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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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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 ...
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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 ...
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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 ...
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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 ...
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35 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 ...
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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 ...
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119 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 $...
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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{...
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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) ...
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Maximum-minimum-satisfiability [closed]

In MAX-SAT, we are given a formula and want to maximize the number of satisfied clauses. I.e., given a formula $\phi = c_1 \cap \cdots \cap c_n$, where each $c_i$ is a disjunction, we want to find the ...
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Proof that MAX-2-SAT is NP-hard [duplicate]

According to Wikipedia, while the 2SAT problem is polynomial, its maximization variant MAX2SAT is NP-hard. But, they do not provide a reference for this claim. Is this obvious? If not, where can I ...
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NP-completeness and reduction of MAX-XOR-SAT and MAX-2-XOR-SAT

It is often stated that the MAX-XOR-SAT problem is NP-hard, and that likewise is the MAX-2-XOR-SAT problem. However, I cannot find a reduction from SAT to either of these problems, nor a proof of NP-...
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General structure of solutions to 3-SAT circuits

Certain special forms of the SAT problem have solution sets of a special form. For example, given any three solutions to a 2-SAT circuit, their bitwise median is also a solution. Likewise, given any ...
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Variance of MAXSAT clause satisfiability

For a given MAXSAT problem, it is trivially easy to compute the mean number of clauses satisfied for all assignments, or equivalently the expected number of clauses satisfied by a random assignment. ...
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2SAT Problem using Implication Graph

I was doing a practice question. As you can see below there is an Implication graph. To check whether the problem is satisfiable, I checked whether there were any 'bad loops'. To do so, for each ...
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Is there a way to convert a program into a Boolean formula?

Let's say I have a program P, in form of a binary code for x86 architecture. I want to find a Boolean formula F (in form of CNF, or something like that), such ...
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How many solutions are there for a XOR-SAT formula? [closed]

Does anyone know how many solutions there are for a XOR-SAT formula? And how do the variables in solutions distribute? For example, if (x0=1, x1=0, x2=1) is a solution for a XOR-SAT formula, how does ...
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what does it mean to extend an assignment?

For a constraint satisfaction problem, what does it mean for an assignment x to extend an assignment a? Sorry if this is super trivial, I did not find an answer e.g here: No Small Linear ...
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49 views

Given a NP Algorithm for SAT, do we expect to have Correct and Incorrect Solutions?

I am reading about Boolean Satisfiability Problem and Nondeterministic Algorithms, in the latter defination it says : In computational complexity theory, nondeterministic algorithms are ones that, ...
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43 views

Number of X3SAT Instances?

Exactly 1 in 3SAT (X3SAT) is known to be NP-Complete. It remains NP-Complete even if we only consider instances that are monotone and linear. Monotone means all of the literals are positive. Linear ...
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What can be a Zero Knowledge Proof of a working SAT Algorithm?

Me and my colleague are exploring new ideas to solve SAT efficiently (i.e. in polynomial time) and it's the case that there is a candidate algorithm. Unfortunately, neither of us can write scripts ...
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SAT for positive CNF clauses with exactly half of the variables being true

I am focusing here on positive even SAT problems, that is a CNF for which all literals are positive, and in which an even number n of variables occur. This is obviously trivial : just set all ...
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Why SAT Requires A Non-determinstic Algorithm?

I am getting started to understand the probelm of Satisfiability and i am reading (Computers and Intractability: A Guide to the Theory of NP-Completeness). I do understand the difference between a ...
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Alternate reduction from 3SAT to 4SAT?

It seems that the standard reduction method you see online from 3SAT to 4SAT is that we let $\phi = (a \lor b \lor c)$ be a 3SAT clause, and so there is an assignment that satisfies $\phi$ iff $\phi' =...
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Proof of the Cook-Levin Theorem - snapshot transitions

I'm trying to understand the proof of the Cook-Levin thereom in Aurora and Barak's "Computational Complexity" text. A snapshot $z_i$ of $M$’s execution on some input $y$ at a particular step $i$ is ...
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101 views

P=NP giving a deterministic algorithm for SAT

I'm trying to prove the following problem: Prove that if $P=NP$ then there is a polynomial time algorithm for the following problem: INPUT: A boolean formula $\phi$. OUTPUT: A satisfying ...
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260 views

Proving special case of SAT is in P

Let SAT-100 be the following problem: Input: Any boolean logic formula Output: True if there exists a combination of exactly 100 input variables that satisfy the formula. This is the description of ...
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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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Is any sudoku solver an SAT solver?

I have recently created a sudoku solver using C#, which outputs the solution to a sudoku after a reasonable amount of time in many cases. I have used the basic sudoku SAT-reduction (i.e. x111 meaning ...
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54 views

When is a 1-in-3 SAT clause satisfied?

