Solution to PSPACE-Complete Problem Using P Systems with Active Membranes with Time-Freeness

Song, Bosheng; Kong, Yuan

doi:https://doi.org/10.1155/2019/5793234

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 5793234 | https://doi.org/10.1155/2019/5793234

Solution to PSPACE-Complete Problem Using P Systems with Active Membranes with Time-Freeness

Bosheng Song¹and Yuan Kong²

Academic Editor: Francesca Vipiana

Received12 May 2019

Revised13 Jun 2019

Accepted18 Jun 2019

Published27 Jun 2019

Abstract

P systems with active membranes are powerful parallel natural computing models, which were inspired by cell structure and behavior. Inspired by the parallel processing of biological information and with the idealistic assumption that each rule is completed in exactly one time unit, P systems with active membranes are able to solve computational hard problems in a feasible time. However, an important biological fact in living cells is that the execution time of a biochemical reaction cannot be accurately divided equally and completed in one time unit. In this work, we consider time as an important factor for the computation in P systems with active membranes and investigate the computational efficiency of such P systems. Specifically, we present a time-free semiuniform solution to the quantified Boolean satisfiability problem (QSAT problem, for short) in the framework of P systems with active membranes, where the solution to such problem is correct, which does not depend on the execution time for the used rules.

1. Introduction

Membrane computing is a member of natural computing that seeks to discover new computational models from the study of biological cells [1, 2]. Since the first computational model was proposed in 1998, many variants of models were constructed [3–7], and all computational models considered in such framework are called P systems. Inspired by the biological structure of cells, there are two kinds of P systems: neural-like P systems [8, 9] or tissue-like P systems [10, 11] (arbitrary graph structure) and cell-like P systems [2, 12] (tree structure). For more information about the area of membrane computing, we refer the reader to [13–18].

A P system with active membranes is one of basic cell-like P systems; all membranes are embedded in a skin membrane [19, 20]. In such P systems, each membrane delimits a region (also called compartment), which contains multisets of objects and evolution rules. If a membrane does not contain any other membrane, then it is called elementary; if a membrane contains at least an inner membrane, then it is called nonelementary. An interesting feature of P systems with active membranes is that there exists one of electrical charge (positive , negative , or neutral ) on each membrane, and such P systems are evolved according to the following types of rules: (a) send-in communication rules, (b) send-out communication rules, (c) evolution rules, (d) division rules, and (e) dissolving rules. When one of such rules is applied, the charge on the involved membrane may be changed.

There are many interesting results obtained by P systems with active membranes and their variants [19, 21, 22]. Because P systems with active membranes contain membrane division rules, which can generate arbitrary numbers of membranes, that is, an exponential workspace in polynomial time has been generated, they provide a way to theoretically solve NP-complete problems or PSPACE-complete problem in feasible time (in polynomial time or in linear time) by a time-space trade-off [23–27]. In [28], the QSAT problem is solved in linear time by P systems with restricted active membranes; however, in [29], the QSAT problem is solved in polynomial time by P systems with active membranes and without polarizations.

Inspired by biological fact that the execution time of a biochemical reaction cannot be accurately divided equally and completed in one time unit, in [30], timed P systems were proposed such that a natural number representing the execution time is associated with each rule. The notion of time-free is also proposed in [30], which is considered to solve hard computational problems [31–37]. In all these time-free solutions to hard computational problems, the correct computation results are obtained when any execution time is given for each rule. Note that, in [32, 36], the QSAT problem was solved by P systems in a time-free uniform way.

In this work, the notion of time-freeness is incorporated into P systems with active membranes, and a time-free semiuniform solution to the PSPACE-complete problem, QSAT problem, is presented, where the solution to such problem is correct, which does not depend on the execution time for the used rules.

2. Preliminary and Model Description

In this section, we first give the basic conception of formal language theory and the notion of P systems with active membranes [20, 38].

For an alphabet (a finite nonempty set of symbols), we denote by the set of all strings over , and by we denote the set of nonempty strings. The length of a string , denoted by , is the total number of symbols in the string.

We denote by a multiset over an alphabet , where is a mapping from onto (the set of natural numbers). If , then the multiset of is denoted by , which can also be represented by or by any permutation of this string.

