Abstract
The paper reviews some aspects of MP grammars, a particular type of P systems (M stands for Metabolic) consisting of multiset rewriting rules, which were introduced in the context of Membrane Computing, for modeling biological dynamics. The main features of MP theory are recalled, such as the control mechanisms based on regulation functions, MP graphs, representation of oscillatory dynamics, regression algorithms, and MP modeling. Finally, the computational universality of MP grammars is proved by means of Minsky’s register machines.
Keywords
Discrete dynamics Multiset grammars MP regression algorithms Metabolic computing1 Introduction
MP systems are discrete dynamical systems introduced in the context of membrane computing [1] and investigated for more than 20 years. Preliminary results were developed since the end of 1990 years [2, 3, 4, 5, 6, 7]. Related approaches to MP theory and first investigations on MP regression (see later on) were investigated in [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Applications of MP systems in modeling biological systems, theoretical foundations, and efficient MP regression algorithms were investigated in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].
MP systems are essentially multiset rewriting rules with functions that define the quantities of transformed elements. The attribute MP comes from P systems (multiset rewriting rules distributed in compartments) introduced by Păun [31, 32, 33, 34], where the focus is on metabolic processes. MP regression is a peculiar aspect of MP theory, which provides methods that determine MP grammars able to generate observed time series of an observed dynamics. MP regression algorithms rely on the methods of algebraic manipulation and Least Square Evaluation, or on statistical methods, or genetic algorithms, by providing very accurate solutions [1, 20, 24, 26, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Public software platforms based on MP theory are available, which provide examples and documentation [44, 45].
In the present paper, we mainly consider MP grammars for their computational aspect, which provides a natural notion of distributed computation where the program is encoded by a graph expressing the transformations and the regulations of a finite number of multiset rewriting rules.
2 MP grammars
An MP grammar G is a discrete dynamical system based on a set X of variables, and a state space constituted by the assignments of values to variables of X. Let \(\mathbb {N}\) be the set of natural numbers. Assuming variables in some order, if X is a finite set of \(n \in \mathbb {N}\) variables, the set of possible states of G coincides with the set \(\mathbb {R}^n\) of real vectors of dimension n. A dynamics function \(\delta _G\) is associated with G that provides a nextstate function, which changes the variable values, according to an increase–decrease variation specified by all the rules (if a variable does not occur in a rule, its value remains unchanged). Namely, a reading of “MP” is the basic Minus–Plus mechanism of the rules of an MP grammar. A multiset over X is a function assigning a natural number, called multiplicity, to every \(x \in X\). The formal definition of MP grammar follows.
Definition 1
 1.

X is a finite set of real variables. The values of variables determine their current state;
 2.

R is a finite set of rules. Each rule \(r \in R\) is expressed by \(\alpha _r \rightarrow \beta _r\) with \(\alpha _r ,\beta _r\) multisets over X, where \(\alpha _r(x)\) and \(\beta _r(y)\) denote the multiplicities of x and y in \(\alpha _r\) and in \(\beta _r\), respectively;
 3.

\(\Phi = \{\varphi _r \;  \; r \in R\}\) is the set of regulators, or flux functions. Functions \(\varphi _r\) assume real values depending on the current values of some variables of X. In this way, a regulator \(\varphi _r\) associates with any state \(s \in \mathbb {R}^n\) a value \(u = \varphi _r(s)\), called “flux”, according to which any variable x occurring in \(\alpha _r\) is decreased by the value \(u \cdot \alpha _r(x)\), and any variable y occurring in \(\beta _r\) is increased by the value \(u \cdot \beta _r(y)\) (in both cases the flux is multiplied by the variable multiplicity).
When variables are equipped with measurement units (related to their interpretation) and time duration of each step, the MP grammar is more properly called an MP system.
The dynamics of an MP grammar can be naturally expressed by a system of (firstorder) recurrent equations, synthetically represented in matrix notation (see [24] for details).

