Satisfiability Modulo Theories and Chiral Heterotic String Vacua with Positive Cosmological Constant
Alon E. Faraggi, Benjamin Percival, Sven Schewe, Dominik Wojtczak
aa r X i v : . [ h e p - t h ] J a n January 2021
Satisfiability Modulo Theoriesand Chiral Heterotic String Vacuawith Positive Cosmological Constant
Alon E. Faraggi ∗ , Benjamin Percival † ,Sven Schewe ‡ and Dominik Wojtczak § Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK Dept. of Computer Sciences, University of Liverpool, Liverpool L69 7ZL, UK
Abstract
We apply Boolean Satisfiability (SAT) and Satisfiability Modulo Theories (SMT)solvers in the context of finding chiral heterotic string models with positive cosmo-logical constant from Z × Z orbifolds. The power of using SAT/SMT solversto sift large parameter spaces quickly to decide satisfiability, both to declare andprove unsatisfiability and to declare satisfiability, are demonstrated in this setting.These models are partly chosen to be small enough to plot the performance againstexhaustive search, which takes around 2 hours 20 minutes to comb through theparameter space. We show that making use of SMT based techniques with integerencoding is rather simple and effective, while a more careful Boolean SAT encodingprovides a significant speed-up – determining satisfiability or unsatisfiability has, inour experiments varied between 0.03 and 0.06 seconds, while determining all models(where models exist) took 19 seconds for a constraint system that allows for 2048models and 8.4 seconds for a constraint system that admits 640 models. We thusgain several orders of magnitude in speed, and this advantage is set to grow witha growing parameter space. This holds the promise that the method scales wellbeyond the initial problem we have used it for in this paper. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected]
Introduction
Although the Standard Model (SM) of particle physics successfully predicts all observa-tional data from particle collider experiments to date, several mysteries are left unan-swered. Among the most important of these mysteries are the origin of dark matter, howto stabilise the Higgs mass and how gravity may be incorporated into the framework. Toaddress these issues we can turn to String theory, which is the leading candidate for atheory beyond the Standard Model and naturally allows for the simultaneous analysis ofgauge and gravitational interactions.The goal of identifying 4D string models that come as close as possible to the SM orMinimal Supersymmetric Standard Model (MSSM) has been pursued within a range ofdifferent approaches (see e.g. ref. [1] and references therein). The main obstacle to thesesearches is the vast space of consistent string vacua in 4D, known as the string landscape.From a geometric perspective, this vast space of string vacua is generated by the enormousnumber of consistent compactifications of the 10D superstring down to 4D. In this paperwe work within the free fermionic worldsheet description [2] of the heterotic string, wherethe large space of vacua comes from the freedom in choosing the boundary conditions ofworldsheet fermions. This construction can be translated into the geometric context of Z × Z orbifolds as detailed in ref. [3].The exploration of subspaces of the string landscape using Machine learning and DeepLearning has become a burgeoning area of research in string phenomenology, see, e.g. ,refs. [4] and for review see, e.g. , ref. [5]. In this paper we make use of a different tool withinthis context: Boolean Satisfiability Checking (SAT) and Satisfiability Modulo Theories(SMT). SAT and SMT solvers use powerful algorithms for determining the satisfiability ofconstraints as well as for finding solutions where the constraints are satisfiable. We expectSAT and SMT solving to have a wide range of applications in string phenomenologyas they have, indeed, proven to have in a range of other fields, such as computationalbiology [6], artificial intelligence [7], combinatorics [8], and mathematical geometry [9].Typically, the input variables needed to specify a string model are positive integersand a priori there are approximately 100 of them. They may, for example, specify thegeometry of the compactified six-dimensional space or, in our case, generate boundaryconditions for world-sheet free fermions. These inputs are required to meet certain con-sistency constraints, such as modular invariance, which reduces the number of independentinput variables. In the case of free fermionic models, the boundary conditions are typ-ically either Ramond or Neveu-Schwarz, and so the input variables can be representedas Boolean variables. Classes of models can be analysed in terms of relations of thesevariables, and constraints related to desirable physical properties can be imposed. Someexamples of typical constraints are the presence of three particle generations, Higgs con-tent and the absence of chiral exotic states. In recent works we have also been interestedin unphysical, but