Questions tagged [sat]

SAT stands for the Boolean satisfiability problem.

Filter by
Sorted by
Tagged with
8
votes
2answers
186 views

Are there analogous works to PPSZ algorithm for #P?

The PPSZ algorithm tells us that we can do SAT-solving for $k-$CNF in time at-most $2^{1-(1-o(1))\frac{\pi^2}{6k}}$. My question is that do we know such results for counting problems in class #P too ? ...
18
votes
2answers
828 views

Lower bounds on #SAT?

The problem #SAT is the canonical #P-complete problem. It's a function problem rather than a decision problem. It asks, given a boolean formula $F$ in propositional logic, how many satisfying ...
0
votes
0answers
58 views

Asymptotically sub-optimal but provably optimal algorithms in a finite range for NP-Hard problems?

Given a formula in propositional logic with $100$ variables. We know that there can be no SAT-checking provably faster than $O(2^n)$, unless P=NP. Now let's say I could find an algorithm which does ...
0
votes
1answer
130 views

Comparing SAT to MCSP reduction class separations and faster SAT class separations?

Assume $SAT$ is in $QuasiP$. We immediately infer $NQuasiP=QuasiP$ and $EXP=NEXP$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $NQuasiP$ is not in $P/poly$ which implies $...
1
vote
0answers
22 views

Clauses structure as quenched random matrix for random $k$-SAT problems

In the Section III of Statistical mechanics of the random K-SAT model, the clauses structure of random $k$-SAT problems are expressed as $M \times N$ quenched random matrix where the numbers of ...
9
votes
2answers
331 views

NP-hardness of a planar SAT variant

Background: An instance of 3-SAT is called monotone if each clause consists only of positive literals or only of negative literals. Given an instance $\phi$ of 3-SAT, we consider the bipartite graph $...
6
votes
3answers
395 views

Can modern SAT-Solvers utilise the symmetry of First Order Logic?

Apologies if the question is trivial or is wrongly stated, I am a Physicist! Assuming that we have a universally quantified first-order logic sentence, all variables are universally quantified, ...
15
votes
2answers
2k views

Random 3-SAT: What is the consensus experimental range of the threshold?

The critical ratio of clauses to variables for random 3-SAT is more than 3 and less than 6, and seems to be commonly described as "around 4.2" or "around 4.25". Mezard, Parisi, and Zecchina prove (in ...
44
votes
0answers
1k views

Problem unsolvable in $2^{o(n)}$ on inputs with $n$ bits, assuming ETH?

If we assume the Exponential-Time Hypothesis, then there is no $2^{o(n)}$ algorithm for $n$-variable 3-SAT, and many other natural problems, such as 3-COLORING on graphs with $n$ vertices. Notice ...
1
vote
1answer
100 views

Complexity of a satisfiability problem

I would like the know the complexity of a specific satisfiability problem. I have a feeling it could be solved in polynomial time, but I am not sure about it. The problem is described below. Given $n$ ...
2
votes
1answer
110 views

Complexity of MAX-ONEs Monotone 2-SAT with $n^{3/2}$ or $C n^2$ clauses?

Let $\phi$ be negative monotone 2-CNF on $n$ variables and $n^{3/2}$ clauses. What is the complexity of finding satisfying assignment with maximum number of ones $k$? Alternatively let $G$ be a graph ...
8
votes
1answer
309 views

On $\text{ETH}$ with $m$ as parameter: consequences of algorithm running in time $2^{\delta m}$ where $\delta \to 0$ as $k \to \infty$

It has been shown in [1] that $k\text{-SAT}$ has a $2^{o(n)}$ algorithm if and only if it has a $2^{o(m)}$ algorithm, $n$ being the number of variables and $m$ being the number of clauses. Being $s_k=\...
3
votes
1answer
71 views

Generating hard satisfiability problems with given constraint graph

Is there a systematic way to tune the hardness of a set of satisfiability problems (say 3-SAT or MAX2SAT) where the constraint graphs are always embeddable into a fixed given graph?
7
votes
1answer
307 views

Is solving the following system of boolean equations NP-hard?

I reduced a problem I'm currently working on to the following system of boolean equations: $$ X_i \iff \begin{cases} \bigvee_{B \in A_i} \bigwedge_{k \in B} X_k \\ true \\ false \end{cases} $$ Where $|...
2
votes
0answers
129 views

Can a tractable sequence distribution prefer satisfying to unsatisfying assignments?