Definition 1. A P system with active membranes is a tuplewhere (i) is an alphabet which includes all objects used in the system;(ii) is a label set which marks for each membrane;(iii) is an initial membrane structure;(iv), are multisets of objects that are placed in membranes at the initial configuration;(v) is a finite set of rules with the following forms:(a)Object evolution rules: , for (b)Send-in communication rules: , for (c)Send-out communication rules: , for (d)Dissolving rules: , for (e)Division rules for elementary membranes or nonelementary membranes: , for

The current membrane structure (including polarization associates with each membrane) and the multisets of objects in each membrane at a moment are considered a configuration of the P system. A P system starts from the initial configuration; using rules in a maximally parallel manner, a sequence of consecutive configurations is archived. If there is no rule used in the system, then halting configuration is reached. The result of a system is encoded by the objects present in the output membrane or emitted from the skin membrane when the system reaches halting configuration.

Next we give the definitions of timed P systems with active membranes and the corresponding recognizer version of such model; time-free solutions to decision problems by such P systems are also introduced [33, 39].

Definition 2. A timed P system with active membranes is a pair , where is a P system with active membranes and is a mapping from the finite set of rules in the system into the set of natural numbers (representing the execution times for rules in such system).

A timed P system with active membranes works in the following way: an external clock is assumed, which marks time-units of equal length; the step of computation is defined by the period of time that goes from instant to instant . If a membrane contains a rule from types selected to be executed, then execution of such rule takes time units to complete. Therefore, if the execution is started at instant , the rule is completed at instant and the resulting objects and membranes become available only at the beginning of step . When a rule from types (b) – (e) is started, then the occurrences of symbol-objects and the membrane subject to this rule cannot be subject to other rules from types (b)–(e) until the implementation of the rule completes. At one step, a membrane can be subject to several rules of type (a).

In timed P systems with active membranes, the evolution rules and division rules also follow the “bottom-up" manner (one can refer to [33, 39] for more details).

Definition 3. A recognizer timed P system with active membranes of degree is a tuple , , such that (i)the tuple is a P system with active membranes, where alphabet includes two elements, yes and no;(ii): the output zone is the environment;(iii) is a time-mapping of ;(iv)all the computations halt;(v)either object yes or object no (but not both) must appear in the output region (environment) when the system reaches the halting configuration.

The present work also uses rule starting steps (RS-steps, for short) as the computation steps; a computation step is called a RS-step if at this step at least one rule starts its execution [33, 39].

A decision problem is a pair () such that is a language over a finite alphabet (whose elements are called instances) and is a total Boolean function (i.e., predicate) over .

Definition 4. Let be a decision problem. We say that is solvable in time-free polynomial RS-steps by a family of recognizer P systems with active membranes if the following items are true: (i)the family is polynomially uniform by a Turing machine; that is, there exists a deterministic Turing machine working in polynomial time which constructs the system from the instance .(ii)the family is time-free sound (with respect to ); that is, for any time-mapping , the following property holds: if for each instance of the problem such that there exists an accepting computation of , we have .(iii)the family is time-free complete (with respect to ); that is, for any time-mapping , the following property holds: if for each instance of the problem such that , every computation of is an accepting computation.(iv)the family is time-free polynomially bounded; that is, there exists a polynomial function such that, for any time-mapping and for each , all the computations in halt in, at most, RS-steps.

We also say that the family provides an efficient time-free semiuniform solution to the decision problem .

3. A Time-Free Solution to QSAT Problem by P Systems with Active Membranes

Quantified Boolean formula problem (QBF, in short) is a well-known PSPACE-complete problem [40]; it asks whether a quantified sentential form over a set of Boolean variables is true or false. For more details about QBF, one can refer to [28].

Theorem 5. A family of P systems with active membranes using rules of types (a), (b), (c), and () can solve the QSAT problem in polynomial RS-steps in a time-free manner, where the solution to such problem is correct, which does not depend on the execution time for the used rules.

