🤖 AI Summary
This work investigates the thermodynamic connection between circuit complexity and functionality, and its implications for program obfuscation. We propose a “circuit thermalization” model, interpreting Boolean circuits as a “gas” of logic gates, where obfuscation corresponds to recursive mixing and entropy-increasing evolution—preserving functionality while driving circuit entropy toward saturation. We introduce the novel concept of “circuit space fragmentation,” proving its equivalence to the NP/coNP equality problem, thereby exposing a fundamental limitation: functionally and size-equivalent circuits are not locally traversable under structural transformations. Integrating thermodynamic analogy, ergodicity analysis, graph-theoretic modeling, and polynomial hierarchy theory, we establish a thermodynamically grounded framework for circuit complexity under functional constraints. Our results provide a physics-inspired paradigm for program obfuscation and rigorously equate circuit fragmentation with standard complexity-class separation assumptions.
📝 Abstract
Circuit complexity, defined as the minimum circuit size required for implementing a particular Boolean computation, is a foundational concept in computer science. Determining circuit complexity is believed to be a hard computational problem [1]. Recently, in the context of black holes, circuit complexity has been promoted to a physical property, wherein the growth of complexity is reflected in the time evolution of the Einstein-Rosen bridge (``wormhole'') connecting the two sides of an AdS ``eternal'' black hole [2]. Here we explore another link between complexity and thermodynamics for circuits of given functionality, making the physics-inspired approach relevant to real computational problems, for which functionality is the key element of interest. In particular, our thermodynamic framework provides a new perspective on the obfuscation of programs of arbitrary length -- an important problem in cryptography -- as thermalization through recursive mixing of neighboring sections of a circuit, which can be viewed as the mixing of two containers with ``gases of gates''. This recursive process equilibrates the average complexity and leads to the saturation of the circuit entropy, while preserving functionality of the overall circuit. The thermodynamic arguments hinge on ergodicity in the space of circuits which we conjecture is limited to disconnected ergodic sectors due to fragmentation. The notion of fragmentation has important implications for the problem of circuit obfuscation as it implies that there are circuits with same size and functionality that cannot be connected via local moves. Furthermore, we argue that fragmentation is unavoidable unless the complexity classes NP and coNP coincide, a statement that implies the collapse of the polynomial hierarchy of computational complexity theory to its first level.
