Questions tagged [propositional-logic]
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153
questions
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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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0answers
27 views
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 ...
0
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1answer
45 views
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 ...
0
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0answers
23 views
About the complexity of deciding if the closed world assumption for renamable Horn CNF is consistent
Let $T$ be a theory that only contains renamable horn formulas. What is the complexity of deciding if the closed world assumption $CWA(T)$ is consistent?
The closed world assumption is defined as ...
0
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1answer
41 views
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 ...
0
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1answer
33 views
Equivalence of Horn formulas tractable?
Assume I have two Horn formulas $\phi_1, \phi_2$. Horn formulas are CNF formulas so that each clause has at most one unnegated literal. For example:
$x_1 \wedge (\neg x_1 \vee \neg x_2 \vee x_3 )\...
2
votes
2answers
69 views
Equivalence of Krom formulas tractable?
Assume I have two Krom formulas $\psi_1, \psi_2$. Krom formulas are propositional formulas in CNF that have 2 literals in every clause. Each literal can be negated or unnegated. In other words, $\...
0
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1answer
40 views
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 ...
1
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1answer
76 views
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....
2
votes
1answer
14 views
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 ...
2
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0answers
47 views
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 ...
3
votes
1answer
39 views
“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 ...
2
votes
1answer
35 views
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 ...
2
votes
1answer
42 views
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 ...
1
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0answers
23 views
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 speciļ¬cation (to compute in r the absolute value of x).
...
2
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0answers
34 views
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 ...
2
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1answer
43 views
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 $[v,i]$...
2
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1answer
42 views
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 ...
0
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2answers
36 views
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 ...
0
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1answer
27 views
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: $$Ļ(x) := x=2 \vee x=7$$ is capable of representing this set....
0
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1answer
34 views
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 ...
1
vote
1answer
35 views
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 ...
2
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1answer
32 views
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
..." ...
0
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1answer
46 views
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 ...
0
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0answers
40 views
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 ...
1
vote
1answer
108 views
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 ...
2
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1answer
41 views
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 ...
0
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2answers
56 views
prove that {$ā,ā$} is incomplete set?
How do i prove that Is {$ā,ā$} not a complete set ? I have no clue how to prove it .
3
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1answer
110 views
Are these 2 equivalent?
Is āxāyāz[Ļ(x,y)ā§p(y,z)->p(x,z)] equivalent to āxāyāz[Ļ(x,y)ā§p(x,z)->p(y,z)] ?
The only thing I can think of is that this question can be answered if we show that ...
0
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2answers
76 views
Does the naive conversion of a Boolean Formula to CNF have a polynomial or exponential complexity?
I am reading the naive conversion to CNF, this procedure is explaining in this book book, but I have not found a conplexity analysis of this algorithm:
elimination of equivalence
Elimination of ...
0
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0answers
53 views
What are necessity and sufficiency?
I was reading deadlock topic from Operating Systems book by Stallings. It states four pre requisites for deadlock:
Mutual exclusion
No preemption
Hold and wait
Circular wait
It then have following ...
1
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1answer
72 views
Context-free grammar for tautologies in one variable
Construct a context-free grammar for the set of tautologies in $p$ - that is, the set of formulae in $\{p, \text{true}, \text{false}, \land, \lor, \lnot, (, )\}$ which evaluate to $\text{true}$ for ...
1
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2answers
109 views
Are $\mathsf{\#P}$ problems harder than $\mathsf{NP}$ problems
I have a method to solve the $\mathsf{\#P}$ version of 3SAT in a way that seemingly reduces it to an $\mathsf{NP}$ problem. - I don't have a formal understanding of these terms so I will just show an ...
2
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1answer
73 views
3-CNF to “independent form”
Is it possible to convert all logical formulae into a form such that each variable ends up in exactly 1 "factor" of the and operation? ($\wedge$). Any combination of operations is allowed, though the ...
2
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1answer
64 views
Conjunctive normal form to simple elementary algebra
I'm curious to know the computational complexity class of each step in this method of converting a CNF formula into simple elementary algebra.
An example:
$$\phi=\left(x_1 \vee x_2 \right) \wedge \...
1
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1answer
89 views
Algorithm for automatic construction of natural deduction proofs
I was wondering if there exists any algorithm for automatic construction of nautral deduction proofs. I'm interested in propositional logic and first order logic.
If there is no algoritm, can you ...
