FFirst-Order Tests for Toricity
Hamid RahkooyCNRS, Inria, and the University of Lorraine, Nancy, France [email protected]
Thomas SturmCNRS, Inria, and the University of Lorraine, Nancy, FranceMPI Informatics and Saarland University, Saarbrücken, Germany [email protected]
February 11, 2020
Abstract
Motivated by problems arising with the symbolic analysis of steady state ideals in Chem-ical Reaction Network Theory, we consider the problem of testing whether the points in acomplex or real variety with non-zero coordinates form a coset of a multiplicative group.That property corresponds to Shifted Toricity, a recent generalization of toricity of the cor-responding polynomial ideal. The key idea is to take a geometric view on varieties ratherthan an algebraic view on ideals. Recently, corresponding coset tests have been proposedfor complex and for real varieties. The former combine numerous techniques from commu-tative algorithmic algebra with Gröbner bases as the central algorithmic tool. The latterare based on interpreted first-order logic in real closed fields with real quantifier eliminationtechniques on the algorithmic side. Here we take a new logic approach to both theories,complex and real, and beyond. Besides alternative algorithms, our approach provides aunified view on theories of fields and helps to understand the relevance and interconnectionof the rich existing literature in the area, which has been focusing on complex numbers,while from a scientific point of view the (positive) real numbers are clearly the relevantdomain in chemical reaction network theory. We apply prototypical implementations ofour new approach to a set of 129 models from the BioModels repository.
We are interested in situations where the points with non-zero coordinates in a given complexor real variety form a multiplicative group or, more generally, a coset of such a group. Forirreducible varieties this corresponds to toricity [23, 16] and shifted toricity [28, 27], respectively,of both the varieties and the corresponding ideals.While toric varieties are well established and have an important role in algebraic geometry[23, 16], our principal motivation here to study generalizations of toricity comes from the sciences,specifically chemical reaction networks such as the following model of the kinetics of intra- andintermolecular zymogen activation with formation of an enzyme-zymogen complex [22], which1 a r X i v : . [ c s . S C ] F e b an also be found as model no. 92 in the BioModels database [9]:Z P + EZ + E
E Z
P + 2 EHere Z stands for zymogen, P is a peptide, E is an enzyme, E—Z is the enzyme substrate complexformed from that enzyme and zymogen. The reactions are labelled with reaction rate constants .Let x , . . . , x : R → R denote the concentrations over time of the species Z, P, E, E–Z, respec-tively. Assuming mass action kinetics one can derive reaction rates and furthermore a system ofautonomous ordinary differential equations describing the development of concentrations in theoverall network [20, Section 2.1.2]:˙ x = p / , p = − x x − x + 21 x , ˙ x = p / , p = − x x + 400 x + 129 x , ˙ x = p / , p = 200 x + 27 x , ˙ x = p / , p = 4000000 x x − x . The chemical reaction is in equilibrium for positive concentrations of species lying in the realvariety of the steady state ideal h p , . . . , p i ⊆ Z [ x , . . . , x ] , intersected with the first orthant of R .Historically, the principle of detailed balancing has attracted considerable attention in thesciences. It states that at equilibrium every single reaction must be in equilibrium with itsreverse reaction. Detailed balancing was used by Boltzmann in 1872 in order to prove his H-theorem [4], by Einstein in 1916 for his quantum theory of emission and absorption of radiation[15], and by Wegscheider [50] and Onsager [42] in the context of chemical kinetics , which leadto Onsager’s Nobel prize in Chemistry in 1968. In the field of symbolic computation, Grigorievand Weber [29] applied results on binomial varieties to study reversible chemical reactions in thecase of detailed balancing.In particular with the assumption of irreversible reactions, like in our example, detailed bal-ancing has been generalized to complex balancing [19, 20, 33], which has widely been used inthe context of chemical reaction networks. Here one considers complexes , like Z, P + E, Z + E,etc. in our example, and requires for every such complex that the sum of the reaction rates ofits inbound reactions equals the sum of the reaction rates of its outbound reactions.Craciun et al. [11] showed that toric dynamical systems [18, 33], in turn, generalize complexbalancing. The generalization of the principle of complex balancing to toric dynamical systemshas obtained considerable attention in the last years [44, 24, 11, 40]. Millan, Dickenstein andShiu in [44] considered steady state ideals with