Questions tagged [propositional-logic]
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135
questions
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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....
0
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1answer
32 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
34 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
25 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
34 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 ...
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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
72 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
30 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
49 views
prove that {$↔,⊕$} is incomplete set?
How do i prove that Is {$↔,⊕$} not a complete set ? I have no clue how to prove it .
3
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1answer
107 views
Are these 2 equivalent?
Is ∀x∀y∀z[φ(x,y)∧p(y,z)->p(x,z)] equivalent to ∀x∀y∀z[φ(x,y)∧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
52 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
49 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
38 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
98 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
70 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
51 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 \...
0
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1answer
59 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
67 views
A Quine–McCluskey variant for conjunctive normal formal?
There is the Quine–McCluskey 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
106 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
39 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
27 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
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1answer
25 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
54 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
133 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
70 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
159 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
63 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
44 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 ...
2
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0answers
40 views
Is there a formula that is not logical equivalent to any Horn clause?
There are of course formulas that can't be transformed into a Horn clause but is it possible to construct an equivalent Horn clause that is true under the same interpretation?
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0answers
64 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
45 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(...
4
votes
1answer
201 views
Knight and knaves
I have these in my lecture notes, its about the rules where knights always tell the truth and knaves always lie:
If A says “The statement ‘there is gold on the island’ and the statement ‘I am a ...
2
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1answer
37 views
Preserving a propositional formula
I know I must be getting stuck on notation. However, I'm having trouble following the logic in Example 1.2 in https://arxiv.org/pdf/cs/0611018.pdf. They define what preserving a propositional ...
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1answer
174 views
How to define the Atomic Propositions in Model Checking
The atomic propositions in Symbolic Model Checking form the state in the state-transition graph (the model $\mathcal{M}$ in Model Checking). The other part of Model Checking is the specification, ...
2
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1answer
70 views
Are there any techniques for checking whether a clause is subsumed by another clause when adding it to a cnf formula?
When doing variable elimination on a formula in cnf form, there is created a lot of new clauses. Is there any efficient way to check if these are subsumed by other, already existing clauses?
1
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1answer
179 views
Resolution Algorithm - No new clauses
The standard resolution algorithm returns false if no new clauses are added.
I know that if $KB \land \neg a \implies []$, it returns true by proof by contradiction, but how can I understand the ...
3
votes
1answer
241 views
What is the connection between combinatorial circuits and finite state automata?
The diagram on the Wikipedia page of FSA shows the hierarchy of the computational devices, in that diagram it is denoted that the finite state machines are superior to the combinatorial circuits.
...
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2answers
78 views
Can any memory-less computer operation be represented in propositional logic?
Take any operation that is done by any type of computer (e.g. a cpu on a modern laptop), which doesn't use any type of temporary memory storage.
I.e. this computer operation computes a function $f(x)=...
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39 views
Give context-free grammars that generate formulas in: [duplicate]
Give context-free grammars that generate formulas in propositional calculus, taking into account:
variables represented by single lowercase letter
Operations are conjunction (∧), disjunction (∨), ...
0
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1answer
35 views
Convert (x + y = z) to propositional logic formula for 8-bit numbers
(Note: this was asked in the cstheory stackexchange, but I was told this is a more appropriate place for it.)
I have three 8-bit 2's-complement numbers (X, Y, and Z). I want to find a propositional ...
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0answers
67 views
What is the definition of the property of Canonicity for a logic?
I'm trying to understand "A Systematic Approach to Canonicity in the Classical
Sequent Calculus" by Kaustuv Chaudhuri, Stefan Hetzl, Dale Miller. The article discusses a property called "canonicity" ...
0
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1answer
64 views
Inference rules for deriving invariants in Hoare logic
The following algorithm is supposed to compare two strings $S_1$ and $S_2$ ("/\" for empty string):
...
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0answers
102 views
How to construct an NFA Ai for the given Regular property P1 and P2?
Let AP = {a; b; c}.
Consider the following regular safety properties:
(a) P1: If a becomes valid, afterwards b stays valid ad infinitum or until c holds.
(b) P2: Between two ...
19
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4answers
5k views
Can proof by contradiction work without the law of excluded middle?
I was recently thinking about the validity of proof by contradiction. I’ve read for the past few days things on intuitionistic logic and Godel’s theorems to see if they would provide me answers to my ...
12
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5answers
3k views
Why does soundness imply consistency?
I was reading the question Consistency and completeness imply soundness? and the first statement in it says:
I understand that soundness implies consistency.
Which I was quite puzzled about ...
5
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1answer
346 views
Implementing mathematical theory of arithmetic in Haskell via Curry-Howard correspondence
I have to ask for forgiveness in advance if the whole question doesn't
make a lot of sense, but unfortunately, I have no better intuition as
of right now and this seems like the best starting point I ...
1
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2answers
968 views
how to solve Hoare logic problems
I'm having trouble proving Hoare logic questions as I'm not sure of the process that is taken to prove them. I understand that they're rules such as assignment axiom, pre-condition strengthening, post-...
0
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1answer
75 views
Does exist NP language that is Cook Levin deterministic reducible to xor satisfiability in polynomial time?
We say that the language $L$ is Cook Levin deterministic reducible to xor satisfiability in polynomial time if and only if for each word $w\in\Sigma^*:w\in L\iff f(w)\in XORSAT$ where $\Sigma=\{0,1\}$ ...
0
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1answer
157 views
Is this possible to solve satisfiability by using Quine McCluskey algorithm to simplify the whole given boolean formula by simplifying subformulas?
In this question
Farewell Stack Exchange suggested to use karnaugh maps to solve the satisfiability problem by simplifying the entire/whole boolean formula by simplifying subformulas until you have ...
