Logical Modelling of Microworld

Date: Nov 8, 2018

Introduction

It is hard to argue that artificial intelligence is far from human cognitive efficiency. However, it is still working in accordance with the rules of human cognition. To be more exact, artificial intelligence follows the same algorithms as human-beings do. What is more, the world around is still of the same nature as it influences humans. Taking this into consideration, it is to be said that artificial intelligence has to obtain at least the basics of human cognition, which does not depend on subjective perception of the environment. Therefore, such basic principles clearly represent rules of logic because logic is a strictly objective phenomenon of human cognition. As a result, following standard logic models is a feasible task for artificial intelligence. In consequence, it is needless to say that operation with the basic logical rules gives an opportunity to simplify and modify the language of programming by applying logic models to a particular algorithm. Thus, it is necessary to describe various aspects of logic in order to identify its ability to be applicable to artificial intelligence.

For starters, it is necessary to construct a semantic model, which consists of ten elements of a chosen universe. Thus, it should be mentioned that the World of Music is chosen. In fact, this model is a basis for the rest of the paper because it presents minimal constituencies of the scope of logic. Then, modelling language is supposed to be specified with a semantic vocabulary. This vocabulary will include five names of individual objects, three unary predicates and two binary predicates, and one attribute function. Needless to say that semantic interpretation should be also provided in order to contextualise vocabulary in correlation with the semantic model. Further, a logical theory will be built. It will include eight axioms, five of them will be factual, and three will be abstract so that they imply variables and quantitative modifiers. As a consequence, satisfiability of the theory should be demonstrated. It will be proved by evidence of three sentences, two of them will be factual, and one will be abstract. Finally, the theory has to be converted into Horn theory, and use of resolution for deductive inference will be demonstrated in order to show the explicitness of the semantic model.

Semantic Model

To begin with, a semantic model of a selected microworld should be developed. In general, any microworld is comprised of certain semantic elements, which form a certain scope. However, it is necessary to derive a scope from a semantic model. In fact, semantic elements of a particular scope are mutually related. These relations are known as a semantic field. Therefore, a semantic model describes both a scope and a mechanism of a semantic field. In other words, a semantic model is a broader phenomenon: it includes a scope and a semantic field. Thus, semantic elements should be enumerated, and their mutual relations have to be described.

As the selected microworld was chosen to be the World of Music, the first element is musical notes. It is a basic element, which actually underpins other semantic elements in the scope. Hence, the following element is a musical instrument. Needless to say, that notes are supposed to be played by an object, which is able to produce musical sounds. In consequence, someone has to produce these sounds with a particular instrument. That is why a musician should be also included. Besides that, it is worth admitting that music is not just an opposite concept to noise. To be more exact, music is a combination of certain tune and rhythm. These elements are also important because they modify the element of note. Except musical instruments, music can be played as a record. As a result, a record and a playing device are also included in the scope. What is more, if music is played, it definitely has a listener. Actually, this element is quite dubious. In case it is music, which is played by a musician, it has at least one listener. On the other hand, a recorded piece of music can have no listeners at all. Finally, it is necessary to provide two more note modifiers: loudness and tempo. As any sound, a note has a certain degree of loudness. Speaking about the tune, it is a sequence of notes; that is why tempo distinguishes the frequency of notes to be played in a particular tune.

Vocabulary

As a consequence, a certain vocabulary of a modelling language should be developed. To begin with, it is necessary to implement several individual objects. In fact, they are the key agents of the process in a sentence. Therefore, they should be very specific so that none of them has any ambiguous implications or additional functions. To begin with, the first individual object is a drummer. It is actually a specified synonym for the semantic element of a musician. The next individual object is a MP-3 player. However, it has a familiar function to a drummer, but it has different modifiers and other semantic government. In consequence, a memory stick can be also considered as an individual object because it is a physical drive, which contains music that can be easily stored and managed on an MP-3 player. Besides, a recording studio should be also implemented. In this model, it is a linking object, which provides a relation between a musician and an MP-3 player. Therefore, it can be considered as an individual object. As the MP-3 player is the only one device of playing recorded music it requires headphones so that a semantic element of a listener is included. That is why headphones are also an individual object in the modelling vocabulary.

Consequently, it is necessary to provide five predicates. Three of them should be unary and two of them should be binary. The first unary predicate is “plays”. It can be used for two objects: the drummer and the MP-3 player. The next is “hears”. It can be referred to the drummer. Furthermore, a unary predicate “contains” is able to be applied to the memory stick. Speaking about binary predicates, they can repeat some of unary ones but with regard to time correlation. Hence, a binary predicate “is playing” also refers to the drummer and MP-3 player, but it implies a process at the current moment. The same rule is applied to the unary predicate “hears”. In a binary form, it can be presented with a correlation of a finished process: “has heard”. Finally, a function attribute should be included, as well. In fact, it modifies an individual model by giving a specific quality to it. Thus, an attribute “famous” emphasizes that the drummer, recording studio, memory stick, or MP-3 player are of a special quality.