binomial generators. They presented a sufficientlinear algebra condition on the stoichiometry matrix of a chemical reaction network in order totest whether the steady state ideal has binomial generators. Conradi and Kahle showed that thesufficient condition is even equivalent when the ideal is homogenous [10, 35, 34]. That conditionalso led to the introduction of MESSI systems [43]. Recently, binomiality of steady states idealswas used to infer network structure of chemical reaction networks out of measurement data [49].Besides its scientific adequacy as a generalization of complex balancing there are practicalmotivations for studying toricity. Relevant models are typically quite large. For instance, withour comprehensive computations in this article we will encounter one system with 90 polynomials
2n dimension 71. This brings symbolic computation to its limits. Our best hope is to discoversystematic occurrences of specific structural properties in the models coming from a specificcontext, e.g. the life sciences, and to exploit those structural properties towards more efficientalgorithms. In that course, toricity could admit tools from toric geometry, e.g., for dimensionreduction.Detecting toricity of varieties in general, and of steady state varieties of chemical reactionnetworks in particular, is a difficult problem. The first issue in this regard is finding suitablenotions to describe the structure of the steady states. Existing work, such as the publicationsmentioned above, typically focuses on the complex numbers and addresses algebraic propertiesof the steady state ideal, e.g., the existence of binomial Gröbner bases. Only recently, a groupof researchers including the authors of this article have taken a geometric approach, focusing onvarieties rather than ideals [27, 28]. Besides irreducibility, the characteristic property for varieties V to be toric over a field K is that V ∩ ( K ∗ ) n forms a multiplicative group. More generally, oneconsiders shifted toricity , where V ∩ ( K ∗ ) n forms a coset of a multiplicative group.It is important to understand that chemical reaction network theory generally takes placein the interior of the first orthant of R n , i.e., all species concentrations and reaction rates areassumed to be strictly positive [20]. Considering ( C ∗ ) n in contrast to C n resembles the strictnesscondition, and considering also ( R ∗ ) n in [27] was another step in the right direction.The plan of the article is as follows. In Section 2 we motivate and formally introduce first-order characterizations for shifted toricity, which have been used already in [27], but exclusivelywith real quantifier elimination methods. In Section 3 we put a model theoretic basis and provetransfer principles for our characterizations throughout various classes of fields, with zero as wellas with positive characteristics. In Section 4 we employ Hilbert’s Nullstellensatz as a decisionprocedure for uniform word problems and use logic tests also over algebraically closed fields. Thismakes the link between the successful logic approach from [27] and the comprehensive existingliterature cited above. Section 5 clarifies some asymptotic worst-case complexities for the sake ofscientific rigor. In Section 6 it turns out that for a comprehensive benchmark set of 129 modelsfrom the BioModels database [9] quite simple and maintainable code, requiring only functionalityavailable in most decent computer algebra systems and libraries, can essentially compete withhighly specialized and more complicated purely algebraic methods. This motivates in Section 7a perspective that our symbolic computation approach has a potential to be interesting forresearchers in the life sciences, with communities much larger than our own, with challengingapplications, not least in the health sector. In this section we set up our first-order logic framework. We are going to use interpreted first-order logic with equality over the signature L = (0 , , + , − , · ) of rings.For any field K we denote its multiplicative group K \ { } by K ∗ . For a coefficient ring Z ⊆ K and F ⊆ Z [ x , . . . , x n ] we denote by V K ( F ), or shortly V ( F ), the variety of F over K . Oursignature L naturally induces coefficient rings rings Z = Z /p for finite characteristic p , and Z = Z for characteristic 0. We define V ( F ) ∗ = V ( F ) ∩ ( K ∗ ) n ⊆ ( K ∗ ) n . Note that the directproduct ( K ∗ ) n establishes again a multiplicative group.Let F = { f , . . . , f m } ⊆ Z [ x , . . . , x n ]. The following semi-formal conditions state that V ( F ) ∗ establishes a coset of a multiplicative subgroup of ( K ∗ ) n : ∀ g, x ∈ ( K ∗ ) n : g ∈ V ( F ) ∧ gx ∈ V ( F ) ⇒ gx − ∈ V ( F ) (1)3 g, x, y ∈ ( K ∗ ) n : g ∈ V ( F ) ∧ gx ∈ V ( F ) ∧ gy ∈ V ( F ) ⇒ gxy ∈ V ( F ) (2) V ( F ) ∩ ( K ∗ ) n = ∅ . (3)If we replace (3) with the stronger condition1 ∈ V ( F ) , (4)then V ( F ) ∗ establishes even a multiplicative subgroup of ( K ∗ ) n . We allow ourselves to lessformally say that V ( F ) ∗ is a coset or group over K , respectively.Denote M = { , . . . , m } , N = { , . . . , n } , and for ( i, j ) ∈ M × N let d ij = deg x j ( f i ). Weshortly write x = ( x , . . . , x n ), y = ( y , . . . , y n ), g = ( g , . . . , g n ). Multiplication between x , y , g is coordinate-wise, and x d i = x d i · · · x d in n . As a first-order L -sentence, condition (1) yields ι ˙= ∀ g . . . ∀ g n ∀ x . . . ∀ x n (cid:18) n V j =1 g j = 0 ∧ n V j =1 x j = 0 ∧ m V i =1 f i ( g , . . . , g n ) = 0 ∧ m V i =1 f i ( g x , . . . , g n x n ) = 0 −→ m V i =1 x d i f i ( g x − , . . . , g n x − n ) = 0 (cid:19) . Here the multiplications with x d i drop the principal denominators from f i ( g x − , . . . , g n x − n ).This is an equivalence transformation, because the left hand side of the implication constrains x , . . . , x n to be different from zero.Similarly, condition (2) yields a first-order L -sentence µ ˙= ∀ g . . . ∀ g n ∀ x . . . ∀ x n ∀ y . . . ∀ y n (cid:18) n V j =1 g j = 0 ∧ n V j =1 x j = 0 ∧ n V j =1 y j = 0 ∧ m V i =1 f i ( g , . . . , g n ) = 0 ∧ m V i =1 f i ( g x , . . . , g n x n ) = 0 ∧ m V i =1 f i ( g y , . . . , g n y n ) = 0 −→ m V i =1 f i ( g x y , . . . , g n x n y n ) = 0 (cid:19) . For condition (3) we consider its logical negation V ( F ) ∩ ( K ∗ ) n = ∅ , which gives us an L -sentence η ˙= ∀ x . . . ∀ x n (cid:18) m V i =1 f i = 0 −→ n W j =1 x j = 0 (cid:19) . Accordingly, the L -sentence ¬ η formally states (3).Finally, condition (4) yields a quantifier-free L -sentence γ ˙= m V i =1 f i (1 , . . . ,
1) = 0 . Let p ∈ N be 0 or prime. We consider the L -model classes K p of fields of characteristic p and A p ⊆ K p of algebraically closed fields of characteristic p . Recall that A p is complete, decidable,and admits effective quantifier elimination [48, Note 16].We assume without loss of generality that L -sentences are in prenex normal form Q x . . . Q n x n ψ with Q , . . . , Q n ∈ {∃ , ∀} and ψ quantifier-free. An L -sentence is called universal if it is of theform ∀ x . . . ∀ x n ψ and existential if it is of the form ∃ x . . . ∃ x n ψ with ψ quantifier-free. Aquantifier-free L -sentence is both universal and existential.4 emma 1. Let ϕ be a universal L -sentence. Then K p | = ϕ if and only if A p | = ϕ. Proof.
The implication from the left to the right immediately follows from A p ⊆ K p . Assume,vice versa, that A p | = ϕ , and let K ∈ K p . Then K has an algebraic closure K ∈ A p , and K | = ϕ due to the completeness of A p . Since K ⊆ K and ϕ as a universal sentence is persistent undersubstructures, we obtain K | = ϕ .All our first-order conditions ι , µ , η , and γ introduced in the previous section 2 are universal L -sentences. Accordingly, ¬ η is equivalent to an existential L -sentence.In accordance with the our language L we are going to use polynomial coefficient rings Z p = Z /p for finite characteristic p , and Z = Z . Let F ⊆ Z p [ x , . . . , x n ]. Then V ( F ) ∗ is a coset over K ∈ K p if and only if K | = ι ∧ µ ∧ ¬ η. (5)Especially, V ( F ) ∗ is a group over K if even K | = ι ∧ µ ∧ γ, (6)where γ entails ¬ η . Proposition 2.
Let F ⊆ Z p [ x , . . . , x n ] , and let K ∈ K p . Then V ( F ) ∗ is a group over K if andonly if at least one of the following conditions holds:(a) K | = ι ∧ µ ∧ γ for some K ⊆ K ∈ K p ;(b) K | = ι ∧ µ ∧ γ for some K ∈ A p .Proof. Recall that V ( F ) ∗ is a group over K if and only if K | = ι ∧ µ ∧ γ . If V ( F ) ∗ is a groupover K , then (a) holds for K = K . Vice versa, there are two cases. In case (a), we can concludethat K | = ι ∧ µ ∧ γ because the universal sentence ι ∧ µ ∧ γ is persistent under substructures. Incase (b), we have A p | = ι ∧ µ ∧ γ by the completeness of that model class. Using Lemma 1 weobtain K p | = ι ∧ µ ∧ γ , in particular K | = ι ∧ µ ∧ γ . Example 3. (i) Assume that V ( F ) ∗ is a group over C . Then V ( F ) ∗ is a group over anyfield of characteristic 0. Alternatively, it suffices that V ( F ) ∗ is a group over the countablealgebraic closure Q of Q .(ii) Assume that V ( F ) ∗ is a group over the countable field of real algebraic numbers, which isnot algebraically closed. Then again V ( F ) ∗ is a group over any field of characteristic 0.(iii) Let ε be a positive infinitesimal, and assume that V ( F ) ∗ is a group over R ( ε ). Then V ( F ) ∗ is group also over Q and R , but not necessarily over Q . Notice that R ( ε ) is not algebraicallyclosed.(iv) Assume that V ( F ) ∗ is a group over the algebraic closure of F p . Then V ( F ) ∗ is a groupover any field of characteristic p . Alternatively, it suffices that V ( F ) ∗ is a group over thealgebraic closure of the rational function field F p ( t ), which has been studied with respectto effective computations [36]. Proposition 4.
