Probabilistic Shoenfield Machines

TL;DR

This work extends the Shoenfield computation framework by introducing Probabilistic Shoenfield Machines (PSMs) as a probabilistic counterpart to deterministic (DSM) and nondeterministic (NSM) Shoenfield machines. It provides formal definitions for DSM, NSM, and PSM, and proves that NSM and DSM are equivalent, then shows PSMs share the same computational power under bounded-error acceptance, via simulations in both directions. A key decidability result (Theorem 1) demonstrates that any PSM can be simulated by a DSM to approximate its acceptance probability to arbitrary precision, linking probabilistic computation to classical computability. The paper also discusses potential applications and future directions, including probabilistic algorithms, cryptography, and extensions to quantum Shoenfield machines, highlighting the theoretical foundation for randomized computation within the Shoenfield framework. Overall, DSM, NSM, and PSM are shown to capture the same class of partial recursive functions, situating probabilistic Shoenfield computation within the traditional Turing-machine landscape while offering new perspectives for modeling randomness in computation.

Abstract

The article provides the theoretical framework of Probabilistic Shoenfield Machines (PSMs), an extension of the classical Shoenfield Machine that models randomness in the computation process. PSMs are introduced in contexts where deterministic computation is insufficient, such as randomized algorithms. By allowing transitions to multiple possible states with certain probabilities, PSMs can solve problems and make decisions based on probabilistic outcomes, thus expanding the variety of possible computations. We provide an overview of PSMs, detailing their formal definitions, the computation mechanism, and their equivalence with Non-deterministic Shoenfield Machines (NSMs)

Figures (4)

• Figure 1: This figure represents a Deterministic Shoenfield Machine (DSM). The machine has a counter set to $2$, which points to the next instruction. Multiple registers hold integer values: Register 1 has $5$, Register 2 has $6$, Register 3 has $2$, and Register 4 has $2$. The instruction list includes operations such as $\mathsf{INC}\ 1$ (increment Register 1) and $\mathsf{DEC}\ 3,2$ (decrement Register 3 and set the counter to $2$ if the result is zero). The current instruction to be executed is $\mathsf{DEC}\ 3,2$.
• Figure 2: This diagram represents a Non-deterministic Shoenfield Machine (NSM). The machine consists of multiple counters ($2,6,5,1,\ldots$), which point to possible next instructions. Registers hold integer values: Register 1 has $5$, Register 2 has $6$, Register 3 has $2$, and Register 4 has $2$. The instruction list includes operations such as $\mathsf{INC}\ 1$, $\mathsf{INC}\ 2$, $\mathsf{DEC}\ 1,2$, $\mathsf{DEC}\ 1,4$, and $\mathsf{DEC}\ 3,2$. This machine can follow different execution paths due to multiple counters pointing to different instructions.
• Figure 3: This diagram compares a Non-deterministic Shoenfield Machine (NSM) and a Deterministic Shoenfield Machine (DSM). The NSM has multiple counters ($2,6,5,1,\ldots$) pointing to possible next instructions, with registers holding values $5$, $6$, $2$, and $2$. Instructions include operations such as $\mathsf{INC}\ 1$, $\mathsf{DEC}\ 3,2$, and $\mathsf{DEC}\ 1,2$, with arrows showing possible transitions. The DSM follows a single deterministic path between instructions.
• Figure 4: The Probabilistic Shoenfield Machine (PSM). In this case, the machine has multiple counters (2, 6, 5, 1, ...) indicating possible next instructions, with registers holding values 5, 6, 2, and 2. Instructions include operations such as Inc 1, Dec 3,2, and Dec 1,2, with arrows showing possible transitions. A graph represents the probabilities of different instructions. The DSM follows a single deterministic path between instructions.