Assistance in making decisions to promote planting and conservation of maize in the state of Puebla of Mexico through Answer Set Programming

Abstract

We present some examples which show how preference ordered disjunction adds useful features to answer set programming, as well as an interesting example of how it can be useful for modeling a real problem about agriculture. In this paper, we present a mathematical formalization for the problem of making decisions to promote planting and conservation of maize in some geographical zone inside the country of Mexico; we present modeling semantics that we consider necessary for the mathematical formalization of this problem using Answer Set Semantics.

Keywords

• Agriculture
• Ordered Disjunction
• Preferences
• Answer Set Programming
• Data Mining

Introduction

At the present time Maize is one of the most important crops around the world, it is

a major source of food for humanity and the livestock sector [ 5 ]. Currently it is very important to support technological development in agriculture in order to increase yields (production per hectare) of crops that are important to satisfy the food demand of humanity in the world [ 20 ].

A problem present in Mexico is that the total production of maize in the country

is not enough to meet food demand of all Mexican population; Mexico has faced the need to import foreign corn mainly from U.S. and Canada since it is cheaper to import corn than to produce national maize in Mexico [ 20 ][ 18 ].

Given experts recommendations and needs [20][18], and experts works [8][15][14] we identify that is possible to de¯ne The maize ¯tting zone problem and give a solution to it in order to support agriculture in the area of corn:

3. Given a geographical area for each land zones belonging to it, to identify the attributes (geographical, climatic, technological, economic, soil, hydrological land use and vegetation) that de¯ne it. 4. To relate this land attributes in function of classes, races and varieties of maize and their yields produced. 5. To determine which land zones report high or low yields for each variety of corn and to explain the necessary and su±cient conditions to obtain high yields in each land zone. 6. To produce corn in land zones selecting the best race of maize that ¯ts each land zone to obtain high yields of production preserving an ecosystem that does not harm nature and that sati¯es the nutrimental needs of producers and population of a given geographical area. 7. To replicate these conditions in land zones with low yields of corn 8. Continuous monitoring of implementation of the model in reality for di®erent classes of producers by experts, and feedback to the mathematical model with the results. Taking care not to a®ect the quality of life for producers who have their sustenance on their own crops.

Recently work [ 8 ][ 15 ][ 14 ] has been done trying to solve the problem The maize ¯tting zone problem. There is great diversity of climatic and geographic attributes, attributes that represent the technologies necessary to grow, control pests and diseases of maize, and attributes of phenotypes according to breeds and varieties of existing corn and classes to which they belong. The data type for these attributes are strings, integers, real numbers, percentages, however in the works we have reviewed percentage and numerical attributes are handled through ranges between values in order to facilitate the categorization of these attributes, because of this it is possible to map, one to one, quantitative information to qualitative data. Due to the complexity in the relation between the attributes involved in these problems we require to handle incomplete knowledge. The problems show variables of uncertainty for example climate and pest or disease control attributes for the corn, and also we require the use of preferences for example preferences according to the classes of seeds to grow depending on the environment. Because we are dealing with these attributes and the type of rules that will arise in these problems, we identi¯ed that it is possible to use

Answer Set Programming [10][9] as a means to formalize this problem in an expressive language. We identi¯ed the need of using an ASP extension that use preferred rules [7][4][21] for handling preference conditions.

There is now software that e±ciently computes ASP [12][19], this is the reason we believe it is possible to propose a real solution that can be implemented based on this programming paradigm.

2

Background

In this section we introduce all the necessary terminology and relevant de¯nitions in

order to make this paper self-contained.1 1 We assume that the reader has familiarity with basic concepts of classical logic, logic programming, answer set semantics, and lattices. For details the reader can refer to [ 1,6,13 ]. 2.1

