Questions tagged [satisfiability]
Satisfiability (SAT) is the problem of determining whether there is a variable assignment that fulfills a given Boolean formula.
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Understanding the Strong Exponential Time Hypothesis
Let $n$ be the number of variables in the input formula and $m$ the number of clauses. Define $s_k = \inf\{\delta : k\text{-SAT can be solved in } 2^{\delta n} \text{ time}\}$. The strong exponential ...
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Problems with proof of NP-completness of SAT following Cooks original paper
I am currently in the process of trying to understand the original proof of NP-completeness of SAT given in the seminal paper by Cook [COOK71] and have struggled with a few of the details of the proof ...
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Finding a 2SAT instance that has a specific solution set
Is there a 2SAT instance of variables $(a,b,c,d,e,f,g)$ that has exactly the solution set $S=\{ (1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(1,1,0,1,0,0,0),(1,0,1,0,1,0,0),(0,1,1,0,0,1,0),(1,1,1,1,...
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Is there an SMT/SAT algorithm for General Predicate Logic (FOL)?
I'm learning how to write my own theorem prover. After skimming Decision Procedures (Kroening & Strichman, 2016), I didn't find any SMT algorithms for solving quantified n-ary predicate formulas. ...
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Variation of 3-SAT
I already know that SAT and 3-SAT are NP-complete.
If in 3-SAT the Boolean expression should be divided to clauses,such that every clause contains at most (in the original problem it says exactly) ...
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How to reduce 3SAT to TwoOrMoreSAT?
I want to prove, that 2OrMoreSAT is NP-complete. It's defined as follows:
A formula is considered strongly satisfiable if there exists a model such that two or more different literals in every clause ...
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MAX-SAT approximation factor
I am stuck on an exercise that ask the approximation factor of a MAX-SAT approximated algorithm generalized from a MAX-3SAT algorithm
MAX-3SAT:
set every variable with a random value ($0$ or $1$ each ...
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Finding a minimal set of package versions in a dependency graph with constraints
Suppose you have a dependency graph of "packages" registered in the ecosystem of a given programming language. We can model each package as a tuple ...
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Question about a proof of the existence of unsatisfiable linear k-CNFs for any k
Today I am reading paper Unsatisfiable Linear k-CNFs Exist, for every k by Dominik Scheder, 2007. But I have some problem to understand the proof of Theorem $3.2$.
I don't know how to understand the ...
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How to get the formal model using propositional logic
Input
There are three chairs (1,2,3) in the same row.
We need to find a seat for three guests (a,b,c).
Constraints
The first guest does not want to be seated next to the third one (neither left nor ...
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Why solving #2SAT in polynomial time implies P = NP?
The wikipedia article for #P states that if we have a polynomial-time algorithm for a #P-complete problem, P = NP is true.
As #2SAT is #P-complete, this would mean that providing a polynomial-time ...
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NP-hard $k$-SAT variant with exactly $\ell$ occurrences per variable
For the purpose of this post, let $k$-SAT be SAT with exactly $k$ literals per clause, as opposed to the more common meaning of at most $k$ literals per clause.
With the purpose of proving some ...
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Is SAT an existential question?
Some sources state that an algorithm that solves the SAT problem not only needs to decide whether a given existentially-quantified formula is satisfiable or not, but, additionally, in the case where ...
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Possible to solve a combinatorial game with integer programming?
I recently had the idea that it would be neat if it were possible to make a SAT solver play combinatorial games. To start, I'm trying a relatively simple case of solving single-stack Misère Nim ...
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CNF – satisfy at most a fixed number of clauses
I'm working on this task:
Prove that the following problem can be solved in time $2^{k} \cdot \Vert \varphi \Vert^{\mathcal{O}(1)}$: given a boolean formula $\Vert \varphi \Vert$ in CNF, decide ...
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Logical Consequence - Equivalent Assertions
I have the following slide in my notes and I'm having trouble understanding how the three assertions are equivalent. I understand to a degree how the 2nd and 3rd assertions are equivalent, but the ...
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Running time of SAT and other EXPTIME algorithms
I need to propose an algorithm for a NP-hard problem. I use dynamic programming which leads to a running time $O(2^s\cdot n^2), s\leq n.$ The algorithm aims to finding a path in a graph $G(V, E)$ (in ...
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Is ANF-SAT P or NP?