How does exactly 1 in 3 sat work given the variables Xi, Xy, Xz if one of the variables in the formula are negative. We know that the results are if they are all positive given that: R(Xi, Xy, Xz) = ...
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General Understanding of SMT Solving Across Multiple Theories

My first question was a little too simplified in that it turned out to be an integer linear programming problem solvable with the Simplex Method. However, what I think I am wondering about is how to ...
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How the Abstract DPLL Algorithm Works in SAT Solving

I have come across many definitions of the DPLL algorithm but haven't been able to follow them. The ones that are closest to making sense to me are the ones based on state-transition systems with ...
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How a SMT / SAT Solver Generates Valuations for this Example

First, an example of a set of constraints which turned out not to be solvable (I don't think): ...
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Circuit satisfiability problem : SAT-C to SAT-2C

I have the following language : $L=\{\langle C_1,C_2\rangle \text{ | } C_1 \text{ and } C_2 \text{ are two circuits that calculate different function}\}$. We can call this language SAT-2C. Prove that ...
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On $\Sigma_kSAT$ algorithms from $\Sigma_{k'}SAT$ where $k'<k$

Define $\Sigma_kSAT$ by 'Given a quantified boolean formula $φ=∃y_1∀y_2\dots Q_ky_k ϕ(y_1,\dots,y_k)$ where $ϕ(y_1,\dots,y_k)$ is boolean predicate with each $y_i$ a vector of variables, $Q_{2j−1}=∃$ ...
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What are known 3SAT to 2SAT reductions?

Is there a way to convert a 3SAT formula into a equisatisfiable 2SAT formula? Each method is of interest, even those that grow exponentially. (So if, for example, my 3SAT formula has 16 variables and ...
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USAT, Arora Barak's book

Here on the page 354 Arora and Barak write below the shaded area "but in fact $f(\phi)$ $\notin SAT$" and not "but in fact $f(\phi) \in SAT$" While in the last line of the shaded area they write $...
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Is my logic correct and is this a new reduction and algorithm from 3 SAT to clique?

Is my logic correct? If so, is this a new reduction and algorithm from 3 SAT to clique? I could only find one SAT to clique reduction; it wasn't this. Definitions: A clause group of a SAT instance ...
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38 views

Lower upper bound on # variables in k-SAT with m clauses

If I have an instance of $k$-SAT with $m$ clauses, then a trivial upper bound on the number of variables $n$ is given by $n \leq mk$. But we can only have $n = mk$ when no variable is repeated and ...
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282 views

Reduce Max-Cut to Max-2SAT

I would like to find a reduction from Max-Cut to Max-2Sat. Could someone shed light on this problem, preferably spiced with some intuition? Thanks, Matan.
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Is there an algorithm to find out if a truthtable can be represented as 2-sat, and if so find its 2-cnf?

I know that not all truthtables have a corresponding 2-cnf representation, but is there a way to find out if a given truthtable has a 2-cnf representation, and if so to find what that is?
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NP-completeness of vertex cover

Show that the following language is NP-complete $$ L = \{ \langle G,k \rangle \mid \text{$G$ is a graph with a set $S$ of $k$ vertices hitting every edge of $G$}\}. $$ I know I should reduce the ...
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594 views

To prove 4-SAT CNF is NP-complete [closed]

To answer the question below, 4-SAT: Given a formula in Conjunctive Normal Form, where each clause contains exactly 4 literals, does it have a satisfying truth assignment? I was going to prove that ...
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$SAT$ and $BPP=P$ conjecture?

$BPP$ is the complexity class that accepts all languages for which there is Poly time TM with at least $1/3$ of their computation paths accept and rejects all languages for which there is Poly time TM ...
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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?
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Reducing INDSET and MAXCUT to 3SAT

Given a graph and an integer $k$ is there an independent set larger than $k$ is INDSET problem and is there a cut larger that $k$ is the MAXCUT problem. Is there standard way to convert to 3SAT from ...
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When using resolution variable elimination to simplify a cnf, does that change the truth values of the other variables?

When you use resolution variable elimination to preprocess/simplify a formula in cnf form the resulting formula is equisatisfiable. What I wonder about is if I can use this technique to remove ...
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Counting (enumerating) minimal solutions of a dual horn formula

Is there an efficient algorithm ("does not necessarily have to be a polynomial time algorithm") to compute all "minimal" solutions for a Dual Horn formula (conjunction of clauses where each clause ...
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165 views

3-SAT reduction to jobs scheduling problem (np-completeness)

In this paper with the title of "NP-Complete Scheduling Problem" by J. D. Ullman, I am trying to understand the reduction from 3-SAT problem to a scheduling problem in order to prove the later is also ...
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Counting solutions of a particular type in HORN SAT

I am interested in counting the number of solutions of a particular type (say #) in HORN SAT. I have 2 questions concerning the same. Suppose we have a HORN SAT -: $(x_1) \land (x_2 \implies x_1)$, ...