Extended Logic Programs We consider extended logic programs which have two kinds of negation, strong negation : and default negation not. A signature L is a ¯nite set of elements that we call atoms, where atoms negated by : are called extended atoms. Intuitively, not a is true whenever there is no reason to believe a, whereas :a requires a proof of the negated atom. In the following we use the concept of atom without paying attention if it is an extended atom or not. A literal is either an atom a called positive literal, or the negation of an atom not a called negative literal. Given a set of atoms fa1; :::; ang, we write not fa1; :::; ang to denote the set of atoms fnot a1; :::; not ang. An extended normal rule (rule, for short) r is a rule of the form a à b1; : : : ; bm; not bm+1; : : : ; not bm+n where a and each of the bi are atoms for 1 · i · m + n. If m + n = 0 the rule is an abbreviation of a à > such that > is the proposition symbol that always evaluates to true; the rule is known as a fact and can be denoted just by a. If n = 0 the rule is an extended de¯nite rule. We denote a rule r by a à B+; not B¡ where the set fb1; : : : ; bmg and the set fbm+1; : : : ; bm+ng are denoted by B+ and B¡ respectively. A constraint is a rule of the form à B+; not B¡. We denote by head(r) the head a of rule r and by body(r) the B+; not B¡ of the rule r. An extended normal logic program P is a ¯nite set of extended normal rules and/or constraints. By LP we denote the signature of P , i.e. the set of atoms that appear in the rules of P . If all the rules in P are extended de¯nite rules we call the program P extended positive logic program. In our logic programs we will manage the strong negation : as it is done in Answer Set Programming (ASP) [ 1 ]. Basically, each atom :a is replaced by a new atom symbol a0 which does not appear in the language of the program and we add the constraint à a; a0 to the program. For managing the constraints in our logic programs, we will replace each rule of the form à B+ not B¡ by a new rule of the form f à B+; not B¡; not f such that f is a new atom symbol which does not appear in LP . 2.2

Logic Programs with Ordered Disjunction Logic programs with ordered disjunction (LPODs) are extended logic programs augmented by an ordered disjunction connector £ which allows to express qualitative preferences in the head of rules [ 3 ]. A LPOD is a ¯nite collection of rules of the form r = c1 £ : : : £ ck à b1; : : : ; bm; not bm+1; : : : ; not bm+n where ci (for 1 · i · k) and each of the bj (for 1 · j · m + n) are atoms. The rule r states that if the body is satis¯ed then some ci must be in the answer set, if possible c1, if not then c2, and so on, and at least one of them must be true. Each of the ci represents alternative, ranked options for problem solutions the user speci¯es according to a desired order. If k = 1 then the rule is an extended normal rule. The semantics of LPODs is based on the following reduction.

De¯nition 1 (£-reduction). [ 3 ] Let r = c1 £ : : : £ ck à b1; : : : ; bm; not bm+1; : : : ; not bm+n be an ordered disjunction rule and M be a set of atoms. Let P be an LPOD and M be a set of atoms. The £-reduct rM is de¯ned as £ r£M := fci à b1; : : : ; bmjci 2 M and M \ (fc1; : : : ; ci¡1g [ fbm+1; : : : ; bm+ng) = ;g. The £-reduct P£M is de¯ned as P£M = Sr2P r£M .

De¯nition 2. [ 3 ] Let P be an LPOD and M a set of atoms. Then, M is an answer set of P if and only if M is a minimal model of P M . We denote by SEMLP OD(P ) £ the mapping which assigns to P the set of all answer set of P .

One interesting characteristic of LPODs is that they provide a mean to represent preferences among answer set by considering the rule satisfaction degree [3].

De¯nition 3. [ 3 ] Let M be an answer set of an LPOD P . Then M satis¯es the rule r = c1 £ : : : £ ck à b1; : : : ; bm; not bm+1 : : : ; not bm+n: { to degree 1 if bj 62 M for some j (1 · j · m), or bi 2 M for some i (m + 1 · i · m + n), { to degree j (1 · j · k) if all bl 2 M (1 · l · m), bi 62 M (m + 1 · i · m + n), and j = minfr j cr 2 M; 1 · r · kg.

The degrees can be viewed as penalties: the higher the degree the less satis¯ed we

are. If the body of a rule is not satis¯ed, then there is no reason to be dissatis¯ed and the best possible degree 1 is obtained [ 3 ]. The satisfaction degree of an answer set M w.r.t. a rule, denoted by degM (r), provides a ranking of the answer set of an

LPOD, and a preference order on the answer set can be obtained using some proposed combination strategies [3].

