Work fast with our official CLI. Learn more. If nothing happens, download GitHub Desktop and try again. If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again.

It is released under the MIT license. Gophersat is rather efficient, i. Gophersat's last stable version is version 1. Gophersat version 1. A vew bugs were also corected, and the support for incremental SAT solving was improved. The certificate will then be printed on the standard output, using the RUP notation. The certificate is generated on the fly, so be aware that a partial, useless certificate will be generated even if the problem is actually satisfiable.

This is common practice in the community, and although the generated clauses are useless noise, in practice this is not a problem. The MUS will the be printed on the standard output. For the moment, these facilities are only available for pure SAT problems i. Since its version 1.

It also includes a new API for optimization and model counting, where new models are written to channels as soon as they are found. Since version 1. In other words, if your program works with version 1. Note that, by "still working", we only mean "will compile and produce the same output", not "will have the same performance memory-wise or time-wise". This is an important distinction: during minor version upgrades, new heuristics or data types can be introduced, meaning some given problems could be solved faster or slower than previously.

Gophersat is also able to read and solve more general boolean formulas, not only problems represented in the user-unfriendly DIMACS format. It also deals natively with cardinality constraints, i. Gophersat can be used as a standalone solver reading OPB files or as a library in any go program.

To solve a pseudo-boolean problem whether a decision one or an optimisation oneyou can call gophersat with the following syntax:. In a few words, a SAT solver tries to find, for a given propositional formula, an assignment for all its variables that makes it true, if such an assignment exists.

Gophersat implements state-of-the-art features and is thus quite efficient, making it usable in practice in Go programs that need an efficient inference engine.

Although this is not always the best decision for practical reasons, any NP-complete problem can be translated as a SAT problem and solved by Gophersat. Such problems include the Traveling Salesman Problem, constraint programming, Knapsack problem, planning, model checking, software correctness checking, etc.

You can also find information about how to represent your own boolean formulas so they can be used by gophersat in the tutorial "SAT for noobs".

Partial MAXSAT means that, although we want to satisfy as many clauses as possible, some clauses called hard clauses must be satisfied, not matter what. For instance, generating a timetable for a school is a partial MAXSAT problem: there are both soft we want to have as little classes as possible that start after 4 PM, for instance and hard two teachers cannot be in two different places at the same time constraints. Weighted MAXSAT means that clauses are associated with a cost: although optional, some clauses are deemed more important than others.

For instance, if clause C1 has a cost of 3 and clauses C2 and C3 both have a cost of 1, a solution satisfying C1 but neither C2 nor C3 will be considered better than a solution satisfying both C2 and C3 but not C1, all other things being equal. Partial weighted MAXSAT means that there are both soft and hard clauses in the problem, and soft clauses are weighted. Pseudo-boolean problems are, in a way, a generalization of SAT problems: any propositional clause can be written as a single pseudo-boolean constraint, but representing a pseudo-boolean constraint can require an exponential number of propositional clauses.

This is equivalent to the propositional formula. The description of a pseudo-boolean problem can be exponentially smaller than its propositional counterpart, but solving a psudo-boolean problem is still NP-complete. Gophersat solves both decision problems is there a solution at all to the problemand optimization problem.In this post, we'll look at how to teach computers to solve puzzles.

Specifically, we'll look at a simple puzzle that can be expressed as a boolean constraint satisfaction problem, and we'll write a simple constraint solver a SAT solver and mention how our algorithm, when augmented with a few optimizations, is used in modern SAT solvers.

The only requisite knowledge is a basic understanding of algorithms and the ability to read Haskell code.

In future posts, we'll extend our puzzle solving abilities beyond boolean constratint satisfaction by writing an SMT solver; after that, we'll look at how these puzzle solving algorithms can be used for software verification. This is an exported IHaskell notebook, and you can download the original to play around with the code.

Let's start by jumping in with an example of a constraint satisfaction problem. Suppose that you need to go grocery shopping, and need to visit three stores: Costco, Home Depot, and Walmart. Costco is open in the morning and evening, Home Depot is open in the evening only, and Walmart is open in the morning only. You can only be in one place at a time, and shopping at a given store takes up the entire morning or evening. Can you go to all three stores in a day?

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To a human, it is intuitively obvious that the answer is no. Since Home Depot and Walmart offer us only one time option evening and morning, respectivelythen we have to go there at those times. However, this leaves no time for a Costco trip, so it's evident that this "puzzle" has no solution.