Given a finite set of equations in ANF, for example:
$$
\begin{cases}
(x_1 \land x_2) \oplus (x_1 \land x_3 \land x_4) \oplus 1 = 0 \\
x_3 \oplus (x_2 \land x_3 \land x_4) = 0 \\
(x_1 \land x_4) \...
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Why can't 3-SAT be solved efficiently if you convert all clauses (x ∨ y ∨ z) into (u ∨ z) by introducing a variable?
Let $a_i$, $b_i$, etc., be a literal, i.e., a variable or the negation of a variable.
3-SAT concerns formulas in CNF form: $(a_1 \vee a_2 \vee a_3) \wedge \dots \wedge (b_1 \vee b_2 \vee b_3)$ (3-CNF)....
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Given an instance of a HORN-SAT problem with at most 3 literals per clause, what context-free grammar is equivalent to deciding the problem?
Given an instance of a HORN-SAT problem with at most 3 literals per clause, what context-free grammar is equivalent to deciding the problem?
For example, here are some HORN clauses:
...
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How to prove that P = NP?
So if someone proves with an algorithm that SAT can be solved in deterministic polynomial time, then P = NP and that's it?
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Simple Skolemization Question
Is it correct that, under a certain signature S, two First Order Logic formulae F and G are equisatisfiable if (F is satisfiable under S iff G is satisfiable under S)? But in Skolemization I’m ...
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Reductions from 3-SAT that won't work directly from SAT
Our prof talked about why it's good to know that 3-SAT is NP-complete because it's easier to craft reductions from it than from plain SAT.
However, all the examples we've seen (reduction to ...
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Polynomial Reduction from 3SAT
Given an undirected graph $G=(V,E)$ where $V$ is a set of vertices, and $E$ is a set of edges and
given a set $D$ where $D \subseteq V $ and $ \forall v \in V \setminus D \: \mid \: \exists w \in D : ...
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Could we know what's the total number of unsatisfiable 3SAT formulas for a given n variables?
given some $n$ variables I would be interested to know what is the count of all 3SAT formulas under $n$ that are unsatisfiable.
An example of all 3SAT forumlas under $n=3$ is the following:
$$
( x \...
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Reducing a mixed Boolean expression containing XOR of conjunctions
I know that XOR-SAT can be solved in polynomial time using arithmetic in $F_2$ and Gaussian elimination.
I have a set of formula that is of the form
$$
G_i := \oplus_{j=0}^{i} \left ( a_j \land b_{i-j}...
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Can these variants of SAT/Tautology be actually pretty simple?
There are 8 (very similiar) languages I'd like to discuss here:
CNF SAT
DNF SAT
CNF No-SAT (Existence of a false assignment)
DNF No-SAT
CNF Tautology
DNF Tautology
CNF Contradiction
DNF Contradiction
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resolving in CDCL
When resolving in Conflict driven clause learning, it is the case that if you resolve a conflicting clause with the reason of the negation of one of his literals, then this results in another ...
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Can you help me find some examples of 3co-SAT for 4 variables?
I've been studying the examples of 3co-SAT recently.
It's easy to find an example of one variable.
$(x_1\lor x_1\lor x_1)\land (\overline{x_1}\lor \overline{x_1}\lor \overline{x_1})$
Examples of 2 ...
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Why don't we consider that NP = co-NP while we can reduce Tautology problem into Satisfiability in polynomail time easily?
Let's determine if an expression is tautological or not and let's try this expression:
((a ⊼ b) ∨ c) ↔ (¬a ∨ ¬b ∨ c).
We can turn this problem into CIRCUIT-SAT decision problem by asking if the ...
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CNF-SAT time complexity and input processing
Boolean Satisfiability (CNF-SAT) problem in $n$ variables may contain a CNF formula with $O(2^n)$ clauses in the worst case.
My question is: Wouldn't a program reading a CNF formula have to ...
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MSAT and IMSAT problems (restricted versions of SAT)
I was reading about about NP-intermediate problems on Wikipedia and saw the IMSAT problem mentioned over there.
There is no Wikipedia page for that problem and they only cite this paper.
In the paper ...
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Probability of a randomised algorithm solving SAT
Let WALKSAT be defined as follows:
Let $σ$ be a random truth assignment to the vars
For $t = 1, 2, . . . , 3n$:
◦ If $\phi$ is satisfied by $σ$, exit the loop
◦ Else, pick an unsatisfied clause $c$ ...
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Complexity of the (Complete/Assign) 3-SAT problem?
