Questions tagged [satisfiability]

Satisfiability (SAT) is the problem of determining whether there is a variable assignment that fulfills a given Boolean formula.

Filter by
Sorted by
Tagged with
2
votes
1answer
31 views

CLIQUE $\leq_p$ SAT

i'm trying to reduce CLIQUE to SAT: Given: Graph G=(Vertices V, Edges E) and $k \in \mathbb{N}$ Output: Formular F such that if G contains a complete subgraph of size k, the formular is satisfiable (...
0
votes
1answer
22 views

A doubt on converting NOT gate to CNF formula

For a NOT gate if $x_1$ is input and $x_2$ is the corresponding output, I see the equivalent CNF (conjunctive normal form) is $(x_1 \lor x_2) \land (\overline x_1 \lor \overline x_2)$. My ...
2
votes
0answers
111 views

Second Order QBF

Consider a universe with two elements 0,1 and a second order formula, i.e. of the form "forall R exists S ... such that F", where R,S are relation symbols of some given arity, and F is some first ...
0
votes
1answer
27 views

Encoding a “not-k-out-of-n” constraint for SAT solvers

I'm finding myself needing to encode a "not-k-out-of-n" constraint in a SAT solver. The "at-most-k-out-of-n" constraint for SAT solvers is something I can find research about -- this paper by Frisch ...
3
votes
0answers
42 views

Determine if a graph has exactly 1 cycle using a SAT solver

I have a connected undirected graph whose edges are either enabled or disabled. I want to create a set of clauses that are SAT iff all enabled edges are part of a single loop. If I assert that each ...
5
votes
1answer
277 views

SAT algorithm for determining if a graph is disjoint

What are some good algorithms to have a SAT (CNF) solver determine if a given graph is fully-connected or disjoint? The best one I can think of is this: Number the nodes 1..N, where N is the number ...
0
votes
2answers
87 views

Reducing Graph Reachability to SAT (CNF)

So I came across this problem in my textbook. I was wondering how to develop a reduction from the Graph Reachability problem to SAT (CNF) problem. (i.e. formula is satisfiable iff there exists a path ...
2
votes
2answers
56 views

Prove that this language is NP-Hard

Given $$\mathrm{\#3SAT} = \{ (w, y) \mid w\text{ is a $\mathrm{3SAT}$ instance with at least $y$ satisfying assignments}\}\,,$$ prove that $\mathrm{\#3SAT}$ is NP-Hard. I am currently stuck with ...
4
votes
0answers
72 views

What is the generating algorithm for the “komb” instances found on satcompetition.org?

For the 2017 and 2018 Random SAT Tracks of the SAT Competition ran by the International Conference on Theory and Applications of Satisfiability Testing there are small, yet difficult, random 3-SAT ...
4
votes
2answers
54 views

Can every Turing Machine be translated into a SAT formula?

For the proof of "Cook-Levin Theorem", for a Turing Machine $M$ that accepts a language $L \in NP$ and input $x \in \{0,1\}^*$, we can create a SAT Formula, that is satisfiable if and only if $M$ ...
3
votes
0answers
30 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 ...
1
vote
2answers
106 views

Reduce 4-SAT to 5-SAT

given is a reduction from 4-SAT to 5-SAT. How is it possible to describe such a function? I found some informations about reduction 3-SAT to 4-SAT here, but it can't help me so much.
0
votes
0answers
28 views

Is retrospective inference NP-hard?

Here is a minimal working example of the question: Consider a network with nodes arranged in a pyramid: $1$ node in the first row, $1+d$ nodes in the second, $1+2d$ nodes in the third, and so on, ...
2
votes
0answers
41 views

Is a “stacked”, “local” version of 3-SAT NP-hard?

In this previous question, I learned that if each variable in a string $C \in 3\text{-SAT}$ appears only "locally", then finding a satisfying assignment is no longer NP-hard. My question below builds ...
2
votes
1answer
33 views

Translation of diagnosis problem to SAT

I have the following diagnosis problem: h(A): z1 = not(in1) h(D): z2 = not(in2) h(B): z3 = z1 or z2 h(C): out1 = not(z3) h(E): out2 = not(z3) This is an image of the system: I have an observation ...
7
votes
1answer
400 views

Is a “local” version of 3-SAT NP-hard?