\(\emptyset \rightarrow A : b\)

\(\emptyset \rightarrow B : d\)

\(\emptyset \rightarrow F : f\)

\(\emptyset \rightarrow L : k\)

\(H \rightarrow \emptyset : g\)

\(C \rightarrow \emptyset : a\)

\(P \rightarrow \emptyset : h\)

\(G \rightarrow \emptyset : s(C)\)

\(A B \rightarrow CD : e(E)\)

\(D \rightarrow G : c(C)\)

\(E F \rightarrow P : p(H)\)

\(L P \rightarrow H : q(P)\)
Example 2
MP grammars have an intrinsic versatility in describing oscillatory phenomena, which are crucial in all processes of life [24, 27]. Here, we give some simple, but interesting examples.
The schema of MP grammars given in Example 3 [24] has an input rule \(r_1\) and an output rule \(r_3\) incrementing and decrementing variables x and y, respectively. Both rules are regulated by the same variable that they change (autocatalysis), while the rule \(r_2\) from x to y is regulated by both variables (bicatalysis). An MP grammar of this type provides a simple model for predator–prey dynamics first modeled in differential terms by Lotka and Volterra [46]. The model represents the growth of the two populations x and y, preys and predators, respectively. Preys grow by eating nutrients from the environment (according to some reproduction rate) while die by predation. Predators grow by eating preys while die by extinction (according to some death rate). If predators increase, then preys decrease; consequently predators have a minor availability of food and decrease. Symmetrically, a decrease of predators implies a consequent increase of preys.
Example 3
Example 4
Even if the intuition underlying MP grammars was initially linked to metabolism, these grammars can be applied to any type of dynamics where some variables change in time. In this sense, metabolic processes are only special cases of a general dynamical framework. We know that ordinary differential equations (ODE) are the most popular mathematical model for dynamical processes. Therefore, a natural question is: “Why using other formalisms different from ODE?”.
The answer to this question is related to the fact that, very often, we observe a dynamics, but we are completely ignorant about what are the forces acting on variables in their changes. In this case, it is difficult to formulate, in differential terms, the rules acting on the observed system. The only data, on which a mathematical model can be based, are the time series of values that variable assume in some given instants. The search for a mathematical model able to reproduce the observed data, within some approximation, is a case of an “inverse problem”. MP regression algorithms provide systematic methods for solving such inverse problems. Now, we shortly recall how MP grammars can be very useful in this regard.
Models obtained by MP regression
Belousov–Zhabotinsky, Prigogine’s brusselator (BZ)  
Lotka–Volterra, Predatorprey dynamics (LV)  
Susceptible–infected–recovered epidemics (SIR)  
Early amphybian mitotic cycle (AMC)  
Drosophila circadian rythms (DCR)  ([50]) 
Nonphotochemical quenching in photosynthesis (NPQ)  ([56]) 
Minimal diabetes mellitus (MDM)  
Bicatalytic synthetic oscillator  ([36]) 
Synthetic oscillators  
Chaotic dynamics  
Gene expression dynamics 
3 Input–output, positive, and reactive MP grammars
Two MP grammars are dynamically equivalent, with respect to a set of variables, when these variables change according to the same dynamics.
An MP grammar is called input–output if its rules have the empty multiset as left or right member [24]. The following theorem holds.
Theorem 5
Any MP grammar is dynamically equivalently to an input–output MP grammar. Moreover, any system of (firstorder) recurrent equations can be expressed in terms of an MP grammar with input–output rules.
Proof
In fact, any rule \(\alpha \rightarrow \beta : \varphi\) with \(\alpha \not = \emptyset\) and \(\beta \not = \emptyset\) can be transformed into the set of rules \(x \rightarrow \emptyset : \varphi\) (output rule) for every \(x \in \alpha\), and \(\emptyset \rightarrow y: \varphi\) (input rule) for every \(y \in \beta\). Of course, applying \(\alpha \rightarrow \beta : \varphi\) is equivalent to applying all the two corresponding input–output rules.
Then, we can consider n variables \(x^1, \ldots , \ldots x^n\) with rules \(\emptyset \rightarrow x^j: a_{i,j}\) for \(j=1, 2, \ldots m\) such that \((a_{i,j})^+ \not = 0\) and rules \(x^j \rightarrow \emptyset : a_{i,j}\) for \(i,j=1, 2, \ldots n\) such that \((a_{i,j})^+ \not = 0\). Of course, these MP rules provide the same dynamics computed by the original system of recurrent equations. \(\square\)
An external variable (called parameter in [24]) is a variable of an input rule without flux. This means that when a time series of values is given the dynamics of the grammar can be computed by taking the values of the series as values of the external variable. An MP grammar with external variables is not a generator of time series, but a function transforming the time series associated with its external variables into the time series of its internal (i.e. not external) variables.
An MP grammar is noncooperative when in it each rule has at most one left variable, and it is monic when its left variables have at most multiplicity one.
Lemma 6
For any MP grammar there exists a monic MP grammar that is dynamically equivalent to it.
Proof
In fact, let us consider a rule with two variables (the extension to more variables is trivial): \(x y \rightarrow z : \varphi\). Then, we can equivalently split this rule into two rules that change the variables in the same manner: \(x \rightarrow z : \varphi\) and \(y \rightarrow \emptyset : \varphi\). Let us apply this transformation to any rule with more than one left variable. Analogously, a rule such as \(2x \rightarrow z : \varphi\) equivalently transforms into the two rules \(x \rightarrow z : \varphi\) and \(x \rightarrow \emptyset : \varphi\). In this way, we get a monic MP grammar that is dynamically equivalent to the original one.\(\square\)
An MP grammar is positive when, starting from a state where all variables are positive, then in all the following states variables and fluxes are always positive. Given an MP grammar G a positive grammar \(G'\), called the positively controlled grammar associated withG, is defined in the following manner. The grammar \(G'\) has the same variables and the same rules as G. Moreover, a regulator \(\varphi '\) is defined in \(G'\) in correspondence to each regulator \(\varphi\) of G in the following way:
let s(x) be the value of variable x in the state s,
let \(\varphi ^+(s) =\)max\(\{\varphi (s), 0\}\),
let \(\Phi ^(x)\) be the regulators of rules decreasing the variable x.
Theorem 7
For any positive MP grammar, there exists a dynamically equivalent reactive MP grammar and vice versa. (a proof is given in [24]).
The computation of \(\sqrt{7}\)
A  B  C  D  Y  W  Z 