3

Important computational aspects for MaizeFitZone problem

In order to show that it is possible to model MaizeFitZone problem througth An

swer Set Programming we need to focus our attention in subproblems (5) and (6). Subproblem (5) presents the problem of determining for a geographical zone its agricultural production potential. Subproblem (6) presents the problem of locating seeds candidates who best ¯t the needs of farmers based on the agricultural production potential of the geographical zone where they are located. Subproblems (5) and (6) describes the use of an integrated database MaizeBioGeoClimAgriTechDB obtained from subproblems (1) to (4). Subproblems (7) and (8) describes the solution in real life once the mathematical model of the main problem has been implemented.

MaizeFitZone problem (1) to (4). We identify that it is possible to link Geo

graphical, Weather and Agricultural techonology databases in order to cross information on the di®erent corn seeds (Creole or Improved), their yields in each municipality and locality, and the relevant attributes (climatic, soil, etc.) involved to obtain certain level of production (low, medium, high, very high for example). Let us call this linked information GeoClimAgriTechDB.

From GeoClimAgriTechDB it is possible relate land attributes in function of maize

classes and their yields produced, however to relate land attributes in function of races and varieties of maize and their yields produced it is necessary to integrate

GeoClimAgriTechDB database with Biological properties of maize database, so it is

possible to know in a general level the candidates of races and varieties of maize seeds that could be presented in the land zone and the probability of appearing in it. Let us call to this integrated database MaizeBioGeoClimAgriTechDB .

To see more detalied information about the MaizeBioGeoClimAgriTechDB database see technical report[16].

MaizeFitZone problem (5). Recent work addresses this problem [8][15][14].

The computer systems developed by INIFAP Puebla [ 8 ] and Chiapas [ 15 ][ 14 ] calculate agricultural production potential in given geographic areas, that is, the level of ¯tness for a plant to be cultivated successfully. The Puebla system provides the potential for various crops (monocultures), however there is no documentation about the mathematical model employed. The Chiapas system provides the potential for varieties of native and improved maize seeds (also monocultures) and uses climatic and soil information; documentation about the mathematical model employed exists [ 15 ][ 14 ].

In order to discern from classes(Creole or Improved), races and varieties of corn seeds yields produced for each municipality and locality the relevant attributes involved to obtain certain level of production (low, medium, high, very high for example) and to explain the necessary and su±cient conditions to obtain high yields in each land zone from MaizeBioGeoClimAgriTechDB, it is possible to run datamining algorithms to discover patterns in the information such as C4.5 [ 17 ] or ID3 [ 2 ], expose these patterns to agronomists experts to discern interesting patterns.

There exists e±cient software that implements data mining algorithms such as

ID3 and C4.5 that it is easy to use such as WEKA software [11].

Let us call to the combination of the subproblems MaizeFitZone problem (4) and

MaizeFitZone problem (5) the maize zone production potential problem .

MaizeFitZone problem (6). Once the potential of agricultural production (the

level of ¯tness for a plant to be cultivated successfully) has been computed, in the

Chiapas System [15][14] a suggestion of improved seeds candidates who best ¯t the needs of the user is given, this computed from a set of geographical, climatic, and technological attributes values given as input by the user.

Once a solution is given for maize zone production potential problem and based on this solution, given a geographical zone, let us call maizes best ¯tting zone problem to the problem of selecting the races and varieties of maize to be planted that ¯ts each land zone to obtain high yields of production preserving an ecosystem that does not harm nature and that sati¯es the nutrimental needs of producers and population of a given geographical area. 4

Computational modeling Answer Set Programming

approach for maize ¯tting zone problem

This section determines the knowledge that is required to provide a knowledge mod

eling, and using the model of recent work [ 15 ][ 14 ] we show that it is possible to model the knowledge that is required to provide the knowledge modeling of MaizeFitZone problem. 4.1

Knowledge Sources We de¯ne the maize agricultural knowledge to cultivate maize plants as the union of the following knowledge: 1. Agronomists experts' knowledge. 2. The knowledge generated by centuries-old traditions and experience of the rural poor 3. Patterns and knowledge discovered from MaizeBioGeoClimAgriTechDB database using knowledge discovering techinques such as data mining.

in order to generate a mathematical model to solve MaizeFitZone problem e±ciently and accurately.

Now we present the argumentation that shows that it is possible to model Maize

FitZone problem using Answer Sets programming using extended semantics handling preferences in disjunctive rules.

To model Agronomist experts' knowledge and experience of the rural poor rep

resent the same problem since experience of the rural poor can be structured in the same way Agronomist experts' knowledge have been structured.