Below is my simplification of part of a larger research project on spatial Bayesian networks: Say a variable is "$k$-local" in a string $C \in 3\text{-CNF}$ if there are fewer than $k$ clauses ...
3
votes
1answer
34 views

Finding a language that is $NP^L$-complete

I'm trying to prove a theorem and as a lemma I would like to identify an $NP^L$-complete language. I was thinking something like a machine that can decide $SAT$ equipped with an oracle for $L$ can ...
4
votes
3answers
388 views

Showing that HALF-2-SAT is in P

I need to show that the following problem is in P: $$\begin{align*}\text{HALF-2-SAT} = \{ \langle \varphi \rangle \mid \, &\text{$\varphi$ is a 2-CNF formula and there exists an assignment} \\ &...
2
votes
1answer
23 views

Satisfiability Toward A Sequential Circuit

Define a sequential circuit model be a directed graph with each vertices being a boolean gate. The difference is that we allow cycles in the boolean circuit. Each cycle will determine a boolean ...
4
votes
1answer
47 views

MAXSAT approximation

We have been studying a 1/2-approximation for MAXSAT which runs in expected polynomial time, by randomly assigning True/False to each variable and repeating until we reach an assignment with at least ...
0
votes
0answers
22 views

What would be a good way to prove NP-Completeness of Min-satisfiablity problem

I am trying to prove the NP-Completeness of Min-satisfiability problem which can be defined as: Given a CNF formula (Set of clauses of 1 or more boolean variables) and a number x, there exists a ...
1
vote
1answer
74 views

3-DNF proves the algorithm is in P class

To understand fully, please read link After, reading the link we will take a look at how we recover our solutions to a constrained Sudoku Puzzle. If we assume that a sudoku puzzle was generated with ...
2
votes
2answers
93 views

Proof that POSITIVE-3-SAT is in the complexity class P

I have the following language: $$\text{POSITIVE-3-SAT} = \{\langle\phi\rangle \mid \phi\text{ is a satisfiable boolean formula in conjunctive normal form,}\\ \text{ in which all clauses consist of ...
8
votes
2answers
2k views

Is “Reachable Object” really an NP-complete problem?

I was reading this paper where the authors explain Theorem 1, which states "Reachable Object" (as defined in the paper) is NP-complete. However, they prove the reduction only in one direction, i.e. ...
1
vote
0answers
128 views

Can someone give me the resolution procedure for 2 SAT, which is O (nˆ2)

According to Wikipedia, Even, S .; Itai, A .; Shamir, A. cited it in "On the complexity of time table and multi-commodity flow problems". The paper can be found here: http://www.cs.technion.ac.il/...
0
votes
1answer
38 views

Resolution when clauses contain more than 1 complementary literals

Let's assume that we have clauses $(l_1 \lor l_2 \lor l_3), (\neg l_1 \lor \neg l_2 \lor l_4), (l_1 \lor l_2 \lor l_5), (\neg l_1 \lor \neg l_2 \lor l_6)$, where both $l_1$ and $l_2$ are complementary ...
0
votes
1answer
23 views

Can someone give me the definition of #Monotone-2SAT?

In the decision problem, I set all variables to true and see if the formula is satisfiable. My question is because I do not understand how there can be multiple solutions, though all variables are ...
1
vote
0answers
59 views

The NP completeness proof for a variation of the 3CNF-SAT problem

There is a variation of 3CNF-SAT which is called 10-3-CNF-SAT = {<$\Phi$>: $\Phi$ is a satisfiable CNF formula with $\textbf{at most}$ 3 literals per clause and every variable occurs in $\textbf{at ...
0
votes
2answers
76 views

Why is Boolean satisfiability such a rare case?

In the space of all K-sat formulas, True and False should have an equal set size. For every un-Satisfiable formula (F), there will an F' (or F-prime) which will be Satisfiable by definition. I cannot ...
3
votes
2answers
88 views

Is every X3SAT instance with no cycles satisfiable?

Exactly 1 in 3 SAT (X3SAT) is a variation of the Boolean Satisfiability problem. Given a set of clauses, where each clause has three literals, is there an assignment such that in each clause exactly ...
1
vote
1answer
65 views

Counting models satisfying a boolean formula

I'm trying to implement the #2-SAT algorithm from the paper "Counting Satisfying Assignments in 2-SAT and 3-SAT" (Dahllöf, Jonsson and Wahlström, Theor. Comput. Sci. 332(1–3):265–...
1
vote
2answers
39 views

Can problems of class P be transformed into SAT formula?

I know that NP problems can be transformed into SAT (due to this fact, SAT is considered to be a NP Complete problem). I am not sure if P problems can too be transformed into SAT.
0
votes
1answer
39 views

Fine-grained complexity of 3-CNF formula evaluation

It's well known that 3-SAT is in NP, which means that one can evaluate a 3-CNF formula in polynomial time. However, I was wondering what the tightest upper bound is for formula verification, expressed ...
2
votes
2answers
41 views

Calculating the number of assignments satisfying a general propositional formula

I know, for a disjunctive clause of the form $x_1 \vee ... \vee x_i$, the number of assignments satisfying it is simply $2^i - 1$, but what about for a general formula? Is the number of satisfying ...
1
vote
1answer
16 views

For a given k-DNF formula, what is the size of the formula for the purpose of complexity?