Proof. Let us consider a quantified Boolean formula:whereWe construct the following P system with active membranes:with the following components: The rules designed for solving the QSAT problem are divided into four parts: generation phase, checking phase, quantifier phase, and output phase. In what follows, we give the set and its explanation. Let be any time-mapping from to (set of natural numbers representing the execution times for the rules).
Generation Phase , . , . , . , , and the clauses contain the literal . , , and the clauses contain the literal . , . , . , . , . , . , . . , . , . , .Initially, we have object in membrane 1, which corresponds to variable , and object no in membrane . At step 1, rules and start to be used at the same step, but they may complete at different steps. By using , membrane 1 is divided, objects (representing the true value true) and (representing the true value false) are generated, and each of the new produced membranes will obtain one of these two objects. Besides, object no exits the membrane by using rule ; polarization of membrane is changed to positive. During the execution of rule , the system takes one RS-step.
Rules and start to be used at the same step only when rule completes; however, due to the fact that execution for rules and may take different times, these two rules may complete at different steps. By using rule (resp., ), object (resp., ) enters membrane 2. After the execution of rule (resp., ), system starts to use rule (resp., ); object (resp., ) enters membrane 3. In this way, rules (resp., ) () will be applied one by one, and object (resp., ) will be transferred to membrane .
If membrane has object (resp., ), rule (resp., ) starts to apply; these rules are used to look for the clauses satisfied by the truth-assignment true (resp., false) of variable . Note that rules and may start to be used at different steps.
If membrane contains object (resp., ), then rule (resp., ) is used; object (resp., ) exits membrane . When membrane contains object (resp., ), rule (resp., ) starts to apply; object (resp., ) exits membrane . In this way, rules (resp., ) () will be applied one by one, and object (resp., ) will be transferred to membrane 1. Note that rules and may start at different steps; rules and () may start at different steps.
When membrane 1 contains object (resp., ), system starts to use rule (resp., ), object (resp., ) exits membrane 1, and the polarization of membrane is changed from neutral to positive (resp., negative). Note that rules and may start to be used at different steps. Rule can be applied only when membrane has object , and there is a membrane 1 having polarization negative; rule can be applied only when both rules and are completed. By applying rule , object evolves to , and object enters membrane 1; polarization of membrane 1 is changed to positive. Hence, rule has a synchronization function because may not be equal to . Hence, after the execution of rule , the system takes at most RS-steps.
After the execution of rule , the system starts to use rule ; object exits membrane 1. Now we have the following two cases: (i)The execution of rule has completed; in this case, rule starts to be used, the polarization of membrane changes to positive, and two copies of object are produced. Each membrane 1 will obtain one copy of object , and polarization of the corresponding membrane is changed to neutral.(ii)The system is still at the execution of rule ; rules and will be applied after the execution of rule . After the execution of rule , the system starts to use rules and .After the execution of rule , rule starts to apply; object enters membrane 2. In all membranes 1, the system starts to use rule at the same step, so rule in all membranes 1 completes at the same step; that is, in each membrane 2, object is generated at the same step. In general, when membrane 2 has object , the system takes at most RS-steps.
When membrane 2 has object , the system starts to assign truth-assignment of variable and looks for clauses satisfied by such variable.
If membrane 2 has object , rule starts to be used; the system starts to assign truth values true and false to variable . In all membranes 1, the system starts and completes rule at the same step. The system starts to use rules and at the same step, but these rules may complete at different steps. Besides, the system starts and completes rules (), (), , , (), (), , and at different steps. Note that rule can be used only when all the above rules have completed their executions. Thus, this process takes at most RS-steps. So we can deduce that if we consider this process in the worse case, then the system assigns from to (); the number of RS-steps decreases by 4. Hence, the system assigns truth-assignment of variable and looks for the clauses satisfied by this variable; this process takes at most 13 RS-steps. Therefore, the system takes at most RS-steps for this process.