7  0  1  1  0  0  0 
6  1  1  1  0  0  0 
3  2  1  3  0  0  0 
3  3  1  0  0  0  0 
3  1  0  0  3  1  0 
3  2  0  0  2  0  1 
An imperative program expressing the MP grammar of Fig. 5
INPUT A;  
\(C := 1\), \(D : = 1\),  \(B := 0, Y := 0, W := 0 ; Z := 0\); 
WHILE \(Z=0\)  DO 
\(A1 := A ; B1 := A ; C1 := C, \; D1 : = D, Y1 := Y, W1 := W, Z1 := Z\);  
\(B := B + C1\);  
IF \(A1 \ge 2B1+C1\) THEN \(A := A  2B1  C1\) AND \(D := 2B1+C1\);  
IF \(C1 \ge C1+D1\) THEN \(C := C  C1  D1\) AND \(W := W \!+\! C1\!+ \! D1\);  
IF \(B1 \ge B1+D1+Y1\)  
THEN \(B := B  B1  D1 + Y1\) AND \(Y := Y +B1+D1+Y1\);  
IF \(Y1 \ge W1\) THEN \(Y := Y  W1\) AND \(Z := Z + W1\);  
ENDDO;  
OUTPUT Y, A;  
HALT. 
The example considered above puts in evidence a deep relationship between MP computations and classical imperative programming, by relating both of them to a representation of computation in terms of “matter” flowing among locations with fluxes determined by the content of these locations. In the next section, we develop this intuition in general terms.
3.1 MP computability
In this section, we follow the papers [30, 57, 58]), by showing that the class of positively controlled MP grammars is computational universal and that computational universality can be obtained by means of regulators having a simple form.

