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    Fluent Calculus: Fundamentals and Applications
    Fluent Calculus: Fundamentals and Applications
    Fluent Calculus: Fundamentals and Applications
    Ebook189 pages2 hoursArtificial Intelligence

    Fluent Calculus: Fundamentals and Applications

    By Fouad Sabry

    ()

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    About this ebook

    What Is Fluent Calculus


    A formalization for expressing dynamical domains in first-order logic, the fluent calculus is a type of calculus. It is a subtype of the scenario calculus, with the primary distinction being that states rather than circumstances are considered to be representations. In order to concatenate the phrases that describe the facts that are true in a circumstance, a symbol for a binary function is utilized. A good illustration of this would be the fact that the formula represents the fact that the box is now located on the table. The solution to the frame problem is to make the assertion that the state of affairs after the completion of an action is the same as it was before, with the exception of the conditions that were altered as a result of the activity. For instance, the action of transferring the box from where it was sitting on the table to where it was standing on the floor can be formalized as follows:


    How You Will Benefit


    (I) Insights, and validations about the following topics:


    Chapter 1: Fluent Calculus


    Chapter 2: First-Order Logic


    Chapter 3: Propositional Calculus


    Chapter 4: Sequent Calculus


    Chapter 5: Kripke Semantics


    Chapter 6: Situation Calculus


    Chapter 7: Fluent (Artificial Intelligence)


    Chapter 8: Event Calculus


    Chapter 9: Epistemic Modal Logic


    Chapter 10: Non-classical logic


    (II) Answering the public top questions about fluent calculus.


    (III) Real world examples for the usage of fluent calculus in many fields.


    (IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of fluent calculus' technologies.


    Who This Book Is For


    Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of fluent calculus.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateJun 30, 2023
    Fluent Calculus: Fundamentals and Applications

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      Book preview

      Fluent Calculus - Fouad Sabry

      Chapter 1: Fluent calculus

      For the purpose of representing dynamical domains in first-order logic, the fluent calculus is a formalism.

      This scenario calculus variation is what it is; The primary distinction is that situations are viewed as models of states.

      A binary function symbol \circ is used to concatenate the terms that represent facts that hold in a situation.

      For example, that the box is on the table in the situation s is represented by the formula {\displaystyle \exists t.s=on(box,table)\circ t} .

      By declaring that the situation following the execution of an action is similar to the one prior to the action, but for the conditions modified by the action, the frame problem is resolved.

      For example, Moving the box from the table to the floor is officially known as:

      {\displaystyle State(Do(move(box,table,floor),s))\circ on(box,table)=State(s)\circ on(box,floor)}

      This formula states that the state after the move is added the term {\displaystyle on(box,floor)} and removed the term {\displaystyle on(box,table)} .

      Axioms specifying that \circ is commutative and non-idempotent are necessary for such axioms to work.

      {End Chapter 1}

      Chapter 2: First-order logic

      First-order logic is a set of formal systems that are used in the fields of mathematics, philosophy, linguistics, and computer science. Other names for first-order logic include predicate logic, quantificational logic, and first-order predicate calculus. In first-order logic, quantified variables take precedence over non-logical objects, and the use of sentences that contain variables is permitted. As a result, rather than making assertions such as Socrates is a man, one can make expressions of the form there exists x such that x is Socrates and x is a man, where there exists is a quantifier and x is a variable. In this way, propositional logic may be thought of as the basis for first-order logic.

      A theory pertaining to a subject, such as set theory, a theory for groups, or a formal theory of arithmetic, is typically a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms that are believed to hold about them. Sometimes, in a more academic meaning, theory is believed to refer to nothing more than a collection of words in first-order logic.

      First-order logic may be differentiated from higher-order logic by the use of the word first-order., which involve predicates that take other predicates or functions as input, quantification may be performed over predicates or functions alternatively, or both, are permitted.: 56 In first-order theories, sets and predicates are often seen working together.

      In higher-order theories with their interpretations, It is possible to see predicates as collections of collections.

      There are a lot of different deductive systems for first-order logic, and most of them are both valid and sound, all verifiable propositions are true in all models) and full (i.e.

      It is possible to prove any proposition if it holds true across all models.

      Despite the fact that the link between logical consequences is merely semidecidable, In first-order logic, automated theorem proving has seen significant advancements in recent years.

      Additionally, first-order logic complies with a number of metalogical theorems, which enables it to be examined via the lens of proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

      The study of first-order logic, which serves as the industry standard for the formalization of mathematics into axioms, is included in the foundations of mathematical research. The axiomatization of number theory into first-order logic is known as Peano arithmetic, while the axiomatization of set theory into first-order logic is known as Zermelo–Fraenkel set theory. A structure having an infinite domain, such as the natural numbers or the real line, cannot be adequately described by any first-order theory since such a theory lacks the power necessary to do so. Stronger logics, such as second-order logic, make it possible to generate axiom systems that categorically describe both of these structures in their entirety. These are known as categorical axiom systems.

      Gottlob Frege and Charles Sanders Peirce both made significant contributions to the development of first-order logic separately.

      For more information on the development of first-order logic and the reasons it came to predominate formal logic, go here, see José Ferreirós (2001).