Let $p(x)$ be a Boolean circuit on $n$ bits $x \in \{0,1\}^n$. Consider a program that computes a probability distribution over all sequences $x$, autoregressively factored as $$\pi(x | p) = \prod_{0 ...
10
votes
1answer
216 views

The number of clauses in an unsatisfiable CNF

I am interested in generalisations of the following observation: An unsatisfiable $k$-CNF has at least $2^k$ clauses. A special case of the observation is when $k=n$, where $n$ is the number of ...
1
vote
1answer
76 views

Common solutions to 3SAT and 2SAT models comprised of the same variables

I have a problem which is a combination of 3SAT & 2SAT instances. Consider $L$ is a set of variables $(x_1 ... x_n)$. $S_3(L)$ is a 3-SAT instance and $S_2(L)$ is a 2SAT instance, both made of ...
2
votes
0answers
68 views

SAT solvers & SAT solving methods admitting minimal satisfying assignments

I'm in need of a SAT solver that outputs minimal assignments - that is, a set of assignments to a subset of all propositional variables such that substitution into the CNF formula makes each clause ...
4
votes
1answer
234 views

3-SAT mixed with 2-SAT formulas

Context: Refering to the question: Complexity of the $(3,2)_s$ SAT problem? and since the paper by Porshen and Speckenmayer : Satisfiability of mixed Horn formulas, we know that even when $F_3$ is ...
3
votes
1answer
59 views

Translating pure literal elimination into rup

I'm exploring building a CDCL SAT-solver with interesting reduction rules. I have two rules based on pure literal elimination, but if either of these rules generate a conflict, I don't know what to ...
0
votes
0answers
75 views

Hardness of approximation by reduction from MAX-E3SAT

For the MAX-E3SAT define $k=\sum_i^n (g_i-1)$, where $n$ is the number of variables and $g_i=\text{max}(\{\text{occurrences of }x_i, \text{occurrences of }\neg x_i\}), \text{ for each }i=\{1,\ldots,n\}...
4
votes
0answers
111 views

On solving Planar Circuit SAT

This enquiry is three-sided. Side 1 - State of the art Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$? Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$ assuming ...
1
vote
0answers
221 views

Disproving $\oplus$ETH by reducing $\oplus k$-SAT with $n$ variables and $m$ clauses to planar graph with $o(m^2)$ vertices?

In this question and its answer, they discuss about reducing CNF-SAT with $n$ variables and $m$ clauses to a (problem on) planar graph $G=(V,E)$ with $|V|$ as small as possible. It is said that the ...
33
votes
5answers
5k views

Fast Reduction from RSA to SAT

Scott Aaronson's blog post today gave a list of interesting open problems/tasks in complexity. One in particular caught my attention: Build a public library of 3SAT instances, with as few variables ...
3
votes
1answer
126 views

Isomorphism preserving transformation CNF to Graph?

In short we are interested in isomorphism preserving transformation CNF to Graph. Let $\phi_1,\phi_2$ be CNF formulas. Define $\phi_1$ and $\phi_2$ to be isomorphic $\phi_1 \cong \phi_2$ if there ...
32
votes
3answers
2k views

Is this variation of TQBF still PSPACE-complete?

Deciding if a quantified boolean formula such as $\forall x_1 \exists x_2 \forall x_3\cdots \exists x_n \varphi(x_1, x_2,\ldots , x_n),$ always evaluates to true is a classical PSPACE-complete ...
-4
votes
1answer
132 views

Is there an algorithm for 3x3 sudokus without backtracking? [closed]

From what I can see on Wikipedia and the Internet at large, all sudoku solving algorithms (including human ones) employ some kind of EXPTIME backtracking search for some sudokus. Are there any SAT ...
20
votes
5answers
28k views

Direct SAT to 3-SAT reduction

Here the goal is to reduce an arbitrary SAT problem to 3-SAT in polynomial time using the fewest number of clauses and variables. My question is motivated by curiosity. Less formally, I would like ...
1
vote
0answers
83 views

At most how many satisfying assignments are there for a 2SAT with n variables?

It is not obvious but easy to see that, for some fixed set of satisfying assignments, there is no 2CNF that can satisfy the set of satisfying assignment exactly, when I discover this, I wonder at most ...
0
votes
0answers
20 views

Smallest/largest number of clauses, of the formula containing them being exact kCNF&there're r(or just r=1) satisfying assignment(s) with n variables?

When I try analysing the property of kCNF. I meet a seemingly inessential problem that is more likely to be a combinatorial math problem. What is the smallest/largest number of clauses where the ...
4
votes
2answers
183 views

What are those deterministic algorithms for k-SAT that are not derandomization of random algorithms like PPSZ and Schöning's local search?

