Model Theory

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Model Theory

• By Ashirul Mubin

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What is a Model

• A Model is a set of statements (either true or
  false) about some system under study.
• A model may be descriptive (as in Physics)
• A model may be a set of specifications (as in
  Engineering)

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Modeling Descriptively

• The model representation is created from a given
  system.
• The interpretation is the description of the
  system
• Such a model is used to analyze the system by
  reasoning

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Modeling as a Specification

• The model is created to meet the intended system
• The interpretation of the specification
  determines the constraints on how a valid system
  may be constructed
• Deduce observable properties from the
  specification

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Model Interpretation

• An interpretation of a model is a mapping of
  elements of the model to elements of the system
  such that the truth value of statements in the
  model can be determined
• If this mapping can be inverted so that elements
  of the system can be mapped to model elements,
  then a model can also be constructed as
  representation of a given system.

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Model Interpretation (cont)

• A sentence S may neither express true nor false,
  because some crucial information is missing about
  what the word means.
• If this information is added so that S becomes a
  true or false statement, it is said to interpret
  S.
• If the interpretation I happens to make S state
  something true, then I is a model of S (I
  satisfies S)

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Model Interpretation (cont)

• Two aspects
• First is the relationship of the model to the
  thing being modeled.
• Second is the relationship of a given model to
  other models derivable from it.

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Theory

• A theory is a way to deduce new statements about
  system from the statements already in some model
  of the system.
• The set of all sentences of a language that are
  true in the first-order language structure is
  called the complete theory.
• The sentence S is extended to a set T of
  sentences. The set T is used to define a class of
  all interpretations that are simultaneously
  models of all sentences in T. Here T is a theory
  or a set of axioms.

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Model Theory

• Model Theory is the study of the interpretation
  of any language (formal or informal) by means of
  set-theoretic structures.
• It defines an evaluation function that determines
  whether a formula is true or false in terms of
  some model.
• Model theory is completely agnostic about what
  kinds of thing exist.

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Modeling

• Modeling a phenomenon is to construct a formal
  theory that
• describes the system
• And explains the system

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Logic

• A logic consists of mathematical objects
• Formal or informal language (part of a natural
  language)
• A deductive system (to capture, codify, or record
  which inferences are correct for the given
  language)
• Model-theoretic semantics (to capture, codify, or
  record the meanings or truth conditions)

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Argument

• Argument is a non-empty collection of formulas in
  the formal language
• One of the formulas is designated as the
  conclusion. And the other formulas in the
  argument are its premises.
• An argument is derivable if there is a deduction
  from some of its premises to its conclusion
• An argument is valid if there is no
  interpretation in which its premises are all true
  and its conclusion is false.

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Formal Language

• It is a math model of a natural language
• Natural language have underlying logical forms
  and these forms are displayed by formulas of a
  formal language
• Its deducibility and validity represent math
  models of correct reasoning in natural languages
• A formal language is a recursively defined set of
  strings on a fixed alphabet.

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Set Theory

• A set is an arbitrary collection of elements
• A set theory is used to define basic mathematical
  structures.
• Set Operators Union, Intersection, complement,
  difference, subset, proper subset, superset,
  empty set, disjoint set
• Rules idempotency, commutativity, associativity,
  distributivity, absorption, double
  complementation, DeMorgans Law

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Relations

• A relation is function of one or more arguments
  whose range is the set of truth values
  true,false
• Predicate is often used as a synonym for relation
• Types of relations reflexive, irreflexive,
  symmetric, asymmetric, antisymmetric, transitive

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Lattice

• Concept A is a subtype of B (Afor B divides the number for A (?)
• It consists of a collection of instances, types,
  formal concepts, embedding map from instances to
  concepts (iff-ont 5)
• Implementation IEEE-SUO-IFF

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Formal Concept Analysis

• Implementation IEEE-SUO-IFF
• Concept lattice is the central data structure in
  FCA

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Propositional Logic

• It deals with statements or propositions and the
  connections between them represents a complete
  statement by a single symbol
• Operators
• Conjunction
• Disjunction
• Negation
• Material implication
• Equivalence

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Propositional Logic (cont)

• Rules of inference
• Modus ponens
• Modus tollens
• Hypothetical syllogism
• Disjunctive syllogism
• Conjunction
• Addition
• Subtraction

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Propositional Logic (cont)

• Derived rules of inference
• Idempotency
• Commutativity
• Associativity
• Distributivity
• Absorption
• Double negation
• De Morgans laws

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Formal Grammars

• A formal grammar is a system for defining the
  syntax of a language by specifying the strings of
  symbols or sentences that are considered
  grammatical
• Grammatical sentences are derived using
  production rules, which are developed as a method
  of transforming strings of symbols.

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Grammar Categories

• Four categories of complexity
• Finite-state (regular) grammar
• Context-free grammar
• Context sensitive grammar
• General rewrite grammar

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Example Modeling

• A UML model makes statements that the software
  system contains certain classes related in
  certain ways.
• Additional Models could make statements on how
  instances of those classes are expected to
  interact resulting a state change over some
  period of time.
• Such models could either be describing the
  structure of an existing system or they may be
  specifying how a planned system is to be built.

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Example Model Interpretation

• The UML class model interpretation
• There are elements of the system that may be
  identified as classes
• They have relationships that may be identified
  with the relationships given in the model.