      First-order logic, as contrast to propositional logic, is concerned with more straightforward declarative statements, although it also addresses predicates and quantification.

      A predicate is a statement that evaluates to either true or false based on one or more things in the domain of discourse. Think about the fact that both Socrates is a philosopher and Plato is a philosopher are true. These statements are thought to have no connection to one another in propositional logic, and as such, they might be represented by variables such as p and q, for instance. Each of these phrases have the basic structure of a is a philosopher, which means that the predicate is a philosopher appears in both of them. In the first phrase, the variable an is represented by the name Socrates, while in the second sentence, the variable an is represented by the name Plato. In this particular example, the employment of predicates, such as is a philosopher, is permissible within the framework of first-order logic, while propositional logic does not.

      Logical connectives allow one to express the nature of the relationships that exist between predicates. Take for instance the first-order rule that states, if an is a philosopher, then an is a scholar. This formulation is an example of a conditional statement, with the hypothesis being that a is a philosopher, and the conclusion being that a is a scholar. The validity of this equation is contingent not only on the nature of the thing signified by a, but also on how one understands the meanings of the predicates is a philosopher and is a scholar..

      In a formula, quantifiers may be used in conjunction with variables. The value of the variable an in the above formula may be quantified globally by using, for example, the first-order phrase For any a, if an is a philosopher, then an is a scholar. The use of the universal quantifier for every in this statement conveys the concept that the assertion if an is a philosopher, then an is a scholar is true for any and all possible combinations of the letter a.

      The logically equivalent opposite of the statement For any a, if an is a philosopher, then an is a scholar, which is the phrase There exists a such that an is a philosopher and an is not a scholar, is the statement There exists a such that an is not a scholar. The concept that the statement a is a philosopher and an is not a scholar is true for some choice of a is what the existential quantifier there exists is trying to communicate to us.

      Both is a philosopher and is a scholar are examples of predicates that accept a single variable as an argument. In most cases, predicates are able to accept a number of variables. The predicate is the teacher of in the first-order phrase Socrates is the teacher of Plato has two variables that it is referring to.

      An interpretation of a first-order formula, also known as a model, clarifies the meaning of each predicate and identifies the entities that are capable of instantiating the variables. The domain of discourse, often known as the universe, is typically expected to be a set that is not empty. These things make up the universe. For instance, in an interpretation in which the domain of discourse consists of all human beings and the predicate is a philosopher is understood as was the author of the Republic, the sentence There exists such a person that such a person is a philosopher is considered to be true, as Plato was the one who witnessed it.

      In first-order logic, there are two fundamental components. In first-order logic, well-formed expressions are determined to be finite sequences of symbols according to the syntax, while the meanings of these expressions are determined according to the semantics of the language.

      The language of first-order logic, in contrast to natural languages such as English, is entirely formal. As a result, it is possible to assess, in a mechanized manner, whether or not a particular sentence is correctly constructed. Well-formed expressions may be broken down into two primary categories: words, which intuitively describe things, and formulae, which intuitively convey propositions that can be either true or untrue. In first-order logic, the words and formulae are represented by strings of symbols, and the alphabet of the language is formed by the symbols themselves in their whole. Formal logic cannot be applied to the nature of the symbols themselves, as is the case with all formal languages; yet, most people consider them to be nothing more than letters and punctuation marks.

      It is usual practice to separate the letters of the alphabet into their corresponding logical symbols, They are usually understood to imply the same thing, in addition to illogical symbols, Its meaning is dependent on how it is interpreted.

      For example, the logical symbol \land always represents and; It is never used to mean or in any context, which is represented by the logical symbol \lor .

      However, One possible interpretation of a non-logical predicate symbol such as Phil(x) is that it means x is a philosopher., the person x is known as Philip, or any other unary predicate, depending on the interpretation that is being applied at the time.

      Logical symbols are a collection of characters that may look different depending on who creates them, but they often consist of the following::

      Quantifier symbols: ∀ for universal quantification, and ∃ for existential quantification

      Logical connectives: ∧ for conjunction, ∨ for disjunction, → for implication, ↔ for biconditional, ¬ for negation.

      Some authors use Cpq instead of → and Epq instead of ↔, especially in contexts where → is used for other purposes.

      Moreover, the horseshoe ⊃ may replace →; the triple-bar ≡ may replace ↔; a tilde (~), Np, or Fp may replace ¬; a double bar \| , + , {\displaystyle \bigwedge pq} , or Apq may replace ∨; as well as an ampersand, Kpq, or the middle dot ⋅ may replace ∧, particularly in the event that certain symbols cannot be used due to technical constraints.

      (Referring back to the previously described symbols Cpq, Epq, Np, Apq, and Kpq are the symbols that are used in Polish notation.)

      Various punctuation marks such as enclosing parentheses and brackets. The selection of such symbols is contingent on the surrounding environment.

      A never-ending list of possible outcomes, a symbol that is often represented by the lowercase letters that come at the very end of the alphabet x, y, z, ..

      .

      Subscripts are often used to distinguish variables: x0, x1, x2,...

      An equal sign (also known as an equality symbol), identity symbol) = (see § Equality and its axioms below).

      In first-order logic, you do not need

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