Playing with Derivation Modes and Halting Conditions

In the area of P systems, besides the standard maximally parallel derivation mode, many other derivation modes have been investigated, too. In this paper, many variants of hierarchical P systems and tissue P systems using different derivation modes are considered and the effects of using di erent derivation modes, especially the maximally parallel derivation modes and the maximally parallel set derivation modes, on the generative and accepting power are illustrated. Moreover, an overview on some control mechanisms used for (tissue) P systems is given. Furthermore, besides the standard total halting mode, we also consider different halting conditions such as unconditional halting and partial halting and explain how the use of different halting modes may considerably change the computing power of P systems and tissue P systems

How to Go Beyond Turing with P Automata: Time Travels, Regular Observer !-Languages, and Partial Adult Halting

In this paper we investigate several variants of P automata having in nite runs on nite inputs. By imposing speci c conditions on the in nite evolution of the systems, it is easy to nd ways for going beyond Turing if we are watching the behavior of the systems on in nite runs. As speci c variants we introduce a new halting variant for P automata which we call partial adult halting with the meaning that a speci c prede ned part of the P automaton does not change any more from some moment on during the in nite run. In a more general way, we can assign !-languages as observer languages to the in nite runs of a P automaton. Speci c variants of regular !-languages then, for example, characterize the red-green P automata

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

P Systems: from Anti-Matter to Anti-Rules

The concept of a matter object being annihilated when meeting its corresponding anti-matter object is taken over for rule labels as objects and anti-rule labels as the corresponding annihilation counterpart in P systems. In the presence of a corresponding anti-rule object, annihilation of a rule object happens before the rule that the rule object represents, can be applied. Applying a rule consumes the corresponding rule object, but may also produce new rule objects as well as anti-rule objects, too. Computational completeness in this setting then can be obtained in a one-membrane P system with non-cooperative rules and rule / anti-rule annihilation rules when using one of the standard maximally parallel derivation modes as well as any of the maximally parallel set derivation modes (i.e., non-extendable (multi)sets of rules, (multi)sets with maximal number of rules, (multi)sets of rules a ecting the maximal number of objects). When using the sequential derivation mode, at least the computational power of partially blind register machines is obtained

Computable functions, quantum measurements, and quantum dynamics

We construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolutions, would contradict the Church-Turing thesis which lies at the foundation of computer science. We conclude that either the Church-Turing thesis needs revision, or that only restricted classes of observables may be realized, in principle, as measurements, and that only restricted classes of unitary operators may be realized, in principle, as dynamics.Comment: 4 pages, REVTE

Instruction sequence processing operators

Instruction sequence is a key concept in practice, but it has as yet not come prominently into the picture in theoretical circles. This paper concerns instruction sequences, the behaviours produced by them under execution, the interaction between these behaviours and components of the execution environment, and two issues relating to computability theory. Positioning Turing's result regarding the undecidability of the halting problem as a result about programs rather than machines, and taking instruction sequences as programs, we analyse the autosolvability requirement that a program of a certain kind must solve the halting problem for all programs of that kind. We present novel results concerning this autosolvability requirement. The analysis is streamlined by using the notion of a functional unit, which is an abstract state-based model of a machine. In the case where the behaviours exhibited by a component of an execution environment can be viewed as the behaviours of a machine in its different states, the behaviours concerned are completely determined by a functional unit. The above-mentioned analysis involves functional units whose possible states represent the possible contents of the tapes of Turing machines with a particular tape alphabet. We also investigate functional units whose possible states are the natural numbers. This investigation yields a novel computability result, viz. the existence of a universal computable functional unit for natural numbers.Comment: 37 pages; missing equations in table 3 added; combined with arXiv:0911.1851 [cs.PL] and arXiv:0911.5018 [cs.LO]; introduction and concluding remarks rewritten; remarks and examples added; minor error in proof of theorem 4 correcte