Logical Theory

Furthermore, it is necessary to develop a logical theory of the microworld. This theory should include eight axioms, five of them are factual, and three are abstract. Abstract axioms presuppose some variables and quantitative modifiers. All in all, the first axiom is “A drummer plays drums”. It excludes any variables because the object is narrowly specified. The next axiom is “A drummer is a musician”. In fact, it is a generalisation of the previous axiom. As the semantic model suggests, a person, who plays a musical instrument, can be considered a musician. Thus, as the drummer plays drums, the drummer is a musician. The next axiom is “a MP-3 player plays MP-3 music only”. Actually, the individual object also excludes any variables. Then, “A recording studio does not play drums”. A recording studio is an individual object, which is also specific that is why it can be referred to only one process. Hence, the process of playing drums is already conducted by the drummer. The last axiom is “a MP-3 player is a device for playing recorded music”. In fact, it is a specification, which is based on the semantic element of the playing device.

Speaking about abstract axioms, the first is “a memory stick contains MP-3 music”. However, the content of the memory stick may vary. In such a way, the memory stick can contain some videos or images. Another abstract axiom, “a memory stick can contain one hundred songs” is evidence of quantitative variation. To begin with, a memory stick can have different memory space, and songs can be of different size so “one hundred songs” is one of the quantitative aspects. Finally, “a recording studio records MP-3 music” is also an abstract axiom because its function may vary, as well. In other words, it can record any other formats of music or just a speech, for example.

Satisfiability

Speaking about satisfiability of axioms in a semantic model, it is necessary to say that two sentences demonstrate factual axioms, and one sentence demonstrates an abstract axiom. Actually, satisfiability means that a certain axiom can have at least one interpretation so that another model can follow this formula and get a relevant outcome. For starters, satisfiability of an axiom with variables should be described. Thus, a sentence “a memory stick contains MP-3 music” is satisfiable because it is possible to convert this formula into another non-related axiom: “a fridge contains food”. Hence, a fridge may contain drinks or any other objects so food is a variable. Taking into consideration factual axioms, it should be said that interpretation has to be expressed through the similarity of modelling vocabulary. As a result, “a drummer plays drums” can be interpreted as “a golfer plays golf”. In other words, evidence of the same semantic relations should be observed in other models. What is more, interpretation of the axiom can be based on objection of a certain function. As a consequence, “a recording studio does not play drums” can be interpreted as “a fridge does not iron the clothes”. In general, interpretation can be based on both similarity or possibility and denial, as well.

Inferences

As a semantic model belongs to a particular semantic field and certain relations, it can be applied to basic rules of inference. In fact, these rules primarily reflect relations while elements of a semantic field are just premise representatives. To begin with, it is necessary to demonstrate the most basic inference rule, which is called “modus ponens”. In other words, this rule means that an individual object produces a certain result, and this result can be produced only by this object. In other words, “The drummer plays the drums” automatically means that “The drums are played by the drummer”. On the contrary, inference rule can depict an objection of a particular conclusion. To be more exact, a certain individual object does not produce a particular result. However, it is worth mentioning that this result does not modify the object because some other individual objects are not able to produce this result. Taking this into account, such inference rule states that “a MP-3 player does not iron clothes”. As a consequence, a lot of other individual objects are known not to be able to iron clothes. The next inference rule is more typical of axioms with variables because it has a linear structure. An individual object produces a certain result, but this result may vary. In consequence, variation of the result depends on the direct object of the sentence. These sentences are usually more complicated because the result produces its own dependable result. In such a way, “a memory stick contains MP-3 files that`s why it contains MP-3 music”. As this axiom implies variables, it is possible to say “a memory stick contains AVI files that`s why it contains videos”. Doubtless, the structure remains the same while variables can be substituted. All in all, derivation of conclusions is obviously seen by application of the basic rules of inference. In fact, it witnesses the presence of the explicit side of the semantic model. However, the theory is supposed to be converted into a Horn theory in order to reveal its other properties.

Horn Theory

To begin with, it is necessary to say that Horn theory is a restriction of first-order logic. To be more exact, Horn theory transforms any model in the simplest implicative formula. In other words, a certain result becomes a core basis of a sentence instead of an individual object. As a consequence, a result modifies an individual object. In such a way, a theory could be enlarged by identifying some extra individual objects or at least their variables. Taking this into consideration, it is necessary to describe the model in accordance with Horn theory.

