# Questions tagged [propositional-logic]

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### To prove (KB ⊭ S) and (KB ⊭ ¬S) is satisfiable

In question (e), I have to prove: A ≡ (KB ⊭ S) and (KB ⊭ ¬S) is satisfiable, where KB and S are propositional variables. I am not able to follow the solution given in the image above as to why it is ...
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### Is there a $L$-complete variant of SAT?

Many complete problem of different class of complexity has SAT variant. Like 3-SAT or $k$-SAT is $NP$-complete, Horn-SAT is $P$-complete, 2-SAT is $NL$-complete, and so on. So I was wondering if there ...
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### Incomplete definition of function- first order logic

Let $\Sigma=\{c,f^1,R_1^2,...,R_k^2\}$ where $c$ is constant, $f$ is one argument function, and $R_i$ are binary relations. Let $\Sigma_2=\{c,g^2,R_1^1,...,R_k^1\}$ where $c$ is constant, $g$ is two ...
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### Problem with formalism in first order logic

This is a general question in first order logic. Assume I have alphabet $\Sigma$ that contains one-argument function (among other symbols). I want a new alphabet, $\Sigma'$, which is the same as the ...
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### equivalence of validity above different alphabet

Given the next alphabets: $\,\,\Sigma_1=\{R^2,P^1,=^2\}\,\,,\Sigma_2=\{c,f^1,=^2\}.$ Prove of Disprove: There's exists an algorithm, that given formula $A$ above $\Sigma_2$, builds formula $A'$ above ...
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### Software for proving tautology with steps

I'm looking for a software which can prove a tautology using logical equivalences. It should be able to show each step. So you can follow the chain of reasoning. Here is an example: \begin{align} (p\...
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### a conjunctive normal form that is a tautology

Are there any examples of CNF formulas that are tautologies? Such that every clause contains different variables so phrases like (a or not a) are rejected?
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### Hoare Triple Logic

I'm having trouble understanding the logic behind Hoare Triples. The question asks for the missing value of the precondition {X} ...
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### What is the Number of Possible Nonequivalent Propositions with $P_1, P_2, P_3$ Using $\iff$ Operator?

A multiple choice question asks this: Number of nonequivalent propositions that only consist of $P_1, P_2, P_3$ and use the $\iff$ logical operator is?$$A)7\text{ }B)8\text{ }C)1\text{ }D)16$$ I am ...
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### Doubt regarding the implications of a 2-SAT constraint

Consider an example 2-SAT instance with the constraint (x1​ ∨ x2)​. This CNF has these two implications: ¬x1​→x2​ and ¬x2​→x1​. "They actually mean, if x1​ is false then x2​ must be true, and if ...
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### Formal Logic - Natural deduction: Problem with assumptions about exists-negation

I'm stuck on how to progress with this proof, despite I have tried, I cannot see the next move. Given this proof without predicate: So far what I've accomplished: My idea is, as I can't see any ...
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### Converting to CNF - Distribution

I am still trying to get my head around what is valid CNF and how distribution works, I am not even sure if the examples I have tried to convert are correct for one, but I am also unsure of the rule ...
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### Boolean logic: why is the "opposite" of and equal to or?

In boolean logic, why is the "opposite" of AND (&) equal to OR (|)? For example, why would ...
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### Propositional Logic: Entailment

Given the sentences, S1, S2, if S1 |= S2 then all models that satisfy S1 also satisfy S2, how is the following statement correct? A ∧ ¬A |= B How can something and ...
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### In propositional logic how is the negation equivalent?

I am unable to understand how the negation of a sentence into CNF actually makes sense, hopefully somebody can explain it to me in understandable terms! Given the sentence: If the battery is charged (...
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### Is Conjunctive Normal Form or not?

I have one formula that I do not understand why it is CNF and one that is not CNF, namely. p && !q (NOT NCF) and !!p(CNF). According to the exercise where I found these examples, 1 is not ...
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### Is it generally possible to convert CNF to Horn clauses?

My intuition is that it is not generally possible, but I cannot think of a proof.
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### When are existential quantifiers in the intuitionistic propositional calculus eliminable and when not

I am so ignorant I don't even know where should I ask this - on FOM? On mathoverflow? On cstheory? So please consider as sort of a meta-question readdressing me in case you think this is a wrong site. ...
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### What is the purpose of learning propositional logic

I am in an AI class where we have extensively learned about propositional logic. I am wondering what the point of learning this is? Are there any uses for propositional logic? Beyond basic logic gates ...
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### Prove (p → ¬q) is equivalent to ¬(p ∧ q)

I need to prove the above sequent using natural deduction. I did the first half already i.e. I proved $(p\rightarrow\neg q)\rightarrow \neg (p \wedge q)$, but I'm stuck on where to start for the ...
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### Proving the validity of a sequent using Modus Tollens

Problem: Prove $p \rightarrow (q \vee r), \neg q, \neg r \vdash \neg p$ using Modus Tollens. I need to prove the validity of the above sequent by using natural deduction. Initially, I didn't read the ...
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### Satisfaction of CTL

I'm trying to compute Sat(Φ), but I can't seem to figure out this question, what i understood is that I first need to solve Sat(req U -busy), but it doesn't make sense, how can it be that req is true ...
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### Computational complexity in Boolean network

An Boolean control networks can be expressed as \begin{equation} \label{ControlBN} \left\{\begin{array}{l}{x_{1}(t+1)=f_{1}\left(x_{1}(t), \cdots, x_{n}(t), u_{1}(t), \cdots, u_{m}(t)\right),} \\ {x_{...
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### Simplify logical expression represented as binary tree

I implemented logical expressions using a binary tree in C++. Now I want to be able to simplify such an expression using rules like e.g. but have issues with the commutativity. Assuming the ...
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### What does it mean to "show algebraically" in propositional logic?