interesting, string vacua, which are free of massless fermions or masslesstwisted bosons that are called Type 0 [10] and Type ¯0 [11] models, respectively. Thesemodels have potential applications within cosmological scenarios of the early universe.As more and more phenomenological constraints are imposed, the frequency of viablemodels can become low enough to evade random searches that take many weeks. Thiswas found, for example, within the classification of Standard-Like Models and Left-RightSymmetric models [12, 13] where, in the latter case, models would satisfy all imposedphenomenological criteria with a probability in the order of 10 − . This scarcity of viable1odels has motivated the introduction of the fertility methodology within these classes ofvacua [12, 14]. Furthermore, in some classes of vacua, important distinct characteristics ofstring models have been found to be incompatible. For example, within the classificationof 10 Flipped SU (5) vacua [15], there were no exophobic models with three generations,whereas such vacua were found within the classification of Pati-Salam models [16]. Asimilar result was found in recent work on Type 0 vacua [10], where it was shown analyt-ically for a minimal class of models that the absence of massless fermions necessitates thepresence of tachyons, and the result persisted within a generalised class of models. As theconstruction of models becomes more complex and realistic, it is not straightforward toprovide an analytic proof of apparent incompatibilities for phenomenological constraintsthat appear in classification statistics. Scenarios of this kind are the strong suit of SATand SMT, as they are powerful and proven tools in both finding rare solutions (and thusuncovering models with interesting properties) and proving unsatisfiability for constraintsand isolating the origin of this unsatisfiability.In this paper we first of all test the efficiency of reducing the problem of findingtachyon-free Type ¯0 models of the sort found in [11] to SAT or SMT. We first employa standard SMT solver, Z3 [17], in an intuitive direct encoding of the input variables asintegers, where all parameters are either 0 (corresponding to NS boundary conditions) or1 (corresponding to R boundary conditions), which results in an efficiency improvementcompared with a random classification: within the class of models we explore, the randomclassification approach finds approximately 500 tachyon-free Type ¯0 vacua in 40 minutes,which is the time it takes the SMT with integer encoding to find all 2048 solutions andto complete the search. We then go further to translate the constraints into a Booleanencoding such that the system is simply a SAT problem and the full power of the SATsolver, also part of Z3 [17], is demonstrated. In this case, establishing satisfiability takes0.04 seconds while constructing all models takes 19 seconds, which is 126 times fasterthan with integer encoding, and around 450 times faster than enumerating and testingall candidate models.Having established this, we then employ the SAT and SMT solvers to confirm a con-tradiction between the presence of spinorial ’s for (tachyon-free) Type ¯0 models. TheSAT solver not only confirms the unsatisfiability of these constraints in 0.06 seconds inthe Boolean encoding (163 seconds for the SMT solver using integer encoding), which ismore than 100,000 times faster than an exhausting search. The SAT solver can also beused to identify a minimal ‘unsatisfiable core’ [18], which isolates where the contradictionarises by giving a (locally) minimal subset of constraints, where dropping either of themresults in a satisfiable constraint system. From this, it is straightforward to apply Op-timisation Modulo Theory (OMT), offered by many SMT solvers, to find models with aminimal number of massless twisted bosons that also contain / ’s. Where OMT isnot offered, one can also add a minimisation/maximisation constraint to manually hone inon contradictions and optimal configurations. Due to the abundance of massless fermionsin their spectra, such models are expected to have a positive cosmological constant at thefree fermionic point and could be candidates for having a necessarily positive one-looppotential once the string moduli are incorporated, as discussed in ref. [19].The models we explore can be regarded as compactifications of the non-supersymmetric10D tachyonic heterotic string vacua [20, 21, 22, 23]. By construction, the class of modelswe analyse here cannot give rise to (quasi-)realistic string vacua but rather is chosen asa simplified set-up, which is perfect for illustrating some key applications of SAT and2MT solvers. We expect the advantages of the SAT/SMT approach to magnify for morerealistic classes of models with larger input spaces.The structure of the paper is as follows: in Section 2 we provide an introduction toSAT/SMT. In Section 3 we define the class of models we will explore and the constraintswe evaluate. In Section 4 we detail how