Question

Monadic predicate calculus has only unary predicate symbols (no identity). Consider \(\mathfrak{A}=\left\langle A, R_{1}, \ldots, R_{n}\right\rangle\) where all \(R_{i}\) are sets. Define \(a \sim\) \(b:=a \in R_{i} \Leftrightarrow b \in R_{i}\) for all \(i \leq n\). Show that \(\sim\) is an equivalence relation and that \(\sim\) has at most \(2^{n}\) equivalence classes. The equivalence class of \(a\) is denoted by \([a]\). Define \(B=A / \sim\) and \([a] \in S_{i} \Leftrightarrow a \in R_{i}, \mathfrak{B}=\) \(\left\langle B, S_{1}, \ldots, S_{n}\right\rangle .\) Show \(\mathfrak{A l} \models \sigma \Leftrightarrow \mathfrak{B} \models \sigma\) for all \(\sigma\) in the corresponding language. For such \(\sigma\) show \(\models \sigma \Leftrightarrow \mathfrak{A} \models \sigma\) for all \(\mathfrak{A}\) with at most \(2^{n}\) elements. Using this fact, outline a decision procedure for truth in monadic predicate calculus.

Short answer

The relation is an equivalence relation with at most \(2^n\) classes, ensuring logical equivalence between structures \(\mathfrak{A}\) and \(\mathfrak{B}\). Decision procedure entails verifying formulas against finitely enumerable structures.

Step-by-step solution

Show that \(\sim\) is Reflexive

Consider any element \(a \in A\). For reflexivity, we need to show \(a \sim a\). By definition, \(a \in R_i \Leftrightarrow a \in R_i\) for all \(i \leq n\). This is trivially true as it holds for all elements. Hence, \(\sim\) is reflexive.

Show that \(\sim\) is Symmetric

Consider any two elements \(a, b \in A\) such that \(a \sim b\). This means \(a \in R_i \Leftrightarrow b \in R_i\) for all \(i\). As this condition is bidirectional, it also implies \(b \sim a\). Therefore, \(\sim\) is symmetric.

Show that \(\sim\) is Transitive

Consider three elements \(a, b, c \in A\) such that \(a \sim b\) and \(b \sim c\). This means \(a \in R_i \Leftrightarrow b \in R_i\) and \(b \in R_i \Leftrightarrow c \in R_i\) for all \(i\). By transitivity of logical equivalence, \(a \in R_i \Leftrightarrow c \in R_i\), which implies \(a \sim c\). Therefore, \(\sim\) is transitive.

Prove \(\sim\) has at most \(2^n\) Classes

Since \(\sim\) is defined by membership in \(n\) sets \(R_i\), each element in \(A\) can be associated with a unique \(n\)-bit string representing membership (0 for not in \(R_i\), 1 for in \(R_i\)). There are \(2^n\) such distinct strings. Hence, \(\sim\) has at most \(2^n\) equivalence classes.

Show \(\mathfrak{A} \models \sigma \Leftrightarrow \mathfrak{B} \models \sigma\)

The structure \(\mathfrak{B}\) is constructed by collapsing \(\mathfrak{A}\) into equivalence classes, preserving the satisfaction of sentences in the monadic language. For any \(\sigma\), the satisfaction is dependent only on whether elements belong to certain \(R_i\). As this is preserved in \(\mathfrak{B}\), satisfaction equivalence \(\mathfrak{A} \models \sigma \Leftrightarrow \mathfrak{B} \models \sigma\) holds.

Demonstrate \(\models \sigma \Leftrightarrow \mathfrak{A} \models \sigma\) with \(|A| \leq 2^n\)

For \(\mathfrak{A}\) having at most \(2^n\) elements, model theory shows any formula \(\sigma\) in the monadic predicate calculus can only differentiate between these equivalence classes. Therefore, for any \(\mathfrak{A}\), if \(\models \sigma\), then \(\mathfrak{A} \models \sigma\), and vice versa.

Outline Decision Procedure

To decide truth in monadic predicate calculus, enumerate all possible structures of size at most \(2^n\) and verify \(\sigma\) against these structures. Since there are finitely many such structures, the decision procedure terminates.

Key concepts

Equivalence Relation

Equivalence relations are a fundamental concept in mathematics. They serve as a way to partition a set into equally sized parts, called equivalence classes. An equivalence relation must satisfy three essential properties: reflexivity, symmetry, and transitivity.
- Reflexivity requires that every element is related to itself. In other words, for any element \( a \), we have \( a \sim a \).
- Symmetry means if one element is related to another, then the second is related to the first, i.e., if \( a \sim b \), then \( b \sim a \).
- Transitivity states if an element is related to a second and the second is related to a third, then the first is related to the third, formally: if \( a \sim b \) and \( b \sim c \), then \( a \sim c \).
For the given exercise, these properties are shown by setting \( a \sim b \) if they share identical membership in the sets \( R_i \). The connection to equivalence classes comes when each \( a \) can belong to an equivalence class \([a]\), where every member shares the same characteristics defined by \( \sim \). This is how we determine that there are at most \( 2^n \) equivalence classes, as each class corresponds to a unique subset of the sets \( R_i \), represented by an \( n \)-bit string.

Decision Procedure

A decision procedure is an established method to determine the truth of a statement within a logical system. In the context of the monadic predicate calculus, a decision procedure helps us decide the truth of sentences. Given the exercise, a finite number of structures \( \leq 2^n \) shows a way to exhaustively verify all possible configurations.
The step-by-step creation includes:

• Enumerating all potential structures that a set \( A \) could form, given its size constraint from the equivalence relation.
  
• Analyzing each structure to see if it satisfies the target sentence \( \sigma \) under consideration in the monadic predicate calculus.
  
• Since this listing is finite, each possible structure can be checked, confirming or denying \( \sigma \)'s truth in a finite amount of time.
  

The practical significance of a decision procedure is that it transforms abstract logical deductions into a concrete, finite process, thus making it accessible for various applications, such as validating the truth of logic-based models.

Logical Equivalence

Logical equivalence in logic means that two or more statements are equivalent in the sense that they yield the same truth value in every model. In the exercise, we explored logical equivalence between the original structure \( \mathfrak{A} \) and its collapsed form \( \mathfrak{B} \).
- To demonstrate their equivalence, we rely on the property that collapsing into equivalence classes in \( \mathfrak{B} \) does not change the satisfaction of logical sentences \( \sigma \).
- Instead, it maintains the same truth conditions, because the necessary conditions—set memberships \(R_i\)—are preserved.
Consider a sentence \( \sigma \) of the monadic predicate calculus: if it holds in \( \mathfrak{A} \), it should also hold in \( \mathfrak{B} \), and vice versa. This logical equivalence forms the bridge that allows reduction of complex structures into simpler, equivalent forms, preserving the logic but reducing complexity.

Model Theory

Model theory deals with the relationship between formal languages (like logical formulas) and their interpretations or models. It helps us understand which types of structures can make a given set of logical statements true. In the given exercise, model theory is used to show \( \models \sigma \Leftrightarrow \mathfrak{A} \models \sigma \) when \(|A| \leq 2^n\).
- Each model \( \mathfrak{A} \) can be seen as a concrete example of the logical sentence in a particular structure.
- With at most \( 2^n \) elements, models enable us to reckon with large, abstract logical concepts by limiting them to finite, manageable structures.
A model must validate logical sentences across all equivalent forms of \( \mathfrak{A} \). Thus, model theory provides a scaffold for understanding logical equivalences and connections within mathematical structures, ensuring that abstract logic adheres to specific, practical models that reflect logic consistently across different scenarios.