It is possible to provide a mathematical model for maizes best ¯tting zone problem

using Answer Sets[ 10,9 ] based on work and documentation form INIFAP Chiapas [ 15,14 ] to show how it is possible to model Agronomist experts' knowledge.

It is possible to run datamining algorithms to discover patterns from the infor

mation contained in MaizeBioGeoClimAgriTechDB such as C4.5 [ 17 ] or ID3 [ 2 ], and from this kind of algorithms is possible to generate a logic program, for instance an

ASP program for example from a classi¯ctation tree to generate in a direct way, at least an ordered disjunctive logic program.

In order to model maize ¯t problem ¯rst part of this problem (maize zone pro

duction potential problem)is modeled from natural language to ASP rules, and the second part, maizes best ¯tting zone problem is modeled using a C4.5 tree obtained from implicit knowledge in [ 15 ][ 14 ] in order to generate a preferred ordered disjunctive program. 4.2

Modeling of maize zone production potential problem : Answer Sets approach

The works [15][14] present the scienti¯c and technological agronomist experts' knowl

edge on how is determined for a given geographical zone if the potential of production in order to cultivate native maize seeds is very good, good, intermediate or low.

As input of the problem, part of extensional data base (this knowledge is included in MaizeBioGeoClimAgriTechDB knowledge base), we have the following facts: { From processing and analysis of climate we have the data: ² precipitation(Zone; Day; V alue): ² evaporation(Zone; Day; V alue): ² temperature(Zone; Day; V alue): ² precipitationByYear(Zone; Y ear; V alue): ² evaporationByYear(Zone; Y ear; V alue): ² temperatureByYear(Zone; Y ear; V alue): { From generation and classi¯cation of images (geographical data): ² zone(Zone): ² locality(Locality): ² localityInZone(Zone; Locality): ² terrainSlope(Zone; V alue): ² soilDepth(Zone; V alue): ² soilTexture(Zone; V ale): { From user input attributes for a locality: ² organicMatter(L; V alue): ² pending(L; V alue): ² rainfedQuality(L; V alue): ² altitude(L; V alue): ² cultureCycle(L; V alue): ² moistureRegime(L; V alue): ² soilTexture(Zone; V ale):

Where value is a cualitative value or an integer in a range inside [0; 1000]

The rules that gives insigth to model this problem are based in this expert knowledge[ 15 ][ 14 ]: potentialProductivity(Z; veryGood) Ã zone(Z); growingSeasonPeriod(Z; middle); soilDepthCentimeter(Z; veryDeep); not abnormal(Z): potentialProductivity(Z; good) Ã zone(Z); growingSeasonPeriod(Z; long); (soilDepthCentimeter(Z; shallow)_ soilDepthCentimeter(Z; veryDeep)); not abnormal(Z): potentialProductivity(Z; intermediate) Ã zone(Z); growingSeasonPeriod(Z; middle); soilDepthCentimeter(Z; shallow); not abnormal(Z): potentialProductivity(Z; intermediate) Ã zone(Z); growingSeasonPeriod(Z; short); soilDepthCentimeter(Z; veryDeep); not abnormal(Z): potentialProductivity(Z; low) Ã zone(Z); (growingSeasonPeriod(Z; short)_ soilDepthCentimeter(Z; shallow) _abnormal(Z)): abnormal(Z) Ã zone(Z); erosion(Z): abnormal(Z) Ã zone(Z); naturalResourcesDegradation(Z): abnormal(Z) Ã zone(Z); droughtChance(Z):

For example the rule:

potentialProductivity(Z; veryGood) Ã zone(Z); growingSeasonPeriod(Z; middle); soilDepthCentimeter(Z; veryDeep); not abnormal(Z):

Can be read as: the agricultural potential of production for a given zone Z is very

good if there is evidence that there exists in the zone Z a middle growing season perdiod and very deep soil depth, and there is no evidence that in zone Z there exists a risk of erosion, and there is no evidence that in zone Z there exists natural resources degradation risk, and there is no evidence that in zone Z there exists drought risk.

