First Order Logic: ∀x,y a(x) ∧ a(y)

In summary, the conversation discusses expressing that "a" is the first and last letter in a word using the set of all words over ∑. The correct expression is ∀x,y ( a(x) ∧ a(y) ∧ x<y ∧ x∈∑* ∧ y∈∑* → ∃z(x<z ∧ z<y)).
  • #1
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Homework Statement
Let sigma = {a,b,c} be a finite alphabet. Construct formulas that specify the following languages over sigma.
Relevant Equations
sigma = {a,b,c}
An example we were given is as follows: {ua|u∈∑*} (where ∑* is set of all words over ∑) so we have ∀x. last(x) → a(x).

I am given {awa|w∈∑*} to do, and I know that I have to express that a is the first letter and last letter in a word. Could I write it as:
∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z<y)), am I on the right track with this, any help would be appreciated, thank you.
 
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  • #2


Hello,

Yes, you are on the right track with your expression. However, there are a few minor changes that can be made to make it more accurate.

Firstly, the expression should be ∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z ∧ z<y)). This ensures that z is between x and y, rather than just being larger than x and smaller than y.

Additionally, you can add in the condition that x and y are words in the set ∑*, so the final expression would be: ∀x,y ( a(x) ∧ a(y) ∧ x<y ∧ x∈∑* ∧ y∈∑* → ∃z(x<z ∧ z<y)).

Hope this helps! Let me know if you have any further questions. Good luck with your work.
 

Related to First Order Logic: ∀x,y a(x) ∧ a(y)

What is First Order Logic?

First Order Logic (FOL) is a formal system used in mathematics and computer science to represent and reason about relationships between objects. It is a type of predicate logic that allows for the quantification of variables and the use of logical operators such as "and" and "or". FOL is a fundamental tool in the field of artificial intelligence and is used in areas such as automated theorem proving and knowledge representation.

What does the notation ∀x,y a(x) ∧ a(y) mean?

The notation ∀x,y a(x) ∧ a(y) is a logical statement in FOL that translates to "for all x and y, a(x) and a(y) are both true". The symbol ∀ (for all) is a universal quantifier that indicates that the statement applies to all possible values of the variables x and y. The ∧ symbol represents the logical operator "and", which means that both a(x) and a(y) must be true for the statement to be true.

What is the difference between First Order Logic and Second Order Logic?

The main difference between First Order Logic (FOL) and Second Order Logic (SOL) is that SOL allows for quantification over sets of objects, while FOL only allows for quantification over individual objects. This means that SOL is more expressive and can represent more complex relationships between objects. However, this also makes it more difficult to reason with and automate, which is why FOL is still widely used in many applications.

How is First Order Logic used in computer science?

First Order Logic is used in computer science for a variety of purposes, including knowledge representation, automated reasoning, and natural language processing. In knowledge representation, FOL is used to represent and reason about objects, relationships, and rules in a formal and logical way. In automated reasoning, FOL is used to prove the validity of logical statements and to automate the process of theorem proving. In natural language processing, FOL is used to analyze and understand the meaning of natural language sentences.

What are some limitations of First Order Logic?

Although First Order Logic is a powerful and widely used formal system, it does have some limitations. One of the main limitations is that it cannot handle self-reference or circular reasoning, which can limit its ability to represent certain types of knowledge. Additionally, FOL is not able to handle uncertainty or incomplete information, which can be important in real-world applications. These limitations have led to the development of more advanced logical systems, such as modal logic and fuzzy logic, to address these issues.

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