Let F ⊆ Z p [ x , . . . , x n ] and let K ∈ K p . Then V ( F ) ∗ is a coset over K if andonly if K | = ¬ η and at least one of the following conditions holds:(a) K | = ι ∧ µ for some K ⊆ K ∈ K p ; b) K | = ι ∧ µ for some K ∈ A p .Proof. Recall that V ( F ) ∗ is a coset over K if and only if K | = ι ∧ µ ∧ ¬ η . If V ( F ) ∗ is a cosetover K , then K | = ¬ η , and (a) holds for K = K . Vice versa, we require that K | = ¬ η and obtain K | = ι ∧ µ analogously to the proof of Proposition 2. Example 5. (i) Assume that V ( F ) ∗ is a coset over C . Then V ( F ) ∗ is a coset over R if andonly if V ( F ) ∗ = ∅ over R . This is the case for F = { x − } but not for F = { x + 2 } .(ii) Consider F = { x − } = { ( x − x + 2) } . Then over R , V ( F ) ∗ = {±√ } is a coset,because V ( F ) ∗ / √ {± } is a group. Similarly over C , V ( F ) ∗ = {±√ , ± i √ } is a coset,as V ( F ) ∗ / √ {± , ± i } is a group.(iii) Consider F = { x + x − } = { ( x − x + 3) } . Then over R , V ( F ) ∗ = {±√ } is acoset, as V ( F ) ∗ / √ {± } is a group. Over C , in contrast, V ( F ) ∗ = {±√ , ± i √ } is nota coset. A recent publication [27] has systematically applied coset tests to a large number for real-worldmodels from the BioModels database [9], investigating varieties over both the real and the com-plex numbers. Over R it used essentially our first-order sentences presented in Section 2 andapplied efficient implementations of real decision methods based on effective quantifier elimina-tion [51, 52, 13, 14, 45, 38].Over C , in contrast, it used a purely algebraic framework combining various specialized meth-ods from commutative algebra, typically based on Gröbner basis computations [8, 17]. Thisis in line with the vast majority of the existing literature (cf. the Introduction for references),which uses computer algebra over algebraically closed fields, to some extent supplemented withheuristic tests based on linear algebra.Generalizing the successful approach for R and aiming at a more uniform overall framework,we want to study here the application of decision methods for algebraically closed fields to ourfirst-order sentences. Recall that our sentences ι , µ , η , and γ are universal L -sentences. Everysuch sentence ϕ can be equivalently transformed into a finite conjunction of universal L -sentencesof the following form: b ϕ ˙= ∀ x . . . ∀ x n (cid:18) m V i =1 f i ( x , . . . , x n ) = 0 −→ g ( x , . . . , x n ) = 0 (cid:19) , where f , . . . , f m , g ∈ Z p [ x , . . . , x n ]. Such L -sentences are called uniform word problems [3].Over an algebraically closed field ¯ K of characteristic p , Hilbert’s Nullstellensatz [31] provides adecision procedure for uniform word problems. It states that¯ K | = b ϕ if and only if g ∈ p h f , . . . , f m i . Recall that A p is complete so that we furthermore have A p | = b ϕ if and only if ¯ K | = b ϕ .Our L -sentence ι for condition (1) can be equivalently transformed into ∀ g . . . ∀ g n ∀ x . . . ∀ x n (cid:18) n W j =1 g j = 0 ∨ n W j =1 x j = 0 ∨ m W i =1 f i ( g , . . . , g n ) = 0 ∨ m W i =1 f i ( g x , . . . , g n x n ) = 0 ∨ m V i =1 x d i f i ( g x − , . . . , g n x − n ) = 0 (cid:19) , b ι ˙= m V k =1 ∀ g . . . ∀ g n ∀ x . . . ∀ x n (cid:18) m V i =1 f i ( g , . . . , g n ) = 0 ∧ m V i =1 f i ( g x , . . . , g n x n ) = 