Chapter 0: Q13P (page 1)

For each let Ƶ_m = {0, 1, 2, . . . , m − 1}, and let = (Ƶ_m, +, ×) be the model whose universe is Ƶ_m and that has relations corresponding to the + and × relations computed modulo m. Show that for each m, the theory Th is decidable.

Short Answer

Expert verified

Answer:

This statement is decidable is given below.

Step by step solution

Turing Machine

A Turing Machine is computational model concept that runs on the unrestricted grammar of Type-zero. It accepts recursive enumerable language. It comprises of an infinite tape length where read and write operation can be perform accordingly.

Problem is decidfable.

$M_{m} = {0, 1, 2, . . ., m - 1} f o r m > 1$ is a universe

$M_{m} = (P_{m'} +, x)$ model whose universe is $p_{m}$ .

+ and x are relations over modulo m.

We need to show that $T h (f_{m})$ is decidable.

We can clear observe that Ƶ_m is finite as up to m terms.

So we can easily enumerate all possible values into formula and check if the formula is right or not.

If it is TRUE, then it belong to $T h (f_{m})$ .
If it is NOT TRUE, then it belong to $T h (f_{m})$ .

Let, Ǝx_i such that $ϕ = (x_{1}, x_{2}, \dots, x_{i})$ Put $X_{i} = 0, 1, \dots, m - |$

So, if formula $ϕ = (x_{1}, x_{2}, \dots, x_{i})$ is true for any value of x_i then the original value is true.

Now, a model of theory is considered to be decidable if the formula which is true and also belong to the model.

So, by above recursive strategy, we say is decidable.

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For each let Ƶ_m = {0, 1, 2, . . . , m − 1}, and let = (Ƶ_m, +, ×) be the model whose universe is Ƶ_m and that has relations corresponding to the + and × relations computed modulo m. Show that for each m, the theory Th is decidable.

Short Answer

Step by step solution

Turing Machine

Problem is decidfable.

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For each let Ƶm = {0, 1, 2, . . . , m − 1}, and let = (Ƶm, +, ×) be the model whose universe is Ƶm and that has relations corresponding to the + and × relations computed modulo m. Show that for each m, the theory Th is decidable.

Short Answer

Step by step solution

Turing Machine

Problem is decidfable.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Problem Solving Techniques

Computer Organisation and Architecture

Data Structures

Algorithms in Computer Science

Databases

Data Representation in Computer Science

Study anywhere. Anytime. Across all devices.

For each let Ƶ_m = {0, 1, 2, . . . , m − 1}, and let = (Ƶ_m, +, ×) be the model whose universe is Ƶ_m and that has relations corresponding to the + and × relations computed modulo m. Show that for each m, the theory Th is decidable.