I am currently in the process of proving some complexity bounds about k-DNF. However I am confused what the $n$ in the time complexity would refer to in this case (that is that I don't know how the "...
1
vote
0answers
42 views

Can I use the Quine-McCluskey to simplify a CNF which is not a product of maxterms?

As I understand it the Quine-McCluskey method allows you to start with a set of maxterms (or minterms), and combine them pairwise in a systematic way into a smaller set of clauses with a smaller set ...
2
votes
1answer
39 views

Refutation in first order logic

Consider the following statement In FOL, we can reduce entailment checking to satisfiability checking: $S \models S' \iff S \land \neg S'$ is satisfiable (This proof strategy is called ...
1
vote
1answer
83 views

Why Cook-Levin thorem's proof can mean SAT's NP-Hardness

I'm studying about Cook-Levin theorem but there is a problem I faced. Cook-Levin theorem shows that any NPTM can be encoded as a boolean formula. About given language $A$, instance $w$, and NPTM $M$ ...
3
votes
0answers
53 views

relationship between SAT and Min-ones SAT

If SAT can be decided in polynomial time, is it clear that Min-ones SAT can be decided in polynomial time? The idea I had was to take a poly decider of SAT and try it on a formula OR'd with all ...
0
votes
2answers
94 views

Seeking nontrivial small SAT/UNSAT instances

I need SAT instances, involving 9 to 20 variables. They need to be hard to solve for humans. Both SAT and UNSAT instances are needed. I tried random-SAT generators on the web, but the results were ...
1
vote
0answers
188 views

Could you show the intractibility of SAT by showing that the number of variables contributing to an arbitrary unsatisfied clause is not constant? [closed]

Preface: This is not an attempted proof at P vs NP Starting with some CNF Boolean expression ϕ, by the rules of logical disjunction, a clause is only unsatisfied if each of the literals in it are ...
1
vote
1answer
93 views

Reducing 3SAT to MAX-3SAT

I have the following problem: Consider the MAX-3-SAT problem: given a Boolean function in Conjunctive Normal Form (CNF) determine the maximum number of clauses that can be satisfied. Prove that ...
0
votes
1answer
44 views

n-DNF boolean formula k satisfiability

Given an n variable boolean DNF formula and a number k, does this formula has satisfying input combination greater than k?. (0<=k<=2^n). Where input is infinite number of n tuples where ...
1
vote
0answers
150 views

Is the solution to Independent Set or Vertex Cover from 3-SAT optimum?

There are plenty of resources online discussing 3-SAT reductions to Independent Set or Vertex Cover problem. I am unable to find a resource which states that a satisfiable assignment to 3-SAT results ...
1
vote
0answers
34 views

3-SAT wher each literal appears at most once [duplicate]

I'm currently following a course and we have to prove that a restricted version of the 3-SAT decision problem where each literal appears at most once is solveable in polynomial time. I think such a ...
1
vote
1answer
48 views

Why not do these checks on the number of clauses in 3-SAT?

I've been writing a 3-SAT solver for fun and comparing its performance against the solver pycosat. My solver vastly outperforms pycosat in two special cases, where I solve by doing simple, obvious ...
1
vote
2answers
60 views

Why is Max SAT in P if SAT in P?

It holds that if SAT could be solved in poly time, one can also find in poly time the assignment that satisfies most clauses of the original formula. Does anyone have any idea how to show this? Let's ...
2
votes
1answer
127 views

Directed HAM Cycles with Additional Constraints to SAT

The $n$ dimensional hypercube $Q_n$ is a graph that has a vertex $v_s$ for each string $s \in \{0, 1\}^n$ and an edge between two vertices $v_s$ and $v_t$ if and only if the Hamming distance between $...
3
votes
1answer
469 views

In SAT, do we require an assignment for arbitrary variables?

I am reading about the Satisfiability Problem, in page (5) the author gives the following example : $(P \lor Q \lor R) \wedge (\bar{P} \lor Q \lor \bar{R}) \wedge (P \lor \bar{Q} \lor S) \wedge (\bar{...
0
votes
1answer
39 views

Sat instance size and definition of TIME(f(n))

Sat usually is defined as the language of a 'reasonable' encoding of satisfable Cnf formulas over n variables. Question: a Cnf formula over n variable with m clauses has a size (as a function of n) ...