these can be evaluated using an SMT and SATsolvers and present the results. Section 5 concludes the paper. Satisfiability Modulo Theories (SMTs) are powerful algorithms used for deciding whethera set of constraints describing a problem is satisfiable. In other words, SMTs determinewhether there exists a ‘satisfying assignment’ of a set of input variables to a system ofconstraints. These constraint formulae are constructed by defining operations over, whatare referred to as, theory variables, and combining them with logical connectives. SMTproblems are more expressive and powerful than the Boolean Satisfiability (SAT) problemsthat restrict all variables to be true or false and operators to be logical connectives. Inparticular, SMTs allow for operations over non-Boolean types such as integers, reals,bitvectors, and arrays. Both SMT and SAT are canonical NP-complete problems [24],which is a class of computational problems for which checking whether a given variableassignment satisfies the constraints can be done in polynomial time, but finding such anassignment is believed to be hard. Despite a lot of effort, no polynomial time algorithm wasshown for any NP-complete problem since the class was defined in 1971. More than that,a widely believed conjecture [25] states that in the worst-case one cannot significantlyimprove over an exhaustive search of all possible assignments. Nevertheless, there hasbeen a tremendous progress over the last decades in efficiency of algorithms solving theseproblems in practice, solving instances with hundreds and even thousands of variables,which shows that truly hard instances are few and far between.A key aspect of how SMT-solvers work so effectively is through following the DPLL orconflict-driven clause learning (CDCL) class of algorithms. These algorithms implementa decision procedure for each theory by adding or subtracting constraints and queryingfor satisfiability as it goes. More detail on DPLL(T) and other decision procedures maybe found in e.g. [26].One of the most efficient and easy to use SMT solvers is Z3, which can be found opensource on Github at https://github.com/Z3Prover/Z3. Z3 was developed by Microsoftprimarily for software verification purposes. It also implements an efficient SAT solverthat we use for the Boolean encoding of our problem. It has bindings for most commonprogramming languages and in our case we used the Python front-end as a means ofinterfacing with Z3. SO (10) ˜ S -models We will utilise the free fermionic construction [2] to define a class of non-supersymmetric Z × Z orbifold models with an unbroken SO (10) observable group. Only the key aspectsof the free fermionic construction will be described here, as our main purpose is theapplication of the SMT solver within this setting. We will be adopting the conventionalnotation used in the free fermionic literature [27, 28, 29, 30, 31, 16, 15, 19, 12, 13, 14].3ver the past two decades systematic methods to classify large numbers of free fermionicheterotic-string models were developed [32, 33, 16, 15, 12, 13, 22, 23]. The initial methodwas developed for the classification of spinorial and anti-spinorial representations of anunbroken SO (10) GUT group [32], and extended to include its vectorial representations[33], which led to the discovery of spinor-vector duality over the space of vacua [33, 34].It was extended to include the entire massless twisted spectrum in models with, SO (6) × SO (4) [16], SU (5) × U (1) [15], SU (3) × SU (2) × U (1) [12], SU (3) × U (1) × SU (2) [13, 14], unbroken SO (10) subgroups. Exophobic three generation models, in which exoticfractionally charged states only appear in the massive string spectrum were discovered inthe case of SO (6) × SO (4) models, whereas all other cases contained non-chiral masslessexotic states in the spectra of three generation models. Over the past year the classificationmethodology was extended to non-supersymmetric heterotic-string vacua [22, 10, 23, 11],in which case the existence of physical tachyonic states in the physical spectrum is ofparticular interest. In the process of such classifications, we are particularly interested inthe presence, or lack thereof, of some specific states in the spectrum of the models. It isevident that SAT and SMT solving are particularly well suited to address such questions.The construction of a free fermionic string model is defined at the free fermionic pointin the moduli space and is generated by specifying two ingredients. The first is a set of N boundary condition basis vectors, v i ∈ B , i = 1 , ..., N , and the second is a set of one-loopGeneralised GSO (GGSO) phases, C (cid:2) v i v j (cid:3) .We will use the basis explored in [11], where Type ¯0 string vacua, i.e. models withoutany twisted massless bosons, were uncovered. This basis is written as = { ψ µ , χ ,..., , y ,..., , w ,..., | y ,..., , w ,..., , ψ ,..., , η , , , φ ,..., } , ˜ S = { ψ µ , χ ,..., | φ , , , } ,T = { y , , w , | y , , w , } ,T = { y , , w , | y , , w , } ,T = { y , , w , | y , , w , } ,b = { ψ µ , χ , y , y | y , y , η , ψ ,..., } , (3.1) b = { ψ µ , χ , y , w | y , w , η , ψ ,..., } ,b = { ψ µ , χ , w , w | w , w , η , ψ ,..., } ,z = { φ ,..., } . We note that the vectors T i , i = 1 , , T tori. The inclusion of ˜ S in the basis means we have no supersymmetry andthese vacua can be considered as compactifications of the tachyonic 10D heterotic string,as discussed in [20, 21, 22, 23].The NS sector vector gauge bosons give rise to a gauge group SO (10) × U (1) × SO (4) × SU (2) (3.2)and may receive additional vector boson enhancements from the sectors ψ µ | z i L ⊗ { ¯ λ i } | z i R ψ µ | z i L ⊗ { ¯ λ i } | z i R ψ µ | z + z i L ⊗ | z + z i R (3.3)4here ¯ λ i are all possible right moving Neveu-Schwarz oscillators and z is the importantlinear combination z = 1 + b + b + b + z = { ¯ φ , , , } . (3.4)Having defined these 9 basis vectors, the other ingredient for defining a model is the GGSOcoefficients, C (cid:2) v i v j (cid:3) = ±
1, which will generate a 9 × ∼ . × independent GGSO phase configurations.With an assignment of GGSO phases the Hilbert space of states | S ξ i can be specifiedthrough H = M ξ ∈ Ξ k Y i =1 (cid:26) e iπv i · F ξ | S ξ i = δ ξ C (cid:20) ξv i (cid:21) ∗ | S ξ i (cid:27) . (3.5)where ξ are sectors (linear combinations of basis vectors) in the additive space Ξ and δ ξ is the spin statistic index. The GGSO projection equation inside the curly brackets of(3.5) will give us the constraints implemented within the SMT solver described in thenext section.The sectors in the model can be characterised according to their left and right movingvacuum separately M L = −
12 + ξ L · ξ L N L (3.6) M R = − ξ R · ξ R N R where N L and N R are sums over left and right moving oscillators, respectively. Physicalstates must additionally satisfy the Virasoro matching condition, M L = M R such thatmassless states are those with M L = M R = 0 and physical tachyons arise for sectors with M L = M R < The first aspect of these models we wish to explore is the massless twisted bosons. In[11], we have described the conditions on the absence of twisted massless bosons withinthis class of models and found tachyon-free GGSO configurations that corresponded totwo distinct partition functions. The details are repeated here as we will apply the SMTsolver to these conditions on the absence of massless twisted bosons.For this class of models, there are 15 vectorial bosonic sectors of the form V pq = b + b + T + pT + qT V pq = b + b + T + pT + qT V pq = b + b + T + pT + qT V = T + T V = T + T V = T + T (3.7)5here p, q = 0 , B pq = b + b + z + T + pT + qT B pq = b + b + z + T + pT + qT B pq = b + b + z + T + pT + qT B pq = + b + z + T + pT + qT B pq = + b + z + T + pT + qT B pq = + b + z + T + pT + qT .B = T + T + z B = T + T + z B = T + T + z B = T + T + z B = T + T + z B = T + T + z (3.8)A Type ¯0 model arises when all of these sectors are projected for an assignment of GGSOphases. For example, taking sector B pq , a projector is constructed of the form P pq = 12 (cid:18) C (cid:20) B pq T (cid:21)(cid:19) (cid:18) C (cid:20) B pq z (cid:21)(cid:19) (cid:18) C (cid:20) B pq b + pT + qT (cid:21)(cid:19) (3.9)and requiring that this is zero ensures its absence from the massless spectrum. Repeat-ing this for all massless twisted bosonic sectors allows for the identification of Type ¯0configurations. The next constraint to impose after the projection of massless bosonic sectors, is the ab-sence of tachyonic sectors. Since our models are non-supersymmetric this is an importantcheck for determining the stability of our models for a 4D Minkowski background. Thesame procedure of encoding the GGSO projections applies to the tachyonic sectors. Thetachyonic sectors in this construction are: { ¯ λ } T i , z , z , z + T i and z + T i , i = 1 , , i = 1 , , λ is some right-moving NS oscillator and we note that the untwistedtachyon from | i L ⊗ { ¯ λ } | i R is projected regardless of the GGSO phase choices in thisbasis.As an example, we can delineate the condition for the projection of the { ¯ λ } T tachyonicsector x ∈ S | x = − > S = (cid:26) C (cid:20) T T (cid:21) , C (cid:20) T T (cid:21) , C (cid:20) T z (cid:21) , C (cid:20) T z (cid:21)(cid:27) . (3.10)which ensures all oscillator cases are projected.6 .3 Spinorial 16 /
16 Sectors