When membrane contains object , the system starts to execute rule . By using rule , objects and are generated, which will be placed in two separate copies of membrane . Note that rule in all membranes starts and completes at the same step. Rules and start to be used at the same step, but they may complete at different steps. Rules and may start and complete at different steps. When both rules and have been completed, rule starts to apply. So, rule has a synchronization function as may not be equal to . If rule completes, rules , , and start to apply one by one. Hence, the system assigns truth-assignment of variable and looks for the clauses satisfied for such variable, this process takes at most 8 RS-steps.
So this phase takes at most RS-steps.
The membrane structure is described in Figure 1 when generation phase finishes.
Checking Phase . , . , . , . .After the execution of rule , all objects in each membrane enter membrane ; polarization for this membrane is changed to neutral at the same step. At that moment, rule is applied; object evolves to object in membrane . After the execution of rule , the system starts to check whether each membrane has object or not.
By applying rule , object evolves to and object enters membranes ; polarization of such membrane is changed from neutral to positive. After the execution of rule , rule starts to be used, object evolves to , and object enters membrane ; polarization of such membrane is changed from positive to neutral. When the charge of membrane changes to neutral, rule is used, object evolves to , and object exits membrane . When membrane has object , it means clause is satisfied, and the system starts to check whether membrane has object or not.
If membrane has object , rule starts to be used. When membrane has object , object will be obtained. If a membrane does not contain an object , then this membrane will stop evolving when is supposed to be used. In general, the system takes at most RS-steps.
If there exists a membrane which contains all objects , then rule starts to apply; object evolves to in membrane .
This phase takes at most RS-steps.
Quantifier Phase . , . , . , .After the execution of rule (if membranes have object ), object presents in membranes at the same step. At that moment, rule starts to be used; object (if it exists) exits membranes . Note that in membranes rule starts and completes at the same step.
The quantifier is simulated by transferring only one copy of object to an upper level membrane; that is, in membrane (), there exists at least one copy of object . Specifically, if membranes () have object , rule starts to be used, object exits membranes , and the polarization of such membrane is changed from neutral to positive. Note that, in each of the immediately lower level membranes, if there exist two copies of object in a membrane (), then in membranes two copies of object appear at the same step. Obviously, there are levels of membranes (), and rule starts to be used at different steps for each level of membranes . Hence, the simulation of all quantifiers takes RS-steps.
The quantifier is simulated by transferring only one copy of to an upper level membrane; that is, there are two copies of object in membrane (). Specifically, if membranes () contain two copies of , rule starts to apply, object evolves to , and object exits membrane; the charge of such membrane is changed from neutral to positive. After the execution of rule , rule starts to be used, object exits this membrane, and the charge is changed from positive to neutral. Note that if each of the immediately lower level membranes of a membrane () has one copy of object , then two copies of appear in membranes at the same step. There are levels of membranes (), and rules start to be used at different steps for each level of the membranes ; thus, the simulation of all quantifiers takes RS-steps.
In general, this phase takes RS-steps.
Output Phase . . .At step 1, object no is sent out of the system by using rule . Note that this operation takes no RS-step. After the quantifier phase, we have the following two cases.
If positive membrane does not contain object , in this case, rules cannot be applied. Hence when computation halts, the environment has object no, which means the formula is not satisfiable.
If positive membrane contains object , in this case, rule will be applied, object evolves to yes, which will be sent to the environment, and the polarization of membrane is changed from positive to negative. After the execution of rule , rule starts to be used; object no enters membrane . Hence when computation halts, the environment has one copy of yes, which means the formula is satisfiable. The output phase takes two RS-steps.
According to the constructed P systems, for any time-mapping , if the computation halts, object yes (resp., no) appears in the environment if and only if the formula is satisfiable (resp., not satisfiable). Thus, the system is time-free sound and time-free complete.
For any time-mapping , the computation takes at most RS-steps when formula is satisfiable, and the computation takes at most RS-steps when formula is not satisfiable. Therefore, the family of P systems with active membrane is time-free polynomially bounded.
The family is polynomially uniform: (i)total number of objects: ;(ii)number of initial membranes: ;(iii)cardinality of the initial multisets: ;(iv)total number of rules: ;(v)maximal length of a rule: .Therefore, P systems with active membranes can solve the QSAT problem in polynomial RS-steps in a time-free manner; this concludes the proof.