0 −→ x d k f k ( g x − , . . . , g n x − n ) n Q j =1 g j x j = 0 (cid:19) . Hence, by Hilbert’s Nullstellensatz, (1) holds in ¯ K if and only if x d k f k ( g x − , . . . , g n x − n ) n Q j =1 g j x j ∈ R for all k ∈ M, (7)where R = p h f i ( g , . . . , g n ) , f i ( g x , . . . , g n x n ) | i ∈ M i .Similarly, our L -sentence µ for condition (2) translates into b µ ˙= m V k =1 ∀ g . . . ∀ g n ∀ x . . . ∀ x n ∀ y . . . ∀ y n (cid:18) m V i =1 f i ( g , . . . , g n ) = 0 ∧ m V i =1 f i ( g x , . . . , g n x n ) = 0 ∧ m V i =1 f i ( g y , . . . , g n y n ) = 0 −→ f k ( g x y , . . . , g n x n y n ) n Q j =1 g j x j y j = 0 (cid:19) . Again, by Hilbert’s Nullstellensatz, (2) holds in ¯ K if and only if f k ( g x y , . . . , g n x n y n ) n Q j =1 g j x j y j ∈ R for all k ∈ M, (8)where R = p h f i ( g ) , f i ( gx ) , f i ( gy ) | i ∈ M i .Next, our L -sentence η is is equivalent to b η ˙= ∀ x . . . ∀ x n (cid:18) m V i =1 f i = 0 −→ n Q j =1 x j = 0 (cid:19) . Using once more Hilbert’s Nullstellensatz, ¯ K | = b η if and only if n Q j =1 x j ∈ R , (9)where R = p h f , . . . , f m i . Hence our non-emptiness condition (3) holds in ¯ K if and only if n Q j =1 x j / ∈ R . (10)Finally, our L -sentence γ for condition (4) is equivalent to b γ ˙= m V k =1 (cid:0) −→ f k (1 , . . . ,
1) = 0 (cid:1) . Here Hilbert’s Nullstellensatz tells us that condition (4) holds in ¯ K if and only if f k (1 , . . . , ∈ R for all k ∈ M, (11)where R = p h i = h i . Notice that the radical membership test quite naturally reduces to theobvious test with plugging in. 7 Complexity
Let us briefly discuss asymptotic complexity bounds around problems and methods addressedhere. We do so very roughly, in terms of the input word length. The cited literature providesmore precise bounds in terms of several complexity parameters, such as numbers of quantifiers,or degrees.The decision problem for algebraically closed fields is double exponential [30] in general, butonly single exponential when the number of quantifier alternations is bounded [25], which cov-ers in particular our universal formulas. The decision problem for real closed fields is doubleexponential as well [12], even for linear problems [51]; again it becomes single exponential whenbounding the number of quantifier alternations [26].Ideal membership tests are at least double exponential [39], and it was widely believed that thiswould impose a corresponding lower bound also for any algorithm for Hilbert’s Nullstellensatz.Quite surprisingly, it turned out that there are indeed single exponential such algorithms [7, 37].On these grounds it is clear that our coset tests addressed in the previous sections can be solvedin single exponential time for algebraically closed fields as well as for real closed fields. Recallthat our considering those tests is actually motivated by our interest in shifted toricity, whichrequires, in addition, the irreducibility of the considered variety over the considered domain.Recently it has been shown that testing shifted toricity, including irreducibility, is also onlysingle exponential over algebraically closed fields as well as real closed fields [27].Most asymptotically fast algorithms mentioned above are not implemented and it is not clearthat they would be efficient in practice.
