DE102019005706A1 - Mechanically implemented procedure for the expansion of sets of representations of boundary conditions for the solution of technical, scientific or economic tasks - Google Patents

Abstract

The invention relates to the task of using a computer program to find solutions for tasks in which formally described boundary conditions must be observed. The invention achieves its purpose by a novel combination of elements of known methods of genetic programming and known methods for solving formal tasks by means of inference machines and / or computer algebra systems. Here, genetic algorithms are used to generate variants of the parameters to complete formula schemes, which are transferred to the computer algebra system or the inference machine. The computer algebra system or the inference machine and the system for generating parameter variants are embedded in an overall solution. In this, the results of the inference machine or the algebra system are used in the sense of a fitness function for the genetic algorithm, which generates the complete formulas from the schemes.

Description

The invention relates to the task of using a computer program to find solutions for tasks in which formally described boundary conditions must be observed. The invention achieves its purpose through a novel combination of implementations of a variant of genetic programming (GP) methods known per se and methods known per se for solving formal tasks using inference machines (IM) and / or computer algebra systems (CAS).

Abbreviations, relevant non-patent literature

GP stands for Genetic Programming in the sense of John R. Koza Genetic Programming. On the Programming of Computers by Means of Natural Selection. ISBN: 9780262527910

• Sterling, Leon; Ehud Shapiro. The Art of Prolog. ISBN 0-262-19250-0.

oder verwandter Systeme.IM stands for inference machine in the sense of:

• Sterling, Leon; Ehud Shapiro. The Art of Prolog. ISBN 0-262-19250-0.

or related systems.

• J. v. z. Garnen, J. Gerhard: Modern Computer Algebra. ISBN: 1107039037

CAS stands for Computer Algebra System in the sense of:

• J. vz Garnen, J. Gerhard: Modern Computer Algebra. ISBN: 1107039037

The invention makes no reference to the specific embodiments of the principles in the aforementioned works, but only to the general principles as such. Special features of the design relevant to the invention are presented below.

background

The background to this is that engineering tasks often require the search for solutions to consider boundary conditions that can be described as a result of scientific research using formal constructs. In this case, the search space, which is traversed by a machine learning process, should be restricted by the representation of the formal boundary conditions.

Structure of the invention

The implementation of a suitable method according to the invention uses a programmable computer of known type. Part of the implementation is the implementation of an IM of a known type, for example Amzi! -Logic Server, and / or a CAS of a known type, for example Wolfram Mathematica. Furthermore, the implementation includes that of a genetic algorithm of a known type.

Genotypes in the sense of the genetic algorithm are interpreted as instantiations of formula schemes using a projection function. It is characteristic of the invention that the formulas that are formed as an interpretation of the genotypes are a subset of the inputs to the CAS or the inference machine, whereby they supplement those formulas which represent either general principles or the special boundary conditions of test cases. Another characteristic of the invention is that the fitness landscape of the genetic algorithm is defined by evaluating the calls to the CAS or the IM.

The generation of inputs to the CAS or IM by the genetic algorithm is fundamentally an embodiment of the GP, in which formula sets with possibly one or more elements are generated in a syntax accepted by CAS / IM. Another special feature of the design is that the genotypes are projected into a space of representations of formula schemes, which can be restricted compared to the set of well-formed formulas in the sense of the CAS or the IM. Furthermore, this projection, together with the rules of mutation and (optional) recombination, defines the paths via which genotypes of other types can be reached. This embodiment of the known approach of the GP is part of the invention but does not in itself characterize the invention.

Characteristic of the invention is primarily that the result of the GP is used as a representation of a boundary condition, which, together with representations of general boundary conditions given by natural laws and other boundary conditions describing special tasks, are used by a CAS or an IM to generate extended sets of representations for the solution used in a task. The results of these calls define the fitness landscape for the GP. The basic idea is to use the GP to search for representations of boundary conditions which are consistent with all or as many of the boundary conditions of the test cases on the one hand and which, combined with the respective boundary conditions of the test cases, enable the CAS (the IM) to find additional boundary conditions that sensibly restrict the solution space, which, however, would not be found from the boundary conditions of the test case alone - without the GP-generated extensions - with the resources of the CAS (the IM). In comparison to the known GP, an additional level of indirection and abstraction is introduced, compared to the known CAS / IM a heuristic component.

Application examples (i)

• Gegeben ist eine Balkenwaage, deren einer Balken B1 die Länge 100mm hat während der andere Balken B2 zwischen 100mm und 200mm verstellt werden kann, wobei die Einstellung jeweils Teil der Definition einer konkreten Aufgabe ist.