4. Conclusions and Future Work

In this work, a time-free way of using rules is considered into P systems with active membranes, and a time-free solution to the QSAT problem by using P systems with active membranes has been given, where the solution to such problem is correct, which does not depend on the execution time for the used rules.

P systems with active membranes presented in this work are semiuniform; that is, the P systems are designed from the instances of the problem. It remains open how we can construct a family of P systems to solve the QSAT problem in a time-free manner in the sense that P systems are designed from the size of instances.

The P system constructed in Section 3 has the polarization on membranes. It is of interest to investigate whether P systems with active membranes without polarization on membranes can still solve the QSAT problem in a time-free context.

QSAT problem is one of the most important issues in many application areas, such as artificial intelligence aspect (e.g., planning, nonmonotonic reasoning, scheduling, model checking, and verification formal verification can be reduced to QSAT [41–45]) and fault localization in digital circuits aspect [46–51]. It is interesting to design new algorithms based on QSAT which can be used in the above-mentioned areas and other areas.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61602192 and 61602188), the Fundamental Research Funds for the Central Universities (531118010355), and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (2017RCJJ068 and 2017RCJJ069).

References

G. Păun, “Computing with membranes,” Journal of Computer and System Sciences, vol. 61, no. 1, pp. 108–143, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
G. Păun, G. Rozenberg, and A. Salomaa, “DNA computing,” in New Computing Paradigms, Springer, Berlin, Germany, 1998.
View at: Publisher Site | Google Scholar
P. Bottoni, A. Labella, and G. Rozenberg, “Reaction systems with infuence on?environment,” Journal of Membrane Computing, vol. 1, pp. 3–19, 2019.
View at: Google Scholar
M. À. Colomer, A. Margalida, D. Sanuy, and M. J. Pérez-Jiménez, “A bio-inspired computing model as a new tool for modeling ecosystems: The avian scavengers as a case study,” Ecological Modelling, vol. 222, no. 1, pp. 33–47, 2011.
View at: Publisher Site | Google Scholar
G. Román, “Inference of bounded L systems with polymorphic P systems,” Journal of Membrane Computing, vol. 1, pp. 52–57, 2019.
View at: Publisher Site | Google Scholar
M. García-Quismondo, M. Levin, and D. Lobo, “Modeling regenerative processes with membrane computing,” Information Sciences, vol. 381, pp. 229–249, 2017.
View at: Publisher Site | Google Scholar
T. Wu, A. Paun, Z. Zhang, and L. Pan, “Spiking neural P systems with polarizations,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 8, pp. 3349–3360, 2018.
View at: Publisher Site | Google Scholar
M. Ionescu, G. Pǎun, and T. Yokomori, “Spiking neural P systems,” Fundamenta Informaticae, vol. 71, no. 2-3, pp. 279–308, 2006.
View at: Google Scholar | MathSciNet
T. Wu, F.-D. Bîlbîe, A. Pǎun, L. Pan, and F. Neri, “Simplified and yet turing universal spiking neural P systems with communication on request,” International Journal of Neural Systems, vol. 28, no. 8, Article ID 1850013, 2018.
View at: Google Scholar
C. Martín-Vide, G. Păun, J. Pazos, and A. Rodríguez-Patón, “Tissue P systems,” Theoretical Computer Science, vol. 296, no. 2, pp. 295–326, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
L. Pan, B. Song, L. Valencia-Cabrera, and M. J. Pérez-Jiménez, “The computational complexity of tissue P systems with evolutional symport/antiport rules,” Complexity, vol. 2018, Article ID 3745210, 21 pages, 2018.
View at: Google Scholar
D. Orellana-Martín, L. Valencia-Cabrera, A. Riscos-Núñez, and M. J. Pérez-Jiménez, “P systems with proteins: a new frontier when membrane division disappears,” Journal of Membrane Computing, vol. 1, no. 1, pp. 29–39, 2019.
View at: Publisher Site | Google Scholar