We have studied 129 models from the BioModels database [9]. Technically, we took our inputfrom ODEbase which provides preprocessed versions for symbolic computation. Our 129 modelsestablish the complete set currently provided by ODEbase for wich the relevant systems ofordinary differential equations have polynomial vector fields.We limited ourselves to characteristic 0 and applied the tests (7), (8), (9), (11) derived inSection 4 using Hilbert’s Nullstellensatz. Recall that those tests correspond to ι , µ , η , γ fromSection 3, respectively, and that one needs ι ∧ µ ∧ ¬ η or ι ∧ µ ∧ γ for cosets or groups, respectively.From a symbolic computation point of view, we used exclusively polynomial arithmetic andradical membership test. The complete Maple code for computing a single model is displayed inFigure 1; it is surprisingly simple.We conducted our computations on a 2.40 GHz Intel Xeon E5-4640 with 512 GB RAM and32 physical cores providing 64 CPUs via hyper-threading. For parallelization of the jobs for theindividual models we used GNU Parallel [47]. Results and timings are collected in Table 1. Witha time limit of one hour CPU time per model we succeeded on 78 models, corresponding to 60%,the largest of which, no. 559, has 90 polynomials in 71 dimensions. The median of the overallcomputation times for the successful models is 1.419 s. We would like to emphasize that our focushere is illustrating and evaluating our overall approach, rather than obtaining new insights intothe models. Therefore our code in Figure 1 is very straightforward without any optimizations.In particular, computation continues even when one relevant subtest has already failed. Morecomprehensive results on our dataset can be found in [27].Among our 78 successfully computed models, we detected 20 coset cases, corresponding to26%. Two out of those 20 are even group cases. Among the 58 other cases, 46, corresponding http://odebase.cs.uni-bonn.de/ ToricHilbert := proc (F: : l i s t (polynom)) 2 uses
PolynomialIdeals ; 34 local
Iota := proc ( ) : : truefalse ; 5 local
R1, s , prod, f ; 6R1 := < op(subs(zip(‘=‘, xl , gl ) , F)) , op(subs(zip((x, g) − > x = g∗x, xl , gl ) , F)) >; 7s := zip ((x, g) − > x = g/x, xl , gl ); 8prod := g ∗ x; 9 for f in subs(s , F) do if not RadicalMembership(numer( f ) ∗ prod, R1) then return false 12 end if end do ; 14 return true 15 end proc ; 1617 local Mu := proc ( ) : : truefalse ; 18 local
R2, s , prod, f ; 19R2 := < op(subs(zip(‘=‘, xl , gl ) , F)) , op(subs(zip((x, g) − > x = g∗x, xl , gl ) , F)) , 20op(subs(zip(‘=‘, xl , zip ( ‘∗ ‘ , gl , yl )) , F)) >; 21s := zip(‘=‘, xl , zip ( ‘∗ ‘ , gl , zip ( ‘∗ ‘ , xl , yl ))); 22prod := g ∗ x ∗ y; 23 for f in subs(s , F) do if not RadicalMembership( f ∗ prod, R2) then return false 26 end if end do ; 28 return true 29 end proc ; 3031 local Eta := proc ( ) : : truefalse ; 32 local
R3, prod; 33R3 := < op(F) >; 34prod := foldl ( ‘∗ ‘ , 1, op(xl )); 35 return
RadicalMembership(prod, R3) 36 end proc ; 3738 local
Gamma := proc ( ) : : truefalse ; 39 local
R4, s , f ; 40R4 := < 0 >; 41s := map(x − > x=1, xl ); 42 for f in subs(s , F) do if not RadicalMembership(f , R4) then return false 45 end if end do ; 47 return true 48 end proc ; 4950 local Rename := proc (base : :name, l : : l i s t (name) ) : : l i s t (name); 51 uses
StringTools ; 52 return map(x − > cat(base , Select(IsDigit , x)) , l ) 53 end proc ; 5455 local X, xl , gl , yl , g, x, y, iota , t_iota , mu, t_mu, eta , t_eta, gamma_, t_gamma, coset , group, t ; 56t := time(); 57xl := convert(indets(F) , l i s t ); 58x := foldl ( ‘∗ ‘ , 1, op(xl )); 59gl := Rename( ’g’ , xl ); 60g := foldl ( ‘∗ ‘ , 1, op( gl )); 61yl := Rename( ’y’ , xl ); 62y := foldl ( ‘∗ ‘ , 1, op(yl )); 63t_iota := time(); iota := Iota (); t_iota := time() − t_iota ; 64t_mu := time(); mu := Mu(); t_mu := time() − t_mu; 65t_eta := time(); eta := Eta(); t_eta := time() − t_eta; 66t_gamma := time(); gamm := Gamma(); t_gamma := time() − t_gamma; 67coset := iota and mu and not eta ; 68group := iota and mu and gamm; 69t := time() − t ; 70 return nops(F) , nops(xl ) , iota , t_iota , mu, t_mu, eta , t_eta, gamm, t_gamma, coset , group, t 71 end proc ; Figure 1: Maple code for computing one row of Table 19able 1: Results and computation times (in seconds) of our computations on models from theBioModels database [9] model m n ι t ι µ t µ η t η γ t γ coset group t Σ