For the following rule:

potentialProductivity(Z; low) Ã zone(Z); (growingSeasonDays(Z; short) _ soilDepthCentimeter(Z; shallow) _ abnormal(Z)):

It can be shown that c à l1; : : : ; lm; (lm+1 _ : : : _ lm+n) where li are literals for n; m ¸ 0 can be replaced by the rules contained in the set fc à l1; : : : ; lm; lm+j j1 · j · ng.

If we have the predicate growingSeasonDays(Z; LengthDays) it is possible to

create a rule that discern the value for growingSeasonPeriod(Z; value) from this predicate. For example: growingSeasonPeriod(Z; long) Ã Zone(Z); growingSeasonDays(Z; LengthDays); LengthDays ¸ 146: (1) (2) (3) (4)

For the sentence [15][14]:

We determined the growing season (duration, start and end), which is the number of days during the year in which there is availability of water or moisture and a suitable temperature for crop development.

we propose the rule: growingSeasonDays(Z; LDaysGSD) Ã LDaysGSD := maxfLengthDaysjZone(Z); growingSeason(Z; StartDay; F inishDay); LengthDays := F inishDay ¡ StartDay; not waterAvailabilityInadequate(Z; StartDay; F inishDay); not temperatureInadequate(Z; StartDay; F inishDay)g: (5) where the last rule can be read as: the growing season days length for a given zone

Z is the maximum growing season days period length for a zone Z from the starting

day to the ¯nishing day of the period, in which for this zone Z and this period, there is no evidence that there exists water availability inadequate, and there is no evidence that there exists temperature innadequate.

Equation (5) can be easily translated to the following rules:

setConditions(Z; LengthDays) Ã Zone(Z); growingSeason(Z; StartDay; F inishDay); F inishDay := LengthDays + StartDay; not waterAvailabilityInadequate(Z; StartDay; F inishDay); not temperatureInadequate(Z; StartDay; F inishDay): betterSetConditions(Z; LnDays) Ã setConditions(Z; LnDays);

setConditions(Z; Y ); LnDays < Y: growingSeasonDays(Z; LnDays) Ã setConditions(Z; LnDays); not betterSetConditions(Z; LnDays): (6)

The condition about water availability inadequate is present for a period of time [StartDay; F inishDay] and a zone Z if there exist evidence that for zone Z water resources are inadequate, or for zone Z in the period [StartDay; F inishDay] the humidity is inadequate. So the following rule is generated:

The de¯nition of temperature inadequate given for a zone Z for a period of time

[StartDay; F inishDay] is de¯ned using the given temperature for every day in the given zone and period. The temperature for a given day and zone is part of extensional data base.

Humidity inadecuate given for a zone Z for a period of time [StartDay; F inishDay]

is present if for the zone Z and period of time [StartDay; F inishDay] there exist excess of humidity or de¯cit of humidity. This gives the following rule: humidityInadecuate(Z; StartDay; F inishDay) Ã Zone(Z); (excessHumidity(Z; StartDay; F inishDay) _ deficitHumidity(Z; StartDay; F inishDay)): (8)

For the sentence [ 15 ][ 14 ]: We estimated the water balance resulting from dividing the precipitation between the ETP, as support for determining the growing seasons and periods of excess and water de¯cit.

waterBalance(Z; Day; V alue) Ã Zone(Z); potentialEvapotranspiration(Z; Day; P ET V alue); precipitation(Z; Day; P recipitationV alue); P recipitationV alue := V alue ¤ P ET V alue: (9)

Rule de¯ned in (9) there is a division that can be rewritten in terms of multipli

cation, in order to simulate the multiplication in answer set, multiplication can be replaced by a predicate multiplication(A,B,AB) in the extensional data base where the third argument is the result of multiplicate the ¯rst two arguments.

De¯nition of predicates growingSeason(Zone; StartDay; F inishDay),

excessHumidity(Zone; StartDay; F inishDay) and deficitHumidity(Zone; StartDay; F inishDay) depends on the de¯nition of waterBalance(Zone; Day; V alue) (rule (9)). Works [ 8 ][ 15 ][ 14 ] does not provide the way how they are related, however we believe that is easy to construct the de¯nition of these predicates if they are de¯ned as an statistical function.

For the sentence [15][14]:

For each year level and estimated daily potential evapotranspiration (ETP) from the evaporation data multiplication by a constant factor of 0.75. The ETP is multiplied by the constant factor of 0.5. We identi¯ed periods of moisture surplus or de¯cit (duration, start and end).