G. Ciobanu, M. J. Pérez-Jiménez, and G. Paun, Eds., Applications of Membrane Computing, Natural Computing Series, Springer, Berlin, Germany, 2006.
View at: MathSciNet
D. Díaz-Pernil, M. A. Gutiérrez-Naranjo, and H. Peng, “Membrane computing and image processing: a short survey,” Journal of Membrane Computing, vol. 1, no. 1, pp. 58–73, 2019.
View at: Publisher Site | Google Scholar
P. Frisco, M. Gheorghe, and M. J. Pérez-Jiménez, Applications of Membrane Computing in Systems and Synthetic Biology, Springer, Switzerland, 2013.
R. Mayne, N. Phillips, and A. Adamatzky, “Towards experimental P-systems using multivesicular liposomes,” Journal of Membrane Computing, vol. 1, no. 1, pp. 20–28, 2019.
View at: Publisher Site | Google Scholar
Gh. Paun, G. Rozenberg, and A. Salomaa, Eds., The Oxford Handbook of Membrane Computing, Oxford University Press, New York, NY, USA, 2010.
E. Sánchez-Karhunen and L. Valencia-Cabrera, “Modelling complex market interactions using PDP systems,” Journal of Membrane Computing, vol. 1, pp. 40–51, 2019.
View at: Publisher Site | Google Scholar
A. Paun, “On P systems with active membranes,” in Proceedings of the Second International Conference on Unconventional Models of Computation, pp. 187–201, Springer, London, UK, 2000.
View at: Google Scholar | MathSciNet
G. Păun, “P systems with active membranes attacking NP-complete problems,” Journal of Automata, Languages and Combinatorics, vol. 6, no. 1, pp. 75–90, 2001.
View at: Google Scholar | MathSciNet
A. Alhazov, L. Pan, and G. Păun, “Trading polarizations for labels in P systems with active membranes,” Acta Informatica, vol. 41, no. 2-3, pp. 111–144, 2004.
View at: Publisher Site | Google Scholar
G. Ciobanu, L. Pan, G. Păun, and M. J. Pérez-Jiménez, “P systems with minimal parallelism,” Theoretical Computer Science, vol. 378, no. 1, pp. 117–129, 2007.
View at: Publisher Site | Google Scholar
L. Pan, D. P. Daniel, and M. J. Pérez-Jiménez, “Computation of ramsey numbers by P systems with active membranes,” International Journal of Foundations of Computer Science, vol. 22, no. 1, pp. 29–38, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
L. Pan and C. Martín-Vide, “Solving multidimensional 0-1 knapsack problem by P systems with input and active membranes,” Journal of Parallel and Distributed Computing, vol. 65, no. 12, pp. 1578–1584, 2005.
View at: Publisher Site | Google Scholar
L. Pan and C. Martín-Vide, “Further remark on P systems with active membranes and two polarizations,” Journal of Parallel and Distributed Computing, vol. 66, no. 6, pp. 867–872, 2006.
View at: Publisher Site | Google Scholar
P. Sosík, “The computational power of cell division in P systems: Beating down parallel computers?” Natural Computing, vol. 2, no. 3, pp. 287–298, 2003.
View at: Publisher Site | Google Scholar
P. Sosík, A. Rodríguez-Patón, and L. Ciencialová, “Standardized proofs of PSPACE-completeness of P systems with active membranes,” in Proceedings of the Eighth Brainstorming Week on Membrane Computing, pp. 301–310, 2010.
View at: Google Scholar
A. Alhazov, C. Martín-Vide, and L. Pan, “Solving a PSPACE-complete problem by recognizing P systems with restricted active membranes,” Fundamenta Informaticae, vol. 58, no. 2, pp. 67–77, 2003.
View at: Google Scholar | MathSciNet
A. Alhazov and M. J. Pérez-Jiménez, “Uniform solution of QSAT using polarizationless active membranes,” Lecture Notes in Computer Science, vol. 4664, pp. 122–133, 2007.
View at: Google Scholar
M. Cavaliere and D. Sburlan, “Time-independent P systems,” Lecture Notes in Computer Science, vol. 3365, pp. 239–258, 2005.
View at: Google Scholar
M. Gheorghe, G. Păun, M. J. Pérez-Jiménez, and G. Rozenberg, “Research frontiers of membrane computing: open problems and research topics,” International Journal of Foundations of Computer Science, vol. 24, no. 5, pp. 547–623, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Y. Niu and Z. G. Wang, “Time-free solution for QSAT by using timed Tissue P systems,” Applied Mechanics and Materials, vol. 568-570, pp. 812–816, 2014.
View at: Publisher Site | Google Scholar