001 12 12 true 7.826 true 7.86 false 4.267 false 0.053 true false 20.007040 5 3 false 1.415 false 0.173 false 0.114 false 0.043 false false 1.746050 14 9 true 1.051 true 2.458 true 0.113 false 0.05 false false 3.673052 11 6 true 3.605 true 1.635 true 0.096 false 0.059 false false 5.396057 6 6 true 0.271 true 0.263 false 0.858 false 0.045 true false 1.438072 7 7 true 0.763 true 0.496 true 0.08 false 0.06 false false 1.4077 8 7 true 0.296 true 0.356 false 0.097 false 0.051 true false 0.801080 10 10 true 0.714 true 1.341 true 0.103 false 0.06 false false 2.219082 10 10 true 0.384 true 0.39 true 0.086 false 0.041 false false 0.902091 16 14 true 0.031 true 0.045 true 0.003 false 0.062 false false 0.142092 4 3 true 0.293 true 0.244 false 0.104 false 1.03 true false 1.671099 7 7 true 0.298 true 0.698 false 0.087 false 0.036 true false 1.119101 6 6 false 4.028 false 10.343 false 0.917 false 0.073 false false 15.361104 6 4 true 0.667 true 0.146 true 0.084 false 0.039 false false 0.937105 39 26 true 0.455 true 0.367 true 0.043 false 0.038 false false 0.905125 5 5 false 0.193 false 0.098 false 0.078 false 0.038 false false 0.408150 4 4 true 0.173 true 0.153 false 0.094 false 0.043 true false 0.464156 3 3 true 2.638 true 0.248 false 0.86 false 0.052 true false 3.8158 3 3 false 0.148 false 0.149 false 0.16 false 0.045 false false 0.503159 3 3 true 0.959 true 0.175 false 0.083 false 0.04 true false 1.257178 6 4 true 0.52 true 1.71 true 0.877 false 1.201 false false 4.308186 11 10 true 31.785 true 1026.464 true 1.956 false 0.095 false false 1060.301187 11 10 true 27.734 true 1023.648 true 0.103 false 0.062 false false 1051.548188 20 10 true 0.075 true 0.079 true 0.04 false 0.047 false false 0.242189 18 7 true 0.035 true 0.02 true 0.002 false 0.062 false false 0.12194 5 5 false 2.338 false 1.922 false 0.612 false 0.05 false false 4.922197 7 5 false 7.562 false 71.864 false 0.485 false 0.05 false false 79.962198 12 9 true 0.397 true 0.793 true 0.077 false 0.042 false false 1.31199 15 8 true 1.404 true 1.531 false 0.215 false 0.054 true false 3.205220 58 56 true 146.146 true 534.832 true 6.921 false 0.964 false false 688.866227 60 39 true 0.273 true 0.485 true 0.01 false 0.077 false false 0.847229 7 7 true 1.917 true 3.348 false 0.131 false 0.062 true false 5.458233 4 2 false 0.16 false 0.44 false 0.17 false 0.557 false false 1.328243 23 19 true 8.598 true 1171.687 true 2.512 false 0.171 false false 1182.97259 17 16 true 1.334 true 1.913 true 0.092 false 0.045 false false 3.385260 17 16 true 2.182 true 0.748 true 0.079 false 0.047 false false 3.057261 17 16 true 3.359 true 2.872 true 0.113 false 0.095 false false 6.44262 11 9 true 0.402 true 0.41 true 0.091 false 0.071 false false 0.975263 11 9 true 0.379 true 0.403 true 0.085 false 0.066 false false 0.934264 14 11 true 1.031 true 2.036 true 0.136 false 0.063 false false 3.268267 4 3 true 1.084 true 0.246 true 0.095 false 0.049 false false 1.475271 6 4 true 0.286 true 0.283 true 0.746 false 0.045 false false 1.361272 6 4 true 0.361 true 0.323 true 0.086 false 0.055 false false 0.826281 32 32 true 20.987 true 29.791 true 0.602 false 0.055 false false 51.437282 6 3 true 0.205 true 0.19 true 0.087 false 0.046 false false 0.528283 4 3 true 0.294 true 0.211 true 0.087 false 0.412 false false 1.005289 5 4 false 2.291 false 1.118 false 0.165 false 0.044 false false 3.619292 6 2 true 0.06 true 0.048 true 0.063 false 0.046 false false 0.218306 5 2 true 0.149 true 0.121 false 0.079 false 0.041 true false 0.391307 5 2 true 0.129 true 0.121 true 0.043 false 0.148 false false 0.441310 4 1 true 0.053 true 0.369 true 0.047 false 0.04 false false 0.509311 4 1 true 0.076 true 0.048 true 0.224 false 0.048 false false 0.397312 3 2 true 0.098 true 0.512 true 0.043 false 0.043 false false 0.697314 12 10 true 0.515 true 1.789 true 0.1 false 0.059 false false 2.464321 3 3 true 0.163 true 0.148 true 0.042 false 0.039 false false 0.393357 9 8 true 0.353 true 1.517 true 0.07 false 0.045 false false 1.986359 9 8 true 1.677 true 3.605 true 0.11 false 0.055 false false 5.448360 9 8 true 0.479 true 0.47 true 0.096 false 0.05 false false 1.096361 8 8 true 1.069 true 2.746 true 0.156 false 0.045 false false 4.017363 4 3 true 0.244 true 0.199 true 0.077 false 0.041 false false 