The fermion generations transforming in the spinorial / of SO (10) arise from thetwisted sectors F pq = b + pT + qT F pq = b + pT + qT (3.11) F pq = b + pT + qT . which have projectors R pq = 12 (cid:18) − C (cid:20) F pq T (cid:21)(cid:19) (cid:18) − C (cid:20) F pq z (cid:21)(cid:19) (cid:18) − C (cid:20) F pq z (cid:21)(cid:19) R pq = 12 (cid:18) − C (cid:20) F pq T (cid:21)(cid:19) (cid:18) − C (cid:20) F pq z (cid:21)(cid:19) (cid:18) − C (cid:20) F pq z (cid:21)(cid:19) (3.12) R pq = 12 (cid:18) − C (cid:20) F pq T (cid:21)(cid:19) (cid:18) − C (cid:20) F pq z (cid:21)(cid:19) (cid:18) − C (cid:20) F pq z (cid:21)(cid:19) which can be used to tell us + ) for any model. Knowing this is sufficient forour purposes here with the SMT solver analysis and we will not implement the chiralityprojection distinguishing the and . Now we turn our attention to the analysis of these model characteristics with an SMTwritten using Z3 in Python. It is convenient to introduce the notation C (cid:20) v i v j (cid:21) = exp [ iπ ( v i | v j )] (4.1)and use the 36 independent phases ( v i | v j ) ∈ { , } as the input variables for our SMTsolver. In terms of these variables, the GGSO projection equation for each sector canbe written in terms of sums of ( v i | v j ) modulo 2. We call this representation the integerencoding for our system of constraints.As mentioned in Section 2, we use the general purpose SMT solver Z3, which allowsus to use mathematical expressions so we do not need to reduce the constraints to purelypropositional logic as in this integer encoding of the GGSO equations. This comes atthe expense of performance because the SMT solver needs to include reasoning for math-ematical theories (such as integer arithmetic). Therefore we will also detail a Booleanencoding of our constraint system, where we rewrite the GGSO projections purely asBoolean propositions.It turns out that the conditions for the projection of massless twisted bosons andtachyonic sectors do not involve the following 9 of the 36 ( v i | v j ) phases:( | ˜ S ) , ( | b ) , ( | b ) , ( | b ) , ( | z ) , ( ˜ S | b ) , ( ˜ S | b ) , ( ˜ S | b ) , ( ˜ S | z ) (4.2)and so our space of models to explore is reduced to 2 (ca. 1 . × ). This is well withinthe reach of a complete enumeration of possible GGSO configurations but our purpose7ere is testing the features and efficiency of the SMT solver within this simple class ofmodels. Moreover, as it does not adversely effect performance, we have re-introducedthese 9 variables as input to our Boolean encoding with the expectation that it will notsignificantly impact the running time (in face, it has proven to actually cut the runningtime by 4%.)As an example of how we can write the GGSO projections in both the integer rep-resentation and the Boolean representation we will take the massless bosonic sector B .The GGSO projection equation is written for generic p, q = 0 , C (cid:2) ·· (cid:3) ’s in eq.3.9. In the integer encoding we can write the B projection condition by first defining P = [( T | b ) + ( T | b ) + ( T | z ) + 1 + ( | T )] mod 2 Q = [( | T ) + ( T | b ) + ( T | b ) + ( T | b ) + ( T | z ) + ( b | b ) + ( b | b ) + ( b | z )] mod 2 R = [1 + ( b | b ) + ( b | b ) + ( b | z ) + ( T | b )] mod 2 (4.3)and then constructing the conjunction( P = 1 ∨ Q = 1 ∨ R = 1) . (4.4)to impose the projection constraint. This can then be translated into the Boolean repre-sentation using the Xor() operator, ⊻ , as follows:˜ P = ( T | b ) ⊻ ( T | b ) ⊻ ( T | z ) ⊻ True ⊻ | T )˜ Q = ( | T ) ⊻ ( T | b ) ⊻ ( T | b ) ⊻ ( T | b ) ⊻ ( T | z ) ⊻ ( b | b ) ⊻ ( b | b ) ⊻ ( b | z )˜ R = True ⊻ ( b | b ) ⊻ ( b | b ) ⊻ ( b | z ) ⊻ ( T | b ) (4.5)where ( v i | v j ) are now Booleans rather than 0 or 1. The constraint for ensuring theprojection of this sector is then ( ˜ P ∨ ˜ Q ∨ ˜ R ) . (4.6)For both representations we can schematically write the steps for the construction ofthe Z3 solver in the case of finding tachyon-free Type ¯0 string models ¶ Define the 27 input variables c , ..., c Add constraints on input variable domain ( c i = 0 ∨ c i = 1) ∀ i = 0 , ..., Add constraints for GGSO projection of all twisted massless bosons Add constraints for GGSO projection of all tachyonic sectors. Check satisfiability OR Find satisfying assignments (print all solutions)Step 5 is perhaps the most fundamental application of SAT/SMT solvers, which canbe used to quickly identify whether the set of constraints permits a solution or not. Inthis case, we can utilise the prior analysis in ref. [11], which found tachyon-free Type ¯0models, to realise that the SMT solver should return sat in this case. ¶ Both the integer and Boolean Z3 codes are available athttps://github.com/thePlumbaked/SMTsType0bar. .1 Results of SMT search for tachyon-free Type ¯0 models Enumerating all tachyon-free Type ¯0 models within the class of models under consider-ation is a good testing ground for the efficiency of the SMT/SAT solver. As mentionedearlier, the space of models is 2 (ca. 1 . × ), which is within the grasp of a completeenumeration approach. We have run a random search in comparison, which is able toanalyse ca. 16,000 sample points per second. An exhaustive enumeration of the modelthus takes around 2 hours 20 minutes, and a random search (with repetitions) needs onaverage around 19 hours to find all solutions and 4 seconds to find the first model.Within the integer representation, Z3’s SMT solver determines satisfiability in 12seconds and finds all 2048 solutions in the full 2 space in 2405 seconds (ca. 40 minutes).A random search found just under a quarter of these models in the same amount of time(500 models). While this is a useful speed