An application example (il) illustrates the nature of the task and its solution by the method according to the invention as follows:

• A beam balance is given, one beam B1 of which is 100mm long while the other beam B2 can be adjusted between 100mm and 200mm, the setting being part of the definition of a specific task.

There is also a container whose weight can be varied continuously between 10g and 20g by filling it with a liquid. Finally, several lead weights are available, both a ball and a cube with 7g, 8g, 9g, ... 50g. Each task consists in finding a combination of the choice of a lead weight with a filling of the container for a given length of the bar B2, which brings the balance into equilibrium when the lead weight on the weighing pan is below bar B2 and the container is below the weighing pan the bar B 1 lies.

• Die Genotypen werden in die Menge der Formeln der Gestalt [T];x=L; projiziert, wobei [T] der Platzhalter für Terme ist, die als Variablensymbole alle Sequenzen von Kleinbuchstaben haben dürfen und L für den Term steht, der in jeder Aufgabe an die Balkenlänge gebunden wird. Also sind beispielshalber

$$\text{y=4;x=L}$$

$$\text{y=6;x=L;}$$

$$\text{y=10;x=L;}$$

$$\text{y=10-4*INT}\left(\text{X/180}\right)\text{;x=L;}$$

mögliche Produktionen, die einem Genotyp korrespondieren. Der optimale Genotyp wird ermittelt, indem für 50 Testfälle mit Balkenlängen zwischen 100mm und 200mm jeweils die Produktion des Genotyps mit einer Gleichung der Gestalt L=[n] für einen Wert von n zwischen 100 und 200, sowie der Gleichung z=y*(L/100) und den Ungleichungen z≧0; x≦20 kombiniert und nach einer Lösung gesucht wird. Die Fitness ist umso höher, je höher die Zahl der Testfälle ist, bei der dies gelingt.An implementation of the invention is structured as follows:

• The genotypes are divided into the set of formulas of the form [T]; x = L; projected, where [T] is the placeholder for terms that may have all sequences of lowercase letters as variable symbols and L stands for the term that is tied to the bar length in every exercise. So are exemplary

$$\text{y = 4; x = L}$$

$$\text{y = 6; x = L;}$$

$$\text{y = 10; x = L;}$$

$$\text{y = 10-4 * INT}\left(\text{X / 180}\right)\text{; x = L;}$$

possible productions that correspond to a genotype. The optimal genotype is determined by producing the genotype for 50 test cases with bar lengths between 100mm and 200mm using an equation of the form L = [n] for a value of n between 100 and 200, as well as the equation z = y * (L / 100) and the inequalities z ≧ 0; x ≦ 20 is combined and a solution is sought. The higher the number of test cases in which this succeeds, the higher the fitness.

For production (i) it is never possible to solve the problem for a test case, since for y = 4 every equation of the form z = y * (L / 100) with a value for L between 100 and 200 contradicts z ≧ 10 results. The production (ii) is valid for beam lengths from (1000/6) mm. On the other hand, (iii) and (iv) correspond to rules for the choice of the weight on bar B2, for which the task can always be fulfilled: In (iii) the weight of the container must be selected between 10g and 20g. With rule (iv) the weight for a length of the bar B2 between 100 and 180-ε between 10g and 18g must be selected, from a length of 180mm between (180/100) * 6 = 10.8g and (200/100) * 6 = 12g.

The application of the method according to the invention can therefore be used to generate one of the representations of boundary conditions x = 10; y = L; or x = 10-4 * INT (x / 180); x = L; lead (or any other equally good solution). The generated boundary condition is then used by the CAS after the learning phase has been completed to perform the real tasks together with the representation of the more general conditions - z = y * (L / 100); z ≧ 10; x ≦ 20; - and the respective concrete task definition - L = [n] - to be solved.

Another application example (i.2) corresponds to the previous one, with the exception that the bar length B2 can now vary between 100mm and 300mm. The genotype encoding formula set (iii) no longer describes a general solution, since a bar with a length> 200m would require a container weight of more than 20g. On the other hand, (iv) still encodes a generally applicable basis for a solution, since the choice of a weight of 6 from a length of 180mm enables balancing with a counterweight between 10.8g and 18g.

Regardless of the triviality of the application examples, these illustrate the background of the invention and the way in which it achieves its purpose: Often there are some boundary conditions for the solution of tasks from natural laws, which are described by theories of natural science (the ratio of lever length, weight and Force in the example). Further boundary conditions for a family of tasks are explicitly known (the selection of possible weights and the possible fillings of the container in the example). However, there is no practicable method to algorithmically generate a solution from these boundary conditions. It is possible to generate representations of additional boundary conditions that do not logically result from the given conditions, but can be fulfilled together with them (the choice of weight on bar B2 in the example). The thus expanded set of representations can be expanded again for each task for each task from the task family together with the concrete boundary conditions of the task (the solution of the system of equations in the example). The result is still not necessarily an unambiguous solution (in the example, the choice between the balls and cubes of the same weight remains open). However, the representation of the boundary conditions is specified in such a way that a constructive solution can be found.