Increment of register R: Inc(R),

Decrement of register R: Dec(R).
About the third type of instructions, when the accumulator has the label of a Halt instruction, the computation of the register machine stops.
Any computation of M is relative to some positive integers as contents of the input registers (all the other registers are assumed with zero content). The computation realized by the machine M is a sequence of application of instructions. The application of any instruction is a change of values in the registers (\(R_0\) included). At any step, the instruction having the label put in the accumulator is applied. The initial integer put in the accumulator determines the first instruction that will be applied. When the computation halts, the results of the computation are the numbers put in the output registers.
 1 :
\(Inc(R_1\))
 2 :
\(Dec(R_2)\)
 3 :
\(Jnz(R_2, 1)\)
 4 :
Halt
Theorem 8
For any Register Machine M,there exists a monic positive MP grammar \(G_{M}\)equivalent to M.
Proof
Given a register machine M with m instructions, we consider an MP grammar \(G_M\) with a variable \(I_h\) for each instruction \(I_h\) of M (h is the instruction label), plus another variable H, and a variable for each register \(R_j\) of M (register variables are denoted in the same way registers are denoted in M). All register variables are initialized with the same values that the registers have in M, and all instruction variables are zero. If M has the program consisting of instructions \(I_1, I_2, \ldots , I_m\) and 1 is the initial content of the accumulator, then the set of rules \(R_M\) of \(G_M\) are produced according to the following procedure.
 1.
\(R_M := \{ \emptyset \rightarrow I_{1} : 1\}\)
 2.
for any instruction I of Mdo
 3.
begin
 4.
if\(I = h : Halt\)then add to \(R_M\) the rule \(I_{h} \rightarrow H : I_{h}\)
 5.
if\(I = h : Inc(R_j)\)then add to \(R_M\) the rules \(\emptyset \rightarrow R_j : I_h\) and \(I_h \rightarrow I_{h+1} : I_h\)
 6.
if\(I = h : Dec(R_j)\)then add to \(R_M\) the rule \(R_j \rightarrow \emptyset : I_h\) and \(I_h \rightarrow I_{h+1} : I_h\)
 7.
if\(I = h : Jnz(R_j, k)\)then add to \(R_M\) the rules specified below.
 8.
end
 1.
\(I_h \rightarrow I_ k : R_j\)
 2.
\(I_h \rightarrow I_{h+1} : (R_j + I_h)^+\)
 1.
\(R_j \rightarrow H_j : I_h\)
 2.
\(I_h \rightarrow L_h : I_h\)
 3.
\(L_h \rightarrow F_k : H_j\)
 4.
\(L_h \rightarrow F_{h+1} : (L_h + H_j)^+\)
 5.
\(H_j \rightarrow R_j : F_k\)
 6.
\(H_j \rightarrow R_j : F_{h+1}\)
 7.
\(F_k \rightarrow I_k : F_k\)
 8.
\(F_{h+1} \rightarrow I_{h +1}: F_{h+1}\)
Theorem 9
For any Register Machine M ,there exists a monic positive MP grammar \(G_M\) (dynamically) equivalent to M where regulators are single variables.
 1.
\(R_j \rightarrow H_j : I_h\)
 2.
\(I_h \rightarrow L_h : I_h\)
 3.
\(L_h \rightarrow F_k : H_j\)
 4.
\(H_j \rightarrow K_h: H_j\)
 5.
\(L_h \rightarrow K_{h} : L_h\)
 6.
\(L_h \rightarrow F_{h+1} : K_h\)
 7.
\(H_j \rightarrow R_j : F_k\)
 8.
\(H_j \rightarrow R_j : F_{h+1}\)
 9.
\(F_k \rightarrow I_k : F_k\)
 10.
\(F_{h+1} \rightarrow I_{h +1}: F_{h+1}\).
4 Conclusions
MP grammars were introduced for modeling metabolic processes. However, they proved to be a natural formalism for expressing a great number of systems. Moreover, MP grammars showed to have a feature very important from the applicative viewpoint, that is, efficient and reliable algorithms for solving inverse dynamical problems: finding grammars that provide a given sequence of observed states for some given variables. This means that, in terms of MP grammars, we can discover a logic that underlies a given time series. MP regression theory concerns with a wide class of algorithms discovering MP grammars that solve dynamical inverse problems, based on initial flux estimation and recurrent solution of linear equations, loggain principle, minimum square methods, linear statistical regression, and evolutionary genetic strategies. Also complex oscillatory dynamics and chaotic processes are naturally obtained by means of MP grammars [27, 48].
The structure of MP grammars is expressed by MP graphs that represent in a very natural way the transformation component of such grammars together with their regulation component. In the visualization of these graphs, the main novelty of MP grammars easily emerges. In fact, the nodes–edges structure of usual graphs is enriched, in MP graphs, by a further level consisting of metaedges connecting nodes to edges. As we showed, this feature allows MP grammars to be a universal computational formalism, with no centralized programs.
A concluding remark intends to point out an important aspect related with the metalevel of MP graphs. In fact, they can be extended for expressing a crucial mechanism of real neural circuits, discovered in the mnemonic consolidation of Aplysia (a marine invertebrate) [60]. Namely, a metalevel mechanism of some neural synaptic connections was discovered, according to which special neural connections are activated for modulating synaptic connections. In this way, new synapses can be developed by some modulation schemata. In MP terms, this kind of synapse modulation corresponds to a special type of metaedge. If our regulators are equipped with the capability of introducing new edges, then MP graphs become metagraphs that express “plastic” structures, and the letters of MP prefix result as the shortcomings for metaplastic graphs. An extension of classical neural networks with the capability of modulations that realize graph structure modifications could be an interesting novelty in the formalization of the neural plasticity, on which many complex behaviors are surely based.
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