0.561364 14 12 true 2.483 true 7.296 true 0.55 false 0.064 false false 10.394413 5 5 false 1.55 false 22.323 false 0.117 false 0.053 false false 24.044459 4 3 true 0.542 true 0.224 false 0.18 false 0.068 true false 1.014460 4 3 false 1.025 false 0.936 false 0.143 false 0.216 false false 2.321475 23 22 true 97.876 true 3377.021 true 0.231 false 0.062 false false 3475.192484 2 1 true 0.384 true 0.143 false 0.099 false 0.048 true false 0.674485 2 1 false 0.564 false 0.354 false 0.209 false 0.042 false false 1.169486 2 2 true 0.119 true 0.106 false 0.073 false 0.041 true false 0.339487 6 6 true 0.475 true 1.008 false 0.099 false 0.045 true false 1.628491 57 57 true 123.138 true 536.865 false 2.067 true 0.007 true true 662.08492 52 52 true 85.606 true 284.753 false 1.123 true 0.003 true true 371.489519 3 3 true 1.357 true 2.367 false 5.142 false 0.097 true false 8.964546 7 3 true 0.327 true 0.338 true 0.109 false 0.042 false false 0.817559 90 71 true 4.742 true 7.525 true 0.19 false 0.053 false false 12.515584 35 9 true 0.4 true 0.655 false 0.095 false 0.043 true false 1.194619 10 8 true 0.411 true 0.443 true 0.087 false 0.052 false false 0.994629 5 5 true 0.209 true 0.197 false 0.079 false 0.046 true false 0.532647 11 11 false 0.854 false 16.436 false 0.165 false 0.051 false false 17.507 o 78%, fail only due to their emptiness η ; we know from [27] that many such cases exhibit infact coset structure when considered in suitable lower-dimensional spaces, possibly after primedecomposition. Finally notice that our example reaction from the Introduction, no. 92, is amongthe smallest ones with a coset structure. We have used Hilbert’s Nullstellensatz to derive important information about the varieties ofbiological models with a polynomial vector field F . The key technical idea was generalizingfrom pure algebra to more general first-order logic. Recall from Section 3 that except for non-emptiness of V ( F ) ∗ the information we obtained is valid in all fields of characteristic 0. Whereverwe discovered non-emptiness, this holds at least in all algebraically closed fields of characteristic0. For transferring our obtained results to real closed fields, e.g., subtropical methods [46, 21, 32]provide fast heuristic tests for the non-emptiness of V ( F ) ∗ there.Technically, we only used polynomial arithmetic and polynomial radical membership tests.This means that on the software side there are many off-the-shelf computer algebra systemsand libraries available where our ideas could be implemented, robustly and with little effort.This in turn makes it attractive for the integration with software from systems biology, whichcould open exciting new perspectives for symbolic computation with applications ranging fromthe fundamental research in the life sciences to state-of-the-art applied research in medicine andpharmacology.We had motivated our use of Hilbert’s Nullstellensatz by viewing it as a decision procedurefor the universal fragment of first-order logic in algebraically closed fields, which is sufficient forour purposes. Our focus on algebraically closed fields here is in accordance with the majority ofexisting literature on toricity. However, it is generally accepted that from a scientific point ofview, real closed fields are the appropriate domain to consider.We have seen in Section 5 that the theoretical complexities for general decision procedures inalgebraically closed fields vs. real closed fields strongly resemble each other. What could nowtake the place of Hilbert’s Nullstellensatz over the reals with respect to practical computationson model sizes as in Table 1 or even larger? A factor of 10 could put us in the realm ofmodels currently used in the development of drugs for diabetes or cancer. One possible answeris satisfiability modulo theories solving (SMT) [41]. SMT is incomplete in the sense that itoften proves or disproves validity, but it can yield “unknown” for specific input problems. Whensuccessful, it is typically significantly faster than traditional algebraic decision procedures. Forcoping with incompleteness one can still fall back into real quantifier elimination. Interest incollaboration between the SMT and the symbolic computation communities exists on both sides[1, 2].
Acknowledgments
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