features under shallow computation. As such, it lies largely outside the scope of classical barriers.
Rather than asking how many resources an algorithm requires, the framework asks what kinds of global geometric structure an efficient algorithm can induce. If P vs NP admits a resolution, it is widely believed that such a nonlocal perspective will be essential.
The work establishes a complete result at constant depth and formulates a precise conjecture at logarithmic depth (NC¹). Progress on this conjecture—either positive or negative—would constitute genuine progress independent of the ultimate status of P vs NP.
Esp:
El problema P vs NP ha resistido durante décadas no por falta de ingenio técnico, sino porque muchas de las herramientas clásicas de la teoría de la complejidad —diagonalización, relativización, algebrización y pruebas naturales— parecen incapaces de capturar restricciones estructurales verdaderamente globales sobre la computación eficiente. La mayor parte de los enfoques fallidos atacan el problema en el nivel equivocado: intentan razonar directamente sobre el tiempo de ejecución o el tamaño de los circuitos, sin disponer de invariantes que sobrevivan a cambios de representación.
El resultado demostrado en este trabajo introduce precisamente ese tipo de invariante.
La contribución central no es una separación completa entre P y NP, sino algo conceptualmente más fundamental: se demuestra que ciertas formas de computación eficiente imponen restricciones topológicas globales inevitables sobre el espacio de soluciones. En particular, se prueba que los circuitos booleanos de profundidad constante solo pueden inducir complejos de compatibilidad topológicamente “planos”, colapsables a dimensión uno, mientras que existen fórmulas CNF explícitas —pertenecientes a NP— cuyo espacio de soluciones fuerza la aparición de homología de dimensión dos.
Local Topological Obstructions, Homological Persistence, and the Structural Boundary of NC
Abstract
This paper develops a topological framework for the analysis of Boolean circuit complexity based on the simplicial compatibility complex associated with a Boolean language. Extending known collapsibility results for constant-depth circuits, a sharp conjectural boundary at logarithmic depth (NC1) is isolated in terms of controlled growth and persistence of second homology. The main contribution is the formulation of a local structural axiom, the Local H2-Triviality Axiom (LHTA), asserting the absence of second homology in links of vertices of the compatibility complex for NC1-computable languages. It is shown that this axiom implies bounded homological persistence and yields polylogarithmic upper bounds on the global second Betti number, reducing the NC1 Homological Growth Conjecture to a local topological statement. While the framework does not resolve P versus NP, it clarifies why local obstruction methods succeed below NC1 and necessarily fail to extend to stronger complexity classes.
Significance for the P vs NP Problem
This work does not resolve the P vs NP problem, nor does it establish new complexity class separations. Its significance lies instead in a structural clarification: it shows that techniques based on local topological obstructions—which are effective for constant-depth circuits and, in a quantitative form, for NC¹—reach an intrinsic limit at this level.
In particular, the analysis suggests that NC¹ is the largest natural complexity class whose computational power can be constrained by fixed-dimensional, local topological invariants. If P ≠ NP holds, any proof must therefore rely on global, non-local phenomena, rather than on obstructions detectable in bounded-radius neighborhoods.
From this perspective, the contribution is not a step toward solving P vs NP directly, but a clarification of why the problem demands fundamentally different methods from those successful in lower complexity classes.
Esp:
Este trabajo no resuelve el problema P vs NP ni establece nuevas separaciones entre clases de complejidad. Sin embargo, aporta una clarificación estructural relevante: muestra que las técnicas basadas en obstrucciones topológicas locales —efectivas para circuitos de profundidad constante y, de forma cuantitativa, para NC¹— alcanzan un límite intrínseco en este régimen.
En particular, el análisis sugiere que NC¹ es la última clase natural cuya potencia computacional puede caracterizarse mediante invariantes topológicos locales de dimensión fija. Si P ≠ NP es cierto, cualquier demostración deberá necesariamente explotar fenómenos globales y no localizables, y no podrá apoyarse únicamente en obstrucciones detectables en vecindarios de tamaño constante.
En este sentido, el trabajo no acerca una solución directa a P vs NP, pero sí contribuye a entender por qué el problema requiere métodos cualitativamente distintos de los que han tenido éxito en clases más débiles.