2
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0answers
80 views
A QuineāMcCluskey variant for conjunctive normal formal?
There is the QuineāMcCluskey algorithm for finding a minimal expression of a boolean expression in dis-junctive normal form. Would applying DeMorgan's rule to the minimal DNF result in the minimal CNF?...
4
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3answers
111 views
Is it a tautology or not? According to my truth table its not
If $\bigr((q\leftrightarrow p)\leftrightarrow s\bigl)$ is a tautology and $p\rightarrow s$ is contingent, does it follow that $q\rightarrow s$ is contingent?
Since I can't show $\bigr((q\...
4
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0answers
44 views
Is there a correspondence of steps between DPLL and sequent-calculus?
Is there a correspondence between the steps in using DPLL to find out that a formula in propositional logic is unsatisfiable and using sequent calculus to prove that its negation is valid?
And given ...
2
votes
1answer
31 views
How do I determine if this argument is valid?
I'm studying propositional logic the section on "valid arguments".
A self-assessment question reads
"Show whether or not the following argument is valid"
$\frac{P}{C}$
I don't know what function ...
1
vote
1answer
26 views
Does “$\forall x\in L, \sigma(\neg x)=\neg \sigma(x)$” hold given that $\sigma(F)\equiv F$ for a CNF formula $F$ built on a set $L$ of literals?
Suppose we have a CNF formula $F$ built on the set of literals $L=\{x_1,\neg x_1,\cdots,x_n,\neg x_n\}$ where each variable is used in at least one clause of $F$. Consider a permutation $\sigma$ of $L$...
0
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2answers
61 views
Why we treat sentence letter $Q$ as conclusion in one form and premise in other?
Sorry for asking such a dumb question.
I am CS student and I am trying to understand the basic tenets of Logic. I am new to the subject and I am lost understanding Implication.
In formal logic, $(P\...
3
votes
1answer
285 views
Matrix Representation for logical gates?
I have been trying to find if there are other matrix type representations logical circuits, in the example below, $$\begin{bmatrix} 1 & 0\\ 1 & 1\\ \end{bmatrix} \equiv \, \, \Rightarrow$$ ...
2
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0answers
85 views
Why Modus ponens works with Horn clauses and Generalized Modus ponens requires definite clauses?
I am reading the Artificial Intelligence: A Modern Approach book and in the chapters about logic i noticed that in propositional logic the Modus ponens inference rule (used by the forward and backward ...
0
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1answer
164 views
Can this set of propositions be represented and proved in Haskell?
I used a set of natural language statements and their formalization from Gries and Schneider. I attempted to transform the propositions into Haskell equations. For example, for S0 : $a \land w \...
1
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1answer
84 views
What is the computational complexity of intuitionistic propositional logic?
Consider the decision problem: given formulas of propositional logic $\phi_1, \ldots, \phi_n, \psi$, determine whether $\phi_1, \ldots, \phi_n \vdash \psi$ intuitionistically. This problem is well-...
0
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1answer
48 views
Finding CNF of $(p \rightarrow q) \rightarrow p$ and $\lnot (q \wedge (\lnot p \rightarrow q)) $
I'm trying to find the CNF of the $(p \rightarrow q) \rightarrow p$ and $\lnot (q \wedge (\lnot p \rightarrow q))$, and afterwards proving it's validity. As i'm new to CNF i wanted to ensure i've ...
1
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0answers
90 views
How to minimize large DNF formula into deeper but smaller formula?
I want to minimize DNF formula which has ~200k products combined into a sum.
The depth of that formula is 2 - first level is the OR, and second level is AND for all conjunctions.
I have tried using ...
-1
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1answer
24 views
C logical consequence of S iff S union {-C} UNSAT
I'm trying to do the following demonstration:
(C and S are CNF)
C is a logical consequence of S iff. S u {-C} UNSAT
And I did the following:
C is a logical consequence of S iff. S u {-C} UNSAT iff.
...
2
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1answer
47 views
Does it hold that $F \equiv \sigma(F)$ for a CNF formula $F$ and a permutation $\sigma$ s.t. $F \vDash \sigma(F)$?
Suppose we have a CNF formula $F$ and a permutation $\sigma$ of its literals such that for any literal $x, \sigma(\neg x)=\neg \sigma(x)$ and $F \vDash \sigma(F)$.
Does it hold that $F \equiv \sigma(...
