[Paper Review] Small Polynomial Time Universal Petri Nets

Dmitry A. Zaitsev|arXiv (Cornell University)|Sep 27, 2013

Petri Nets in System Modeling7 references3 citations

TL;DR

This paper demonstrates that small universal Petri nets with 14 places and 29 transitions, originally thought to have exponential time complexity, can be modified using inhibitor arcs and an inverse control flow with moving zero to achieve polynomial time complexity when interpreted as timed Petri nets with multichannel transitions. The resulting deterministic arithmetic Petri net (DAPN) simulates weakly universal Turing machines in linear time, proving small universal Petri nets are efficient models for high-performance computation.

ABSTRACT

The time complexity of the presented in 2013 by the author small universal Petri nets with the pairs of places/transitions numbers (14,42) and (14,29) was estimated as exponential. In the present paper, it is shown, that their slight modification and interpretation as timed Petri nets with multichannel transitions, introduced by the author in 1991, allows obtaining polynomial time complexity. The modification concerns using only inhibitor arcs to control transitions' firing in multiple instances and employing an inverse control flow represented by moving zero. Thus, small universal Petri nets are efficient that justifies their application as models of high performance computations.

Motivation & Objective

• To address the inefficiency of previously constructed small universal Petri nets, which exhibited exponential time complexity.
• To demonstrate that minor modifications to existing small universal Petri nets can drastically improve their time complexity.
• To establish that inhibitor arcs and an inverse control flow with moving zero enable polynomial-time computation.
• To introduce and formalize the concept of a deterministic arithmetic Petri net (DAPN) as a model for efficient computation.
• To show that small universal Petri nets can simulate universal Turing machines and other computational models in polynomial time.

Proposed method

• The paper modifies existing small universal Petri nets (UPN(14,29)) by introducing a new place, LEFT, to explicitly represent left moves of a Turing machine's control head.
• It employs only inhibitor arcs to control multiple firings of transitions, ensuring deterministic behavior and avoiding exponential blowup.
• The control flow is inverted by using moving zero as a mechanism to check conditions, replacing traditional token-based checks.
• The modified net is interpreted as a timed Petri net with multichannel transitions, where each transition's firing time is unit and channels are unlimited.
• A new class of Petri net, the deterministic arithmetic Petri net (DAPN), is formalized as a quadruple (P, T, F, μ⁰), combining deterministic inhibitor Petri nets with timed multichannel transitions.
• The dynamics are defined via a sequential firing rule that prioritizes transitions by index and computes the number of firings via a via operation involving division and zero checks.

Experimental results

Research questions

• RQ1 Can small universal Petri nets with 14 places and 29 transitions be modified to achieve polynomial time complexity?
• RQ2 What role do inhibitor arcs and inverse control flow with moving zero play in enabling efficient computation in Petri nets?
• RQ3 How does interpreting a Petri net as a timed net with multichannel transitions affect its time complexity?
• RQ4 Can the resulting model simulate weakly universal Turing machines in linear time?
• RQ5 Is it possible to construct a universal Petri net that simulates other computational models like cellular automata or tag systems in polynomial time?

Key findings

• The modified Petri net PolyUPN(15,29), derived from UPN(14,29), simulates WUTM(2,4) in time O(14·k) and space O(k), where k is the number of Turing machine steps.
• The time complexity of the modified net is polynomial, specifically linear in the number of steps of the simulated weakly universal Turing machine.
• The use of inhibitor arcs and an inverse control flow with moving zero enables the net to simulate Turing machine transitions in a deterministic and efficient manner.
• The net's behavior is formalized as a deterministic arithmetic Petri net (DAPN), which allows synchronous firing of the maximal valid set of transitions at each step.
• The construction demonstrates that small universal Petri nets can simulate universal Turing machines and other models like cellular automata, cyclic tag systems, and 2-tag systems in polynomial time.
• The results confirm that small universal Petri nets are efficient models for high-performance computation, justifying their use in Petri net-based computational paradigms.