The presented invention does not claim to convert a mathematical theorem into a technical solution. Although it integrates mathematical-logical concepts (IM, CAS, GP), the invention itself is an engineering solution for using a combination of implementations of these concepts for a class of task families.

Heuristic search for solutions for subclasses of the task family The procedure can be extended in an obvious way to a method of the heuristic search for solutions for parts of the task family. For this purpose, as set out in the description of the basic method, genotypes are first searched for, which generate representations of boundary conditions that can be fulfilled in as many test cases as possible together with the original boundary conditions. If no universal solutions are found, an attempt is made to identify a subset of the tasks for which a solution (a suitable genotype) exists. Then a solution is sought for the tasks that do not belong to this set. If this succeeds for the entire remaining quantity, a division into two sub-classes is successful. Otherwise, the process is repeated until either a usable classification is found or the search is terminated as unsuccessful.

If a manageably small amount of genotypes is found that are suitable for subsets of the test cases, these can be tried out as extensions of the boundary conditions until a solution is found.

An attempt can optionally be made to identify the classes on the basis of characteristics that can be determined independently of the genotypes. For this purpose, neural networks in particular can be used.

Application example (ii)

An application example (ii.1) can be constructed from application example (i.1) by introducing an additional variable for the size of the container. This example assumes that a set of tasks should be solved for a vessel with a weight between 10g and 20g, another for a vessel with a weight between 30g and 40g. It also assumes that the vessel size cannot be referenced in the formula scheme. Therefore there is no extension of the boundary conditions which can be fulfilled for both classes of tasks together with the original boundary conditions. However, there is such a solution for each of the classes, which can be found using the genetic algorithm.

A simple version of a method for solving all of the tasks in the example is to first try the extension after the first partial solution and then, in the event of failure, to try the alternative partial solution. One possible optimization is to subdivide the tasks using a method that can see the different vessel sizes and then to use the appropriate extension in each case.

Claims (6)

Machine implementation of a method for expanding sets of representations of boundary conditions for the solution of technical, scientific or economic tasks, - characterized in that the representation of boundary conditions takes place in a format which can be processed by an inference machine and / or a computer algebra system, - further characterized in that an implementation of a genetic algorithm generates codes that can be understood as genotypes in the sense of the genetic algorithm, - further characterized in that representations of boundary conditions are generated by implementing a method for projecting the space of genotypes into a space of instantiating schemes for representing boundary conditions, - further characterized in that the representations of boundary conditions generated by the genotype-defined instantiation of schemes are passed on as inputs to the implementation of a computer algebra system or to the implementation of an inference machine, together with other representations of boundary conditions functioning as test cases, - further characterized in that the fitness landscape for the genetic algorithm is defined by evaluating the results of calling up the computer algebra system and / or the inference machine for the test runs. Machine implementation of a method for expanding sets of representations of boundary conditions for the solution of technical, scientific or economic tasks according to Claim 1 - additionally characterized in that the evolution of genotypes is simulated for several sets of test cases. Machine implementation of a method for expanding sets of representations of boundary conditions for the solution of technical, scientific or economic tasks according to Claim 2 - additionally characterized in that the division of a set of test cases is carried out dynamically as a function of previously performed simulation runs. Machine implementation of a method for expanding sets of representations of boundary conditions for the solution of technical, scientific or economic tasks according to Claim 2 and optional Claim 3 - additionally characterized in that a method for classifying boundary conditions is implemented, which determines the affiliation of boundary conditions to the sub-classes according to the Claim 2 automatically decides. Machine implementation of a method for expanding sets of representations of boundary conditions for the solution of technical, scientific or economic tasks according to Claim 4 - additionally characterized in that the method for classifying boundary conditions accordingly Claim 4 is generated from the test case classes through the implementation of a method for machine learning, in particular the training of a neural network. Extension of the machine implementation of a method for the extension of sets of representations of boundary conditions for the solution of technical, scientific or economic tasks according to Claim 1 alone or Claim 1 combined with one or more of the Claims 2 - 5 - additionally characterized in that the extension of the implementation accepts the representation of boundary conditions as inputs, that it forwards them to the implementation of the computer algebra system and / or the inference machine together with at least one representation of additional boundary conditions generated by the implementation of the basic method and by calling the implementation of the computer algebra system and / or the inference machine, further representations of boundary conditions are generated.