We model the rule:

potentialEvapotranspiration(Z; Day; V alue) Ã Zone(Z) evaporation(Z; Day; EvaV alue); V alue := EvaV alue ¤ 0:75 ¤ 0:5: (10)

In (10) there is a multiplication of rational numbers, since 0:75 ¢ 0:5 = 0:375, it is possible to handle rational numbers knowing the range value of evaporation for every day in a similar way that in rule (9) and mapping each rational number to an integer proportional value. Potential evapotranspiration for a year is de¯ned in terms of potential evapotranspiration of a day by a statistical function.

For the sentences [ 15 ][ 14 ]: { If the slope is less than 15 percent to prevent soil erosion and degradation of natural resources in general. { If there are no droughts lasting over 45 days. { If the probability of occurrence of a drought is not greater than 66 percent. the following rules are generated: soilErosion(Z) Ã zone(Z)

terrainSlopePercent(Z; S); S > 15: naturalResourcesDegradation(Z) Ã zone(Z)

terrainSlopePercent(Z; S); S > 15: droughtChance(Z) Ã zone(Z) longerDroughtLengthDays(Z; S); L > 45 droughtProbabilityPercent(Z; E); E > 66: (11) (12)

Rules in (11) are de¯ned in terms of terrain slope, however this de¯nition can

be extended with more predicates related with this de¯nition. In (12) de¯nition of longerDroughtLengthDays can be declared in a similar way than in rule (6).

The predicate droughtProbabilityPercent can be de¯ned as the top value of

the range probability for the number of years a drought may occur for a given zone

Z. For example:

droughtProbabilityPercent(Z; 45) Ã zone(Z); numberOfYearsDroughtMayOccur(Z; 4): (13)

Modeling of maizetech ¯t zone problem using Answer Set In [ 15 ][ 14 ] once is determined for a given geographical zone the potential of production in order to cultivate native maize seeds, given some diagnosis data by the user (culture cycle, moisture regime, altitude, rainfed quality, pending, organic matter, soil texture), the system computes what seeds are more appropriate to plant in the geographical zone. Since in [ 15 ][ 14 ] is not clearly stated how this part is modeled, we evaluated of possible chances from the system by manual inspection and using data mining techniques (C4.5 algorithm) a decision tree, from this tree we can build rules as the following one: seed(v537c) £ seed(v538c) £ seed(hv521c) Ã locality(L); organicMatter(L; OrgMatP ercent); OrgMatP ercent > 3; pending(L; P enP ercent); P enP ercent < 5; rainfedQuality(L; highRiskOfDrought); altitude(L; AltV alue); AltV alue < 1200; cultureCycle(L; springSummer): (14)

Note that there is a preference depending of the class of maize seed.

5

Conclusions

In this work we provide ASP modeling for AgriFitZone problem. We propose to use an ASP approach using preferences. Once necessary and su±cient conditions have been determined to give solution for AgriFitZone, it is possible to provide assistance to farmers in making decisions to promote planting and conservation of maize through the implementation of the mathematical model (in this case using ASP) according to the classes of producers, environmental conditions and geographical location. In future work we plan to research about planning on agricultural cultivation of maize using possibilistic logic programs.

References

1. Chitta Baral . Knowledge Representation, reasoning and declarative problem solving with Answer Sets . Cambridge University Press, Cambridge, 2003 .

2. Elizabeth M. Boll and Daniel C. St. Clair . Analysis of rule sets generated by the cn2, id3, and multiple convergence symbolic learning methods . In CSC '95: Proceedings of the 1995 ACM 23rd annual conference on Computer science , pages 48 { 55 , New York, NY, USA, 1995 . ACM.

3. Gerhard Brewka , Ilkka NiemelÄa, and Tommi SyrjÄanen. Logic Programs with Ordered Disjunction . Computational Intelligence , 20 ( 2 ): 333 { 357 , 2004 .

4. Gerhard Brewka , Ilkka NiemelÄa, and MirosÃlaw Truszczyn¶ski. Preferences and Nonmonotonic Reasoning . AI Magazine , 29 ( 4 ): 69 { 78 , 2008 .