up, it is not that impressive. This is expectedsince, in this representation, the SMT solver deals with mathematical theories that createa significant overhead. Figure 1 depicts the accumulation of solutions over time for theSMT solver within the integer representation.Figure 1: Rate at which the Integer representation finds all tachyon-free Type ¯0 models. The performance is significantly improved when implementing a Boolean encoding. Inthis case, Z3’s SAT solver determines satisfiablitiy in 0.04 seconds. It can find and print all2048 solutions in 19 seconds. This is 126 times faster than for the integer representation,and 450 times faster than exhaustively enumerating and checking the statespace. Figure 2depicts the accumulation of solutions over time for the SAT solver, where we see that thesolver is not slowed down by the creation of new lemmas as solutions are enumerated. Itcan be seen that only 1850 compact solutions are recorded on the graph in Figure 2. Thisis because some of the compact solutions include variables labelled with
None , meaningthat this variable/s can be set to true or false; they can be trivially expanded to the 2048solutions (without omissions and without multiple occurrences of individual solutions).9igure 2:
Rate at which the Boolean representation finds all tachyon-free Type ¯0 models,depicting the number of compact solutions found; the 1850 compact solutions contain all2048 explicit solutions (exactly once). We have repeated the experiment using all 36 original input variables. Our expectationwas that the SAT solver would essentially ignore them, as they do not enter into thereasoning at any point, though it will need to output a few additional
None in the compactrepresentation. We expected an insignificant increase in the overall running time. Wefound that the solver worked 4% faster, which is likely due to different decisions made bythe heuristics. But it shows an interesting effect: overlooking the variables that do notmatter did not slow the SAT solver down, whereas an exhaustive search would have taken6 orders of magnitude longer. ¯0 Vacua
In the analysis from ref. [11] it was found that no Type ¯0 vacua include the fermiongenerations from the spinorial / sectors (3.11). Since these sectors are phenomeno-logically desirable, we aim to ensure at least one remains in the Hilbert space after GGSOprojections.The sectors giving rise to the / were given in eq. (3.11) and the projectors ofeq. (3.12) can be rewritten in the integer and Boolean representations in a similar way asdelineated above for B .With the addition of this condition the SMT structure summary can be updated to10 : Define the 27 input variables c , ..., c Add constraint on input variable domain ( c i = 0 ∨ c i = 1) ∀ i = 0 , ..., Add constraints for GGSO projection of all twisted massless bosons Add constraints for GGSO projection of all tachyonic sectors. Add constraint on presence of at least 1 / sector Check satisfiability OR Find satisfying assignments (solutions)As expected from the findings of ref. [11], with this added constraint on the spinorial / the Z3 solver returns unsat . This takes only 0.06 seconds in the Boolean encod-ing (163 seconds in the integer encoding), underlining the feasibility of combing largeparameter spaces.In such cases where constraint systems are unsatisfiable there are several tools withinZ3 that are helpful to understand and isolate where the inconsistency arises from. Usingits proof() method, Z3 will output a proof of inconsistency. Unsatisfiability proofs canbe long and tedious because, while satisfiability can be shown by providing a model,unsatisfiability needs to make a mathematical argument that establishes the contradiction.Such proofs are often long and tedious—though this is no comparison to the tedious workof manually distilling the contradiction, but it is fair to say that they often do not providemuch accessible further insight.There are, however, several other helpful tools that can be used to isolate inconsisten-cies. In particular, a minimal ‘unsatisfiable core’ of constraints can be returned by mostSMT solvers. Additionally, a more manual approach of using push and pop methods onconstraints allows for pinning down the source of an inconsistency. Using this approach,the presence of the spinorial / ’s from F ipq is found to contradict the projection of thevectorial V ipq (under the presence of the remaining constraints). Since the Higgs bidou-blet representation would reside within the vectorial of SO (10) coming from the V ipq , i = 1 , ,