Now suppose instead of three stores, you were given three thousand each with its own scheduleand instead of two times, you were given all the hours of a day? At this point, the problem becomes intractable for a human. Luckily, though, cruching numbers and analyzing thousands of different options are what computers excel at. Computers are built upon boolean algebra, operating on true and false values. Thus, a natural way to encode our problem is to try to rewrite it as an expression involving boolean variables, which can either be true or false.

For example, using the example of three stores and two times, let's make six variables:. Each of these variables if true or 1 if we visit the store at the corresponding time, and false otherwise. Next, we form some constraints on these variables, and express them in a unified form we could feed to a computer. First, we know that we can only be in one place at a given time.

Thus, the constraint that we only go to one place in the evening can be represented as. We have similar constraints for Walmart and Home Depot, yielding the following constraint to represent that we must go to each store:.

To find out whether we can complete our shopping trip, we must find a set of true or false values for all our boolean variables such that the constraints are satisfied. This type of problem is known as the boolean satisfiability problemoften abbreviated to just "SAT". A program that finds solutions to these problems is known as a SAT solver. Let's write a simple SAT solver in Haskell.

Once we have a simple solver, we'll discuss heuristics for speeding up solving. Note the Var Char constructor: in this simple expression data type, we use Char to represent variables. We also allow for Const constructors to represent concrete true or false values; this is particularly useful for intermediate computations. The most basic algorithm for SAT solving is a backtracking search. This search has the following steps:. Next, implement step 2: replacing free variables with a constant true or false guess value.

At this point, we introduce Const constructors into our expression tree, replacing the appropriate Var constructors with them. Now that we've introduced Const constructors, our expression is no longer simplified. Thus, we introduce simplifywhich returns either a Const constructor or a simplified expression; if the result is not a Const constructor, it guarantees that there are no Const constructors in the Expr tree further down.SAT is short for "satisfiability".

Chances are you have heard of it or one of its variants like 3-SAT in passing, especially in discussions of complexity and NP-completeness. In this post, we will go into details of what it is all about, why it is of such importance from both a theoretical and practical perspective, and how to approach solving it by developing a simple Python SAT solver. By the end of this post, we will have a working SAT solver with a command-line interface.

Feel free to fork and contribute improvements. On the other hand, the rest of the introduction section below can be skipped if you already know the problem definition and relevant technical terms.

### Satisfiability and SAT solvers

Before we start with the definitions, you might be asking why SAT is written in all capitals if it is not an acronym. Well, great question. SAT happens to fall under what are called decision problems in computer science. What that means is that the answer to a particular instance of the problem is either "yes" or "no".

Decision problems are often simply identified with the set of inputs for which the answer is "yes", and that set is given a capitalized name. This naming style, as far as I know, is mainly due to Garey and Johnson's classic textbook on complexity theory. So, back to SAT. Now, if you happen to know your Boolean logic and already know all about satisfiability and CNF expressions, then feel free to skip ahead to next section.

The rest of this section assumes no prior knowledge of logic. Like many other interesting problems, there are a variety of ways of describing SAT, some more technical and some less. Here I will provide a very non-technical description of the problem that nonetheless is an accurate description. Assume you are in charge of elections in a society. Elections in this society work as follows: there are n n n candidates, and any number of them, from 0 0 0 nobody to n n n everybody can be elected as the result of the elections.

Each voter provides a list of candidates they want elected and candidates they want not elected. For example, the voter with the "A, B, not C" vote will be satisfied if either A or B is elected, or if C is not elected.

To be clear, that voter will be happy even if nobody is elected anarchy! It's also possible to receive an empty vote. We take this to mean that the voter will not be satisfied regardless of who is elected.

You are given all the votes, and your job is to determine if all the voters can be satisfied or not, and if yes, provide at least one possible pick of candidates that would satisfy everybody.

We assume that each candidate is represented by a unique identifier that will be a string in the input. Now that the problem makes sense, let's define the technical vocabulary. First, what we called "candidates" are called variables. A variable can be assigned true or false. A literal is a variable or its negation. A set of literals is called a clause.

An assignment is a mapping of variables to true or false. A clause is satisfied by an assignment if at least one of its unnegated literals is assigned true by the assignment, or one of its negated literals is assigned false in the assignment. So SAT can be summarized as follows: given a list of clauses, determine if there exists an assignment that satisfies all of them simultaneously.