I am doing a survey on k-SAT where time complexity is in terms of n, i.e. the number of variables in a formula. As for the fast algorithms for k-SAT, we see biased-PPSZ, PPSZ, Schöning's local search,...
47
votes
9answers
8k views

Best Upper Bounds on SAT

In another thread, Joe Fitzsimons asked about "the best current lower bounds on 3SAT." I'd like to go the other way: What's the best current upper bounds on 3SAT? In other words, what is the time ...
1
vote
1answer
294 views

Potentially stronger form of non-$ETH$

If we have a $2^{n^a}$ algorithm to $K$-$SAT$ where $a<1$ for all $K>2$ then $ETH$ fails and literature gives consequences. What are the consequences if $a=o(1)$?
23
votes
3answers
4k views

Minimum Unsatisfiable 3-CNF Formulae

I am currently interested in obtaining (or constructing) and studying 3-CNF formulae which are unsatisfiable, and are of minimum size. That is, they must consist of as few clauses (m = 8 preferably) ...
3
votes
2answers
124 views

Understanding non-equivalence of proof lengths according to proof systems

Here, in section 4.3, Fortnow says: But to prove P != NP we would need to show that tautologies cannot have short proofs in an arbitrary proof system. I am ...
14
votes
3answers
1k views

Checking formulas with two quantifiers ($\forall \exists$) - 2QBF

SAT solvers give a powerful way to check the validity of a boolean formula with one quantifier. For instance, to check the validity of $\exists x . \varphi(x)$, we can use a SAT solver to determine ...
12
votes
1answer
909 views

Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
1
vote
1answer
152 views

Reducing 3-XOR-SAT to HORN-SAT

In this question - XOR-SAT to Horn-SAT reduction, two algorithms are described for reducing any XOR-SAT formula to a HORN-SAT formula. My question is: say I limit the clauses of an XOR-SAT formula to ...
9
votes
1answer
213 views

What is the complexity of checking equivalence of two boolean formulae without NOT symbol?

Suppose I have two boolean formulae (propositions) $P_1$, and $P_2$ (can be assumed to be in CNF) over the same variables and such that there are no "NOT" symbols used. I.e. only conjunction and ...
6
votes
1answer
129 views

SMT solving with less-than theory and monotonic functions

I am attempting to solve a less-than theory within an SMT paradigm that involves variables assigned to reals and assumes that all the functions used in the theory are monotonic. The theory's signature ...
14
votes
1answer
636 views

Consequences of sub-exponential proofs/algorithms for SAT

Would there be any major consequences if SAT had at most subexponential unsat proofs or even more strongly, SAT had subexponential-time algorithms?
5
votes
1answer
244 views

What are the consequences of a faster algorithm for $CIRCUIT$-$SAT$?

What is the best algorithm known for $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates? What is the consequence if there is an $\alpha\in(0,1)$ such that $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates ...
1
vote
1answer
624 views

What languages can be reduced to a NP-complete problem in polynomial time

NP-complete: Language is NP-complete, when it is in NP and every problem in NP is reducible to it in polynomial time. But what languages are reducible to a NP-complete problem (for example SAT) in ...
9
votes
1answer
315 views

Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses?

We think that $\mathsf{PH}$ does not collapse, and that $\mathsf{PP}$ is not in $\mathsf{P}$. Suppose on the contrary that $\mathsf{PH}$ does collapse, say even $\mathsf{P}= \mathsf{NP}$. $\mathsf{...
4
votes
1answer
260 views

Complexity of SAT parameterized by treewidth

Many papers state that Boolean satisfiability is in FPT when parameterized by primal, dual, or incidence treewidth. What are the best known time complexities of these parameterized algorithms? In ...
8
votes
1answer
365 views

On theoretical aproaches for solving $\mathsf{SAT}$ in special cases

In what cases $\mathsf{SAT}$ can be solved in polynomial time? I know two cases: $2$-$\mathsf{SAT}$ and Horn-$\mathsf{SAT}$. Question 1: Is there a reference with algorithms for solving $\mathsf{...
21
votes
10answers
3k views

#SAT Solver download

Could anyone please point to one or more websites where is possible to download a working implementation of a #SAT solver? I'm interested in those returning the exact solution count, not an ...
15
votes
0answers
240 views

Does small circuits for a NP-complete problem contradict ETH?

The remarks of the Theorem 4 in the paper "On the complexity of circuit satisfiability" claims that: if circuit satisfiability (CktSat) problem can be decided by deterministic circuits of $2^{o(n)}$ ...
6
votes
0answers
271 views

Reverse Skolemization?

I'm wondering if there are any references on "reverse skolemization", that is, converting a formula with functions into one purely consisting of quantifiers by eliminating function applications. I'm ...
5
votes
0answers
52 views

Hard family for degree-$D$ MAX-3LIN

Max 3LIN is the problem where one is given a linear system of equations $C$ over $\mathbb{F}_2$ with $3$ variables per equation, and needs to determine the maximum number of equations that can be ...

1
2 3 4 5 6