3, there is physical motivation to keep at least one of these twisted bosonic sectorsin the Hilbert space. Demanding that at least one sector from F pq remains, and at leastone of V pq makes the SMT return sat , which takes 0.03 seconds for the Boolean constraintsystem and 8.4 seconds for the integer encoding. Creating all 640 satisfying assignmentsin 7.5 seconds for the Boolean constraint system (1661 seconds for the integer encoding)of the 27 input variables with these conditions. One such model is defined by C (cid:20) v i v j (cid:21) = ˜ S T T T b b b z − − − S − − − − T − − − − T − − − T − − b − − − − − − b − − − − − b − − − − − − z − − which has partition function Z = 2¯ q − + 1280 q − / ¯ q / + 48 q / ¯ q − / − − q / ¯ q / + · · · (4.8)11here we can see that N b − N f = − − . λ , via the rescaling λ = − M Λ, where M = M String / π . The application of Machine Learning techniques within the string landscape is alreadya burgeoning field. In this work, we open the door to the application of SAT and SMTsolvers within this context and have demonstrated their power and efficiency within a sim-plified class of string models and expect their benefits to only increase as more generalisedand (quasi-)realistic classes of vacua are studied.We have demonstrated how SAT and SMT solvers can be used to help isolate incon-sistent constraints and be used to optimise desired characteristics. This method was thenemployed to find string vacua with positive cosmological constant and desirable SO (10)representations. Furthermore, this approach essentially meant minimising the number ofmassless twisted bosons, which is generally enough to ensure a large abundance of masslessfermions resulting in the largest contribution to cosmological constant (the massless level)being positive. Although the massive and off-shell contributions also need accounting for,a large abundance of massless fermions can effectively guarantee a positive cosmologicalconstant. Such vacua are then ripe for analysis away from the free fermionic point inthe moduli space by exploring the one-loop potential in terms of the moduli fields as waspursued in [19], where a model with N f > N b at the free fermionic point and N b = N f ata generic point in the moduli space resulted in guaranteeing a positivity of the one-looppotential. Using the SAT and SMT solvers described in the current work makes it veryquick and easy to find string models at the free fermionic point with desired propertiesat the massless level, even when the models are rare in the full space.There are a couple of obvious limitations, from a phenomenological perspective, in theclass of models that we examined in this paper. First of all, a F pq and V pq could not giverise to desired coupling as they reside on the same orbifold plane. Secondly, due to onlyemploying the T i , i = 1 , ,
3, in the basis we do not allow for shifts around the 6 circlesof the internal T and there is a multiplicity factor of 4 attached to each sector, making3 generation models impossible. However, our focus here is on the introduction of SATand SMT solving and the illustration of its application, while the application of SAT andSMT solving to more realistic constructions is left for future work. It is evident, however,that as we seek to construct satisfiability criteria for non-supersymmetric [35] and moredetailed phenomenological string models, for which the SAT and SMT algorithms areparticularly well suited, these algorithmic approaches are of immense interest and utility. Acknowledgments
We would like to thank Viktor Matyas for helpful discussions. The work of BP is supportedin part by STFC grant ST/N504130/1. The work of SS and DW are supported in partby EPSRC grant EP/P020909/1. 12 eferences [1] L. Ib´a˜nez and A. Uranga (2012),
Cambridge University Press ,doi:10.1017/CBO9781139018951;K. S. Choi and J. E. Kim, Lect. Notes Phys. (2006), 1-406doi:10.1007/b11681670.[2] I. Antoniadis, C. Bachas, and C. Kounnas,
Nucl. Phys.
B289 (1987) 87;H. Kawai, D.C. Lewellen, and S.H.-H. Tye,
Nucl. Phys.
B288 (1987) 1;I. Antoniadis and C. Bachas,
Nucl. Phys.
B298 (1988) 586.[3] A.E. Faraggi,
Phys. Lett.
B326 (1994) 62;E. Kiritsis and C. Kounnas,
Nucl. Phys.
B503 (1997) 117;A.E. Faraggi, S. Forste and C. Timirgaziu,
JHEP (2006) 057;R. Donagi and K. Wendland,
J.Geom.Phys. (2009) 942;P. Athanasopoulos, A.E. Faraggi, S. Groot Nibbelink and V.M. Mehta, JHEP (2016) 038.[4] see e.g. and references therein :Lara B. Anderson, Mathis Gerdes, James Gray et al (2020), arXiv:2012.04656;J. Carifio, J. Halverson, D. Krioukov and B. D. Nelson,
JHEP (2017) 157;A. Mutter, E. Parr and P. K. S. Vaudrevange,
Nucl. Phys.
B940 (2019) 113;Y.H. He,
Phys. Lett.
B774 (2017) 564;S. Abel and J. Rizos,
JHEP (2014) 010.[5] For a comprehensive review see e.g. and references therein :F. Ruehle (2020),
Phys. Rep. (2020) 1.[6] Paoletti, Nicola, et al (2014), Computer Aided Verification. Springer InternationalPublishing.[7] Luca Pulina and Armando Tacchella (2012), AI Commun. 25, 2, 117–135;Kautz, H., McAllester, D. and Selman, B. (1996), Proc. of KR-96, Boston, MA.[8] B. Konev B. and A. Lisitsa (2014), doi:10.1007/978-3-319-09284-3 17.[9] C. L¨uders (2020), arXiv:2004.07058.[10] A.E. Faraggi, V.G. Matyas and B. Percival, arXiv:2010.06637.[11] A.E. Faraggi, V.G. Matyas and B. Percival, arXiv:2011.12630.[12] A.E. Faraggi, J. Rizos and H. Sonmez,
Nucl. Phys.
B927 (2018) 1.[13] A.E. Faraggi, G. Harries and J. Rizos,
Nucl. Phys.