It is also worthy of mention that there is a variation of SAT called 3-SAT with the restriction that each clause consists of at most 3 distinct literals. Even though SAT is NP-complete and therefore no known polynomial-time algorithm for it is yet known, many improvements over the basic backtracking algorithms have been made over the last few decades.

However, here we will look at one of the most basic yet relatively efficient algorithms for solving SAT.Work fast with our official CLI. Learn more. If nothing happens, download GitHub Desktop and try again. If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again. It is to our knowledge the first ever performant pure-Go SAT solver made available. I recommend however building the package github.

This package is all about anything-goes performance and is the workhorse behind most of the gini sat solver. It is also extensively tested and well benchmarked, so it should not pose any safety threat to client code. In practice, many SAT problems are quite easy but not decryption problems The solvers are used in software verification, hardware verification and testing, AI planning, routing, etc.

The SAT problem is a Boolean problem. All variables can either be true or false, but nothing else. The SAT problem solves systems of Boolean constraints, called clauses. There are many ways to efficiently code arbitrary logic into CNF, so this is not so much a restricting factor. Nonetheless, we present CNF and the problem below in a brief self-contained fashion which we find useful. N] is either a Boolean variable such as xor the negation of a Boolean variable such as not y.

An expression which is either a Boolean variable or its negation is called a "literal". Numerical negation indicates logical negation, and spaces are disjunctions "or". The SAT problem is determining whether or not a model exists for a given set of clauses. Resolution is a form of logical reasoning with conjunctions of clauses. Given 2 clauses of the form. Resolution proof of unsatisfiability is a derivation of the empty disjuction false by means of resolution.

Resolution proofs, even minimally sized ones, can be very large, exponentially larger than the input problem. Modern SAT solvers mostly rely on performing operations which correspond to bounded size in terms of number of variables number of resolutions. Given this fact together with the fact that the minimal proofs can be exponentially large in the number of variables, some problems can take an exponential amount of time.

Nonetheless, many SAT solvers have heuristics and are optimised so much that even hard problems become tractable.

With up to several tens of millions of resolutions happening per second on one modern single core CPU, even problems with known exponential bounds on resolution steps can be solved. Gini provides a simple and efficient logic modelling library which supports easy construction of arbitrary Boolean formulas.

The library uses and-inverter graphs, structural hashing, constant propagation and can be used for constructing compact formulas with a rich set of Boolean operators. The circuit type implements an interface which makes it plug into a solver automatically.

In fact, the circuit type uses the same representation for literals as the solver, so there is no need to map between solver and circuit variables.

Additionally, sequential circuits are supported.We review the notion of satisfiability and SAT formula via an example. Also pointers to SAT solvers are provided.

Recall that the Satisfiability problem is to decide, given a SAT formula we will assume it is in CNFwhether it is satisfiable or consistent or not. If you need a refresher, the following short video might be useful:.

These solvers are typically binaries which accept input in the form of a text file with the CNF formula and write the relevant output to the console. If you are interested, Lingeling is the exact SAT solver we are using to grade your submission. Note that we will not be able to provide any support for local execution and testing, please use the SAT Solver submission on Autolab created specifically for this purpose to test your generated SAT formulas out.

We will present this format via an example, where we first present things in propositional logic and then present its corresponding representation for SAT solvers. Consider the following data provided to you regarding course codes and their corresponding prerequisites. And if so, what would be a set of courses to take to satisfy all the pre-requisites.

Once could argue that this might not be the most natural objective but it is simple enough so that the SAT formulas that we have to talk about are not too complicated.

These set of clauses are fed into a SAT solver, which outputs a satisfying assignment : i. A set of constraints is satisfiable if and only if the assignment as a whole evaluates to true. That is, at least one of the literals inside the clauses being ANDed together must be set to true. For an example, look below. And naturally, this is one way to satisfy the prereqs, although probably not the one you will want to choose. The 0 is used by the solver to indicate the end of the assignment sequence.

And indeed, this is the bare minimum requirement for a set of clauses to be satisfiable. Satisfiability Satisfiability problem Recall that the Satisfiability problem is to decide, given a SAT formula we will assume it is in CNFwhether it is satisfiable or consistent or not. The 0 does not represent a variable.Every drop makes the ocean. If you'd like to participate in transforming society, join hands.

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## Writing a SAT Solver

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