[Paper Review] How to Obtain Computational Completeness in P Systems with One Catalyst

Rudolf Freund, Gheorghe Pǎun|Sep 5, 2013

DNA and Biological Computing3 citations

TL;DR

This paper establishes computational completeness in P systems using only one catalyst by introducing specific control mechanisms—such as rule selection via labels, targets, or time-varying rule sets—enabling universal computation in one membrane with minimal resources. The key contribution is proving that computational completeness is achievable with a single catalyst when combined with appropriate control strategies, resolving an open problem under these constraints.

ABSTRACT

Whether P systems with only one catalyst can already be computationally complete, is still an open problem. Here we establish computational completeness by using specific variants of additional control mechanisms. At each step using only multiset rewriting rules from one set of a finite number of sets of multiset rewriting rules allows for obtaining computational completeness with one catalyst and only one membrane. If the targets are used for choosing the multiset of rules to be applied, for getting computational completeness with only one catalyst more than one membrane is needed. If the available sets of rules change periodically with time, computational completeness can be obtained with one catalyst in one membrane. Moreover, we also improve existing computational completeness results for P systems with mobile catalysts and for P systems with membrane creation.

Motivation & Objective

• To resolve the open problem of whether P systems with only one catalyst can achieve computational completeness.
• To investigate whether additional control mechanisms can compensate for the absence of multiple catalysts in achieving universality.
• To improve existing results on computational completeness in P systems with mobile catalysts and membrane creation by reducing resource requirements.
• To demonstrate that computational completeness can be achieved with only one membrane and one catalyst using time-varying rule sets or rule selection mechanisms.
• To clarify the minimal requirements for universality in catalytic P systems under restricted catalyst and membrane configurations.

Proposed method

• Employing time-varying rule sets where the available rules cycle periodically with a fixed period (e.g., period six), enabling maximal parallel derivation in a single membrane.
• Using rule labeling with a finite alphabet H, where in each step only rules with the same label are applied, ensuring controlled and sequential-like behavior in a maximally parallel setting.
• Applying target-based rule selection (e.g., rules targeting 'in', 'out', 'here') across multiple membranes to guide rule application and achieve computational completeness.
• Designing a simulation of register machines using catalysts and object evolution rules, where register contents are encoded via object counts and control flow via label propagation.
• Introducing trap objects (#) to prevent invalid configurations and ensure halting only in correct derivations, enforcing correctness through irreversible rule application.
• Constructing a P system with one membrane, one catalyst, and a time-varying rule set of period six, showing that halting occurs only when no rules can be applied, thus achieving universality.

Experimental results

Research questions

• RQ1 Can computational completeness be achieved in P systems with only one catalyst using additional control mechanisms?
• RQ2 Is it possible to achieve universality in a single-membrane P system with one catalyst by using time-varying rule sets?
• RQ3 Can target-based rule selection in multi-membrane systems with one catalyst simulate universal computation?
• RQ4 Does the use of rule labeling with a finite alphabet enable computational completeness in one-membrane P systems with a single catalyst?
• RQ5 Can the need for membrane creation or permeability control be eliminated when achieving computational completeness with one catalyst?

Key findings

• Computational completeness is achieved in a P system with one catalyst, one membrane, and a time-varying rule set of period six using the maximally parallel derivation mode.
• A P system with one catalyst and rule selection via labels from a finite alphabet H can simulate universal computation in a single membrane, proving computational completeness.
• Target-based rule selection enables computational completeness with one catalyst, but requires more than one membrane, thus resolving the multi-membrane case.
• The use of trap objects (#) ensures that incorrect configurations lead to non-halting computations, enforcing correctness in the simulation of register machine instructions.
• The result improves upon earlier constructions by eliminating the need for membrane creation and permeability control operations δ and τ, reducing resource usage.
• The Parikh image of the language generated by the constructed P system equals the Parikh image of the target recursively enumerable language, confirming computational completeness.