B936 (2018) 472.[14] A.E. Faraggi, G. Harries, B. Percival and J. Rizos
Nucl. Phys.
B953 (2020) 114969.[15] A.E. Faraggi, J. Rizos and H. Sonmez,
Nucl. Phys.
B886 (2014) 202;H. Sonmez,
Phys. Rev.
D93 (2016) 125002.[16] B. Assel et al , Phys. Lett.
B683 (2010) 306;
Nucl. Phys.
B844 (2011) 365;C. Christodoulides, A.E. Faraggi and J. Rizos,
Phys. Lett.
B702 (2011) 81.1317] L. de Moura and N. Bjørner (2008), In Conference on Tools and Algorithms for theConstruction and Analysis of Systems (TACAS ’08).[18] A. Cimatti, A. Griggio, and R. Sebastiani,
Journal Artificial Intelligence Res. , 1(2001) 701–728.[19] I. Florakis and J. Rizos, Nucl. Phys.
B913 (2016) 495.[20] A.E. Faraggi,
Eur. Phys. Jour.
C79 (2019) 703.[21] A.E. Faraggi, V.G. Matyas and B. Percival
Eur. Phys. Jour.
C80 (2020) 337.[22] A.E. Faraggi, V.G. Matyas and B. Percival,
Nucl. Phys.
B961 (2020) 115231.[23] A.E. Faraggi, V.G. Matyas and B. Percival, arXiv:2011.04113.[24] S. Cook (1971), doi:10.1145/800157.805047;L. Levin (1973), Problems of Information Transmission (in Russian). 9(3): 115–116.[25] R. Impagliazzo and R. Paturi (1999), doi:10.1109/CCC.1999.766282.[26] A.R. Bradley and Z. Manna.
The calculus of computation: decision procedures withapplications to verification.
Springer, Oct. 2007. isbn: 3-540-74112-7;D. Kroening and O. Strichman.
Decision procedures . Springer, 2008. isbn: 978-3-540-74104-6[27] I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos,
Phys. Lett.
B231 (1989) 65.[28] A.E. Faraggi, D.V. Nanopoulos and K. Yuan,
Nucl. Phys.
B335 (1990) 347;A.E. Faraggi,
Phys. Rev.
D46 (1992) 3204;G.B. Cleaver, A.E. Faraggi and D.V. Nanopoulos,
Phys. Lett.
B455 (1999) 135.[29] I. Antoniadis. G.K. Leontaris and J. Rizos,
Phys. Lett.
B245 (1990) 161;G.K. Leontaris and J. Rizos,
Nucl. Phys.
B554 (1999) 3.[30] A.E. Faraggi,
Phys. Lett.
B278 (1992) 131;
Nucl. Phys.
B387 (1992) 239;A.E. Faraggi, E. Manno and C.M. Timirgaziu,
Eur. Phys. Jour.
C50 (2007) 701.[31] G.B. Cleaver, A.E. Faraggi and C. Savage,
Phys. Rev.
D63 (2001) 066001;G.B. Cleaver, D.J Clements and A.E. Faraggi,
Phys. Rev.
D65 (2002) 106003.[32] A.E. Faraggi, C. Kounnas, S.E.M Nooij and J. Rizos, hep-th/0311058;
Nucl. Phys.
B695 (2004) 41.[33] A.E. Faraggi, C. Kounnas and J. Rizos,
Phys. Lett.
B648 (2007) 84;
Nucl. Phys.
B774 (2007) 208;
Nucl. Phys.
B799 (2008) 19.[34] T.Catelin–Julian et al , Nucl. Phys.
B812 (2009) 103;C. Angelantonj, A.E. Faraggi and M. Tsulaia,
JHEP (2010) 314;A.E. Faraggi, I. florakis, T. Mohaupt and M. Tsulaia,
Nucl. Phys.
B848 (2011) 332.1435] L.J. Dixon, J.A. Harvey,
Nucl. Phys.
B274 (1986) 93;L. Alvarez–Gaume, P.H. Ginsparg, G.W. Moore and C. Vafa,
Phys. Lett.
B171 (1986) 155;H. Kawai, D.C. Lewellen and S.H.H. Tye,
Phys. Rev.
D34 (1986) 3794;H. Itoyama and T.R. Taylor,
Phys. Lett.
B186 (1987) 129;K.R. Dienes,
Phys. Rev. Lett. (1990) 1979; Phys. Rev.
D42 (1990) 2004;S. Abel, K.R. Dienes and E. Mavroudi,
Phys. Rev.
D91 (2015) 126014;M. Blaszczyk et al , JHEP (2015) 166;J.M. Ashfaque et al , Eur. Phys. Jour.
C76 (2016) 208;H. Itoyama and S. Nakajima,
Nucl. Phys.