5. Pedro Reyes C. El Maiz y su cultivo . AGT Editor, S. A., Mexico , D.F. , ¯rst edition , 1990 .

6. B. A. Davey and H. A. Priestly . Introduction to Lattices and Order. Cambridge University Press, second edition, 2002 .

7. James Delgrande , Torsten Schaub, Hans Tompits, and Kewen Wang . A classi¯cation and Survey of Preference Handling Approaches in Nonmonotonic Reasoning . Computational Intelligence , 20 ( 2 ): 308 { 334 , 2004 .

8. Hilario Garc ¶³a- Nieto , Oscar J . Garc¶ ³a-Silva, Ceferino Ortiz-Trejo, Rom¶an de Jesu¶s Barajas, and Germ¶an S¶anchez- Altamirano . Sistema de Consulta del Potencial Agroproductivo del Estado de Puebla , 2002 . December, 2002 . INIFAP and FUPPUE and SDR .

9. Michael Gelfond and Nicola Leone . Logic programming and knowledge representation - the a-prolog perspective . Arti¯cial Intelligence 138 ( 1-2 ), pages 3 { 38 , 2002 .

10. Michael Gelfond and Vladimir Lifschitz . The stable model semantics for logic programming . In R. Kowalski and K. Bowen, editors, 5th Conference on Logic Programming , pages 1070 { 1080 . MIT Press, 1988 .

11. Mark Hall, Eibe Frank, Geo®rey Holmes, Bernhard Pfahringer, Peter Reutemann , and Ian H. Witten . The weka data mining software: an update . SIGKDD Explor . Newsl., 11 ( 1 ): 10 { 18 , 2009 .

12. Nicola Leone , Thomas Eiter, Cristinel Mateis, Gerald Pfeifer, Georg Gottlob, Simona Perri, Wolfgang Faber, Christoph Koch, and Francesco Scarcello . The dlv system for knowledge representation and reasoning, 2002 . http://www.dbai.tuwien.ac.at/proj/ dlv/.

13. John W. Lloyd. Foundations of logic programming . Springer-Verlag New York, 1987 .

14. Walter Lopez-Baez and et al. Zonas potenciales y recomendaciones tecnicas para producir maiz en chiapas, mexico . Technical report , INIFAP Chiapas, Mexico, March 2007 .

15. Walter Lopez-Baez, Aurelio Lopez-Luna, Bulmaro Coutin~o- Estrada , Robertony CamasGomez, Bernardo Villar-Sanchez , Jaime Lopez-Martinez, Victor Serrano-Altamirano, Ausencio Zamarripa-Moran, Pedro Cadena-In~iguez , and Carlos Sandoval-Morado. Sistema de Zonas Potenciales y Recomendaciones tecnicas para la Produccion Sustentable de Maiz en el estado de Chiapas, Mexico, 2007 . INIFAP and Fundacion Produce Chiapas , A. C.

16. Luis A. Montiel , Claudia Zepeda, Pedro Wesche, and Javier A. Cervantes . Assistance in making decisions to promote planting and conservation of maize in the state of puebla of mexico through asp . Technical report , Universidad de las Americas Puebla, August 2010 .

17. J. Ross Quinlan . C4 . 5: Programs for Machine Learning . Morgan Kaufmann, San Mateo, CA, 1993 .

18. Gerardo Reyes , Jimena Guerra, and Gilberto Calder¶on. Condiciones de cultivo del ma¶ ³z criollo en comunidades de puebla, tlaxcala e hidalgo: Un an¶alisis de las economias de autoconsumo . In BUAP, editor, Aportes, volume 10 , pages 63 { 82 . Benemerita Universidad Auto¶noma de Puebla, Puebla, Mexico, ¯ rst edition , May 2005 .

19. Tommi Syrjnen and Ilkka Niemel . The smodels system , 2001 . http://www.tcs.hut. fi/Software/smodels/.

20. D. Vega-Valdivida and P. Ram ¶³rez-Moreno. Situaci¶on y perspectivas del ma¶³z en m¶exico . Research report , Universidad de Chapingo, March 2004 .

21. Claudia Zepeda , Mauricio Osorio, Juan Carlos Nieves, Christine Solnon, and David Sol . Applications of Preferences using Answer Set Programming . In Proc. of the 3rd Intl. Workshop in Advances in Theory and Implementation , volume 142 of CEUR Workshop Proceedings. CEUR-WS.org , 2005 .