B. Song and L. Pan, “Computational efficiency and universality of timed P systems with active membranes,” Theoretical Computer Science, vol. 567, pp. 74–86, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
B. Song, M. J. Pérez-Jiménez, and L. Pan, “Computational efficiency and universality of timed P systems with membrane creation,” Soft Computing, vol. 19, no. 11, pp. 3043–3053, 2015.
View at: Publisher Site | Google Scholar
B. Song, M. J. Pérez-Jiménez, and L. Pan, “An efficient time-free solution to SAT problem by P systems with proteins on membranes,” Journal of Computer and System Sciences, vol. 82, no. 6, pp. 1090–1099, 2016.
View at: Publisher Site | Google Scholar
B. Song, M. J. Pérez-Jiménez, and L. Pan, “An efficient time-free solution to QSAT problem using P systems with proteins on membranes,” Information and Computation, vol. 256, pp. 287–299, 2017.
View at: Publisher Site | Google Scholar
B. Song, T. Song, and L. Pan, “A time-free uniform solution to subset sum problem by tissue P systems with cell division,” Mathematical Structures in Computer Science, vol. 27, no. 1, pp. 17–32, 2017.
View at: Publisher Site | Google Scholar
G. Paun, Computing with Membranes: An Introduction, Springer-Verlag, Berlin, Germany, 2002.
View at: Publisher Site | MathSciNet
B. Song, T. Song, and L. Pan, “Time-free solution to SAT problem by P systems with active membranes and standard cell division rules,” Natural Computing, vol. 14, no. 4, pp. 673–681, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. H. Papadimitriou, Computational Complexity, Addison-Wesley, Massachusetts, USA, 1994.
View at: MathSciNet
M. Cashmore, M. Fox, and E. Giunchiglia, “Planning as quantified boolean formula,” Frontiers in Artificial Intelligence and Applications, vol. 242, pp. 217–222, 2012.
View at: Google Scholar
X. Gu and W. Ma, “On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials,” Mathematical Methods in the Applied Sciences, vol. 41, no. 10, pp. 3779–3789, 2018.
View at: Publisher Site | Google Scholar
Y. Li, Y. Sun, F. Meng, and Y. Tian, “Exponential stabilization of switched time-varying systems with delays and disturbances,” Applied Mathematics and Computation, vol. 324, pp. 131–140, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
H. Mangassarian, A. Veneris, and M. Benedetti, “Robust QBF encodings for sequential circuits with applications to verification, debug, and test,” IEEE Transactions on Computers, vol. 59, no. 7, pp. 981–994, 2010.
View at: Publisher Site | Google Scholar
P. Marin, M. Narizzano, L. Pulina et al., “Twelve years of QBF evaluations: QSAT is PSPACE-hard and it shows,” Fundamenta Informaticae, vol. 149, no. 1-2, pp. 133–158, 2016.
View at: Publisher Site | Google Scholar
B. Hu, T. Xia, and W. Ma, “Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line,” Applied Mathematics and Computation, vol. 332, pp. 148–159, 2018.
View at: Publisher Site | Google Scholar
J. Ha, H. Zhang, and Q. Zhao, “Exact solutions for a Dirac-type equation with N-fold Darboux transformation,” Journal of Applied Analysis and Computation, vol. 9, no. 1, pp. 200–210, 2019.
View at: Google Scholar | MathSciNet
L. Liu, F. Deng, and T. Hou, “Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition,” Applied Mathematics and Computation, vol. 330, pp. 201–212, 2018.
View at: Publisher Site | Google Scholar
H. Riener and G. Fey, “Exact diagnosis using boolean satisfiability,” in Proceedings of the the EEE/ACM International Conference on Computer-Aided Design, pp. 1–8, Austin, Tex, USA, November 2016.
View at: Publisher Site | Google Scholar
Z. Tu and X. Zeng, “Two classes of permutation trinomials with Niho exponents,” Finite Fields and Their Applications, vol. 53, pp. 99–112, 2018.
View at: Publisher Site | Google Scholar
H. Wu and J. Song, “Mixed lump–stripe soliton solutions to a dimensionally reduced generalized Jimbo–Miwa equation,” Applied Mathematics Letters, vol. 90, pp. 181–187, 2019.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2019 Bosheng Song and Yuan Kong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

886

Downloads

715

Citations