A complete $k$-CNF formula on $n$ variables $(k\le n)$ is a $k$-CNF formula which contains all clauses of width $k$ or lower it implies.
Let us define the (Complete/Assign) 3-SAT problem: Given $F$, a ...
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what does **input** mean for the $3SAT$ question? Is it the number of variables $n$ or the number of clauses $m$
We know that $3SAT \in NP$,
and the definition of $NP$ is as follows:
$NP$ is the class of languages that have polynomial time verifiers.
But I have a question:
what does input mean for the $3SAT$ ...
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Time-complexity of evaluating a CNF formula
Given a Boolean formula over $n$ variables in CNF and a partial assignment to it, all the algorithms I can think of to evaluate the assignment run in time $\Theta(n^2)$. Is it possible to do it in $O(...
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Correctness implication graph for 2-sat
i want to proof some stuff for the 2-sat problem.
So we have something like this:
$\varphi$ = (x$_{1}$ $\lor$ y$_{1}$) $\land$ (x$_{2}$ $\lor$ y$_{2}$) $\land$ ... $\land$ (x$_{n}$ $\lor$ y$_{n}$). We ...
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I need to show that the problem is NP-complete
Double-SAT = {𝜓: 𝜓 has at least two satisfying truth assignments}. Hint: reduce from SAT. Start with a formula 𝜑 and modify it to get a formula 𝜓 so that 𝜑 is satisfibale if and only if 𝜓 has at ...
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Effecient encoding of sum equality in cnf+xor
I am wondering as to how to efficiently encode the following subcircuit for a binary satisfiability solver (cnf, and optionally xor clauses, if this helps):
...
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reducing the word problem for dtm to sat / cnf-sat / 2-sat
word problem: given a language L through a deterministic turing machine, is the word w in the language L?
the problem should be decidable, since if there is a deterministic turing machine i can simply ...
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Why can't $QBF$ be reduced to $SAT$
Let $QBF_k$ be the problem of determining the satisfiability of a formula of the form
$Φ = Q_1x_1Q_2x_2 . . . Q_kx_k φ(x_1, . . . , x_n)$.
where each $Q_i$
is one of the quantifiers $∀$ or $∃$. So, $Φ$...
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Exponential Time Hypothesis and the input size vs number of variables
According to Exponential Time Hypothesis there does not exist a deterministic algorithm to solve SAT over $V$ variables in time $o(2^V)$. However, let's say the number of literals $n = \omega(poly(V))$...
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Is there an alternative method to using Gaussian elimination in order to solve 3-XORSAT
I have a large system of $3$-$XORSAT$ constraints (i.e. up to $3$ variables per constraint) and this can be represented in matrix form as a linear algebra problem $Ax=b$ $mod$ $2$. Solvability (i.e. ...
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Proving the NP hardness of two variants of SAT
$k$-$\text{RSAT}$ is a variant of $k$-$\text{SAT}$ where we restrict our attention to formulae in
which each variable occurs at most $3$ times, and each literal occurs at most twice. The language
$k$-$...
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4-SAT but two literals per clause must be true
I'm trying to show that a modified 4-SAT in which at least two literals per clause must be true is NP-complete. I'll call it $4_2$-SAT. I understand the reduction from 3-SAT to 4-SAT, and I know why $...
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Proving 2SAT is in P vs algorithm for finding a satisfying assignment
I want to understand the proof in the following link that 2SAT is in P. What is the need for the last corollary? Wouldn't be enough to just prove the case for the graph with the help of the path ...
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How overapproximation of boolean abstraction function works?
I just found that if original formula is unsat, using boolean abstraction function can result in sat. Clearly this is overapproximate. I wonder if I have contingent formula, does that means my boolean ...
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What are efficient approaches to implement unit propagation in DPLL-based SAT solvers?
I'm trying to decompose deduction steps of DPLL algorithm -- unit propagation and pure literal elimination -- for parallelization. However, I want a baseline and asymptotic analysis to compare to my ...
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Find the flaw in the 3SAT solver algorithm
I consider decision version of 3SAT problem.
Main idea is to find congruent clauses and construct such maximum formula,
which satisfiability/truth table won't be changed.
In case of unsatisfiable ...
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3-SAT with atmost 3 variables and variable occuring once per clause
I've stumbled across this problem on CSES
https://cses.fi/345/task/E/
and was wondering is it somehow reducible to 2-SAT with given constraints?
So, the problem states that you need to solve a 3-SAT ...