The biconditional operator $\iff$ of Propositional Logic can be defined by the identity $p \iff q \equiv (\lnot p \lor q) \land (\lnot q \lor p) \quad (1.1)$ Use the identity $(1.1)$ and identities ...
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### How are prime implicates of HORN-Formulas defined?

I'm confused about the definition of prime implicates in Horn formulas. For example in the paper of Kira 2012 on page 109 it is stated: Now in the paper of Boros 2010 on page 82 the following ...
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### Consistent theory based on L and not(A->A) is a theorem

I am working on this problem in which I have a theory $T$ based on language $\mathcal{L}$ and the only information we have is that T is consistent and $\vdash \lnot(A \rightarrow A)$. Given this ...
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### Is Monotone 3-SAT with exactly 3 distinct variables untractable?

I have given the following SAT variation: Given a formula F in CNF where each clause C has exactly 3 distinct literals and for each C in F either all literals are positive or all literals are negated....
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### Circuits and formulas for Clique

Is it correct to say that the Clique Problem is in $P$ iff there exists a family of Boolean circuits $C$ to decide Clique whose sizes are bounded by a polynomial? And based on this question, does that ...
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### An optimization version of 2QBF: is it $\mathsf{NP}^{\mathsf{NP}}$-hard?

I am studying the computational complexity of the following decision problem related to 2QBF: Input: a 3-CNF formula $\varphi$ over $X \cup Y$, where $X$, $Y$ are disjoint sets of propositional ...
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### "State of the art" algorithms deciding entailment of propositional formulas?

I fail to find much about how to efficiently calculate whether a propositional formula entails another. Considering the following two points... We can check, for each truth assignment which makes the ...
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### Unification Algorithm without Occur Check

I have been reading about Unification algorithm here https://en.wikipedia.org/wiki/Unification_(computer_science)#A_unification_algorithm . And I wonder about the importance of occur check. I know ...
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### Convert propositional logic formulas to mathematical constraints

Brief introduction In all boolean (or more generally mixed-integer) linear programs, constraints are represented as a matrix $A$, a support vector $b$ and is computed by $A^T x \leq b$, where $x$ is ...
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### Correctness of Simple Programs

For a homework assignment I am asked to provide claim sequences to verify that the given conditional does indeed satisfy its speciﬁcation (to compute in r the absolute value of x). ...
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### Large Conjunctive Normal Form Examples

I'm currently learning about conjunctive normal form in a course on logic for Computer Science. I was reading the Wikipedia entry on the subject and encountered this: Typical problems in this case ...
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### Programming in Propositional Logic article notation question

I was reading this article about propositional logic and transforming problems to SAT. The author often uses the following notation (taken from Dominating set section): I don't understand what $[v,i]$...
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### Definition of 2-CNF (a.k.a Krom) formula

In my lecturer's notes, the following definition for a 2-CNF wff is given: A 2-CNF formula, or Krom formula is a CNF formula F such that every clause has at most two literals. However, there is ...
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### What if P implies Q is false when both P and Q are false?

This is actually a problem that our professor gave us, and I'm clueless of how to answer this. I browsed through various sources, but none were helpful regarding this question. The question is, In ...
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### Embedded arithmetic set expressions

In set builder notation, we can represent the set: $$\{ 2, 7\}$$ as: $$\{ x | x=2 \vee x=7 \}$$. Therefore, the PA arithmetic predicate: $$φ(x) := x=2 \vee x=7$$ is capable of representing this set....
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### Representing a sentence using propositional logic

I am confused regarding a propositional logic representation of a sentence. Please note that this sentence is not realistic: "A person who is male (M) is smart (S) if he is tall (T), but otherwise ...
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### Complexity of negation cancellation

Consider propositional logic over the connectives $\land$, $\lor$, and $\lnot$. Notation: $| \alpha |$ is the length of formula $\alpha$. We are given a formula $\phi$. Cancel all cancellable ...
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### Significance of quantifier ordering in quantified boolean formulas (kQBF vs. QBF)

I am studying solvers of quantified boolean formulas (QBF) as a generalization of SAT solving. The standard DIMACS format of SAT specification is extended to QDIMACS, which adds "a ..." and "e ..." ...
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### How to correctly negate a predicate bounded by some quantifiers?

this is a problem which was asked in GATE CS 2010. This is question statement: Q: Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. which ...
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### How to write negation of an 'AND' statement in logic

Suppose the given proposition is: "Zach blocks emails and texts from Jennifer" Where, P implies "Zach blocks emails from Jennifer". Q implies "Zach blocks texts from Jennifer". So it's P AND Q in ...
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### The optimized numbers of variables and clauses to encode a graph coloring problem in CNF

Problem Statement Given a finite graph $G = \langle V, E\rangle$, consisting of vertice set $V$ and edge set $E$, and a finite set of colors $C$, a problem instance of graph coloring is to assign ...
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### Non-Boolean SAT

I was wondering about the complexity of SAT tests with variables $x_i = 0 \lor 1 \lor 2 \dots \lor n$, with clauses being of the form $x_i = a \implies x_j \neq b$. When $n=2$, we have 2SAT, which has ...
### prove that {$↔,⊕$} is incomplete set?
How do i prove that Is {$↔,⊕$} not a complete set ? I have no clue how to prove it .