Discrete Structures Lecture Notes: Translating English Sentences into Propositional Logic, Study notes of Computer Science

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CSC102 - Discrete Structures By Mahwish Waqas

Department Of Computer Science, CUI

Lahore Campus

Lecture Outline

• Applications of Propositional Logic
• Logic and Bit Operations
• Logical Equivalence

Translating English Sentences

• Steps to convert an English sentence to a statement in propositional logic - Identify atomic propositions and represent using propositional variables. - Determine appropriate logical connectives

Translating English Sentences

• “I have neither given nor received help on this exam” Let p = I have given help on this exam q = I have received help on this exam
• Rephrase: It is not the case that either I have given or received help on this exams

Translating English Sentences

• “If I go to Harry’s or to the country, I will not go shopping.”

Translating English Sentences

• “If I go to Harry’s or to the country, I will not go shopping.”
  • Let p = I go to Harry’s
  • q = I go to the country.
  • r = I will go shopping.

Translating English Sentences

• Let p = It is below freezing q = It is snowing a) It is below freezing and it is snowing b) It is below freezing but not snowing c) It is not below freezing and it is not snowing d) It is either snowing or below freezing (or both) e) If it is below freezing, it is also snowing f) It is either below freezing or it is snowing (not both), but it is not snowing if it is below freezing g) That it is below freezing is necessary and sufficient for it to be snowing

Translating English Sentences

• “You can access the Internet from campus only if you are a computer science major or you are not a freshman.”

Translating English Sentences

• “You can access the Internet from campus only if you are a computer science major or you are not a freshman.”
• Let a = You can access the Internet from campus c = You are a computer science major and f = You are a freshman” respectively
• a only if c or not f a → (c ∨￢f ).

Exercise

• Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express each of these compound propositions as an English sentence.

1. ¬p
2. p ∨ q
3. ¬p ∧ q
4. q → p
5. ¬q → ¬p
6. ¬p → ¬q
7. p ↔ q
8. ¬q ∨ (p ∧ q)

System Specifications

• System and Software engineers take requirements in English and express them in a precise specification language based on logic.
• The automated reply cannot be sent when the file system is full p = The automated reply can be sent q = The system is full

Consistency

• System specifications should be consistent , They should not contain conflicting requirements that could be used to derive a contradiction.
• When specifications are not consistent, there would be no way to develop a system that satisfies all specifications.
• A list of propositions is consistent if it is possible to assign truth values to the proposition variables so that each proposition is true.

• Determine whether these system specifications are consistent:

1. The diagnostic message is stored in the buffer or it is retransmitted.
2. The diagnostic message is not stored in the buffer.
3. If the diagnostic message is stored in the buffer, then it is retransmitted.

• p = The diagnostic message is stored in the buffer
• q = The diagnostic message is retransmitted
• 1. 𝒑∨𝒒 2. ¬𝒑 3. 𝒑→𝒒

Reasoning

• An assignment of truth values that makes all three specifications true must have p false to make ￢𝑝 true.
• Because we want 𝑝 ∨ 𝑞 to be true but 𝑝 must be false, q must be true.
• Because 𝑝 → 𝑞 is true when 𝑝 is false and 𝑞 is true
• we conclude that these specifications are consistent
• Let us do it with truth table now