As a drummer plays drums, an atom of this sentence is “plays drums”. Thus, according to Horn theory, “Playing drums is conducted by the drummer”. The next sentence is “A drummer is a musician”. Therefore, “Musician is a drummer”. However, it is to be mentioned that this sentence is also converted into an abstract theory because it becomes able to have variables. In general, it is possible to state, for example that “Musician is a pianist”. The same peculiarity of converting is also evident in the next sentence. “A MP-3 player plays MP-3 music” corresponds to a Horn sentence “MP-3 music is played by a MP-3 player”. Though, MP-3 music can be played by some other device, as well. The next sentence is peculiar because it contains a denial of the result. Thus, according to Horn theory, a “denied atom” has a huge number of variables, because denial can be modified in a lot of ways. In general, each way corresponds to a particular variable. In such a way, “a recording studio does not play drums” in accordance with Horn theory can be stated as “Not playing drums is done by a recording studio”. As a consequence, “not playing drums” can be conducted by any other agent, except a certain group.

Speaking about abstract theories, it is to be admitted that Horn theory converts predicates of a first-order logic sentences into a variable, as well. What is more, it should be also mentioned that these variables are not directly related to the standard variables. Taking this into consideration, “a memory stick contains MP-3 music” can be converted into “MP-3 music is downloaded by a laptop”. In such a way, an individual object becomes an atom, but conclusions may vary in many different ways. On the contrary, quantitative atoms do not influence variables. Therefore, “a memory stick contains one hundred songs” can be converted into “One hundred songs are downloaded by a laptop” so that quantity does not have a primary influence. Besides that, as it is the atom of Horn sentences, it cannot be regarded as a quantitative premise any longer. All in all, Horn theory implies an inversion of the government in a sentence so that the explicitness of the model is presented as a sign of ability to apply another theory.

Resolutions

Speaking about resolutions of converted theories, it is necessary to mention that these resolutions have to be based on a semantic field of the model because it does not presuppose any extra vocabulary elements. In other words, resolutions are based on inference, which is comprised only of the outlined elements of a semantic field. Hence, it means that variables are limited by the inference because not all variables belong to the semantic field. In such a way, “The drums are played by the drummer” can have the following resolution. As Horn theory restricts logic of the first order, a conclusion is in the initial position in the sentence so that “playing drums” is modified by “the drummer”. In general, A is conducted by B if B modifies C. On the contrary, “drums are played with a MP-3 player” is also satisifiable because the MP-3 player plays music and music certainly has a rhythm, except for some odd examples. In consequence, it should be added that a resolution can be presented as “A is conducted by B if B modifies C, and A may be conducted by D if D equals to D and is able to modify C”. A more complicated resolution can be used in the sentence “MP-3 music is contained in a memory stick”. As this sentence has two variables, the resolution can be described as “A is conducted by B if B corresponds to C”. However, it is also possible to state that “A is conducted by D if it is equal to B and is able to correspond to C. As a consequence, E can be equal to C if it is able to correspond to D”. Finally, it is to be said that variables are the key factor of application of a particular resolution.

Conclusion

To sum up, it is necessary to say that the basic rules of logic have been reviewed in accordance with computer sciences and artificial intelligence. To be more exact, the semantic model of a microworld was developed concerning the chosen universe of music. As a result, an appropriate vocabulary was developed in order to contextualise the language of the semantic model. The specific vocabulary consists of five names of individual objects, three unary predicates, two binary predicates, and one attribute function. In such a way, the basis of the developed theory was presented. In consequence, the relevant theory was developed. It demonstrates eight axioms; five of them are factual, and three are abstract. What is more, a satisfiability of the theory was also proved. Satisfiability is the ability of a theory to be interpreted. Taking this into consideration, several examples have been given in order to describe particular interpretations. In order to show different levels of explicitness of the semantic model, the use of rules of inference was demonstrated. In addition, the developed theory has been converted into a Horn theory. Besides that, several conclusions of a deductive inference were demonstrated.

All in all, logic is a constituent part of dealing with artificial intelligence. It helps to understand and enlarge a programming language and simplify it to the level of linear algorithms. Taking this into consideration, it is possible to say that the basic rules of logic can be applied to artificial intelligence as a tool of interpretation and contextualising of particular scopes or concepts. Finally, logic is suggested to draw a link between artificial intelligence and semiotics. It is a science, which studies signs; that is why logic is potentially able to link internal architecture of artificial intelligence to its external representation.

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