Part 1 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.
Sunday, March 4 Wednesday, March 7 Sunday, March 18 This particular problem set is hard, so start early.
You can add new public methods and classes to the code we have provided you. But do NOT change the signatures, specifications, or representations of any existing methods and classes that we provided, and do NOT add new packages or change any existing packages' names.
Your code will be tested automatically, and will break our testing suite if you do so. Otherwise you will break our testing suite and your grades will be affected.
To get started, pull out the problem set code from SVN Admin. Background A Sudoku puzzle is a kind of Latin square. The aim is to complete a 9x9 grid with a digit between 1 and 9 in each square, so that -- as in a Latin square -- each digit occurs exactly once in each row and in each column.
In addition, the grid is divided into 9 blocks, each 3x3, and each digit must also occur exactly once in each block. Sudoku is normally solved by reasoning, determining one step at a time how to complete an additional square until the entire puzzle is finished.
Solving Sudoku by SAT is not very appealing for human players but works well on a computer.
There are two classes of high-performance algorithms for solving instances of SAT in practice: A modern Parallel SAT solver is ManySAT. It can achieve super linear speed-ups on important classes of problems. SAT Game - try solving a Boolean satisfiability problem yourself SAT problem format. Sep 13, · I am kinda proud of the parallel performance of CMS as it can showcase the heterogeneity of the system and the different capabilities of the solver. It’s basically doing a form of acrobatics where the solver can behave like a very agile SAT solver with one set of parameters or like a huge monolith with another set of parameters. SAT Solvers: Theory and Practice Clark Barrett [email protected] • Solving SAT The Language of SAT solvers: Propositional Logic A SAT solver solves the Boolean satisﬁabiliy problem. In order to understand the satisﬁability problem, we must ﬁr st deﬁne the language in which the problem is phrased.
The satisfiability problem is to find an assignment of truth values to the variables that makes the formula true. A SAT solver is a program that solves the satisfiability problem: A formula in conjunctive normal form CNF consists of a set of clauses, each of which is a set of literals.
A literal is a variable or its negation. Each clause is interpreted as the disjunction OR of its literals, and the formula as a whole is interpreted as the conjunction AND of the clauses.
So an empty clause represents false, and a problem containing an empty clause is unsatisfiable. But an empty problem containing no clauses represents true, and is trivially satisfiable. The basic idea is just backtracking search: DPLL adds a powerful but simple optimization called unit propagation: There's actually another optimization included in the original algorithm for 'pure literals', but it's not necessary and doesn't seem to improve performance in practice.
Wikipedia articles cover these topics nicely: Overview The theme of this problem set is to solve a sudoku puzzle. To do this, we'll: Read in a text file with an incomplete sudoku puzzle, and represent the puzzle using an immutable abstract datatype.
Traslate the puzzle into a propositional formula, represented using immutable list data structures. Solve the formula with the SAT problem solver. Translate the formula back into a solution to the Sudoku puzzle.
We're also providing you with packages that include implementations of immutable list data types and some of the propositional formula data types, as well as skeleton implementations of the Formula and Sudoku datatypes, which are in the sat. You only need to fill in the skeleton code to complete this problem set.
Your program should be efficient and should solve the 9x9 puzzles within 5 minutes when assertions are turned off. Loading Sudoku Puzzles Sudoku is an immutable datatype representing a Sudoku puzzle, with creator methods and observer methods.
We have given you the specification of its methods, and its representation. It's your job to determine the representation invariant and implement the methods.
The datatype also has a factory method for loading a puzzle from a file.
The file format is one line for each row of the puzzle, consisting of numbers for known squares and blanks for squares to be filled separated by " ". You will use assertions in this assignment to check representation invariants see Programming with Assertions.
Assertions are turned off by default in Java, which means that assert statements are completely ignored.Read our expert strategies on scoring a perfect on the SAT Writing section. Get help from a perfect SAT scorer. Read our expert strategies on scoring a perfect on the SAT Writing section.
How to Get on SAT Writing: 9 Strategies From a Perfect Scorer. Posted by Allen Cheng | Oct 19, PM. SAT Strategies. 4 Question pop-quiz Jump into a short quiz to get a quick read on your SAT performance.
SAT Solvers: Theory and Practice Clark Barrett [email protected] • Solving SAT The Language of SAT solvers: Propositional Logic A SAT solver solves the Boolean satisﬁabiliy problem.
In order to understand the satisﬁability problem, we must ﬁr st deﬁne the language in which the problem is phrased. performance results on a wide range of solvers [Œ, ].
In the competition, over 30 solvers competed on instances selected from thousands of benchmark problems. Most of these SAT solvers can SAT Solver TechnologyŠComplete Methods A complete solution method for the SAT problem is . Empirical Study of the Anatomy of Modern Sat Solvers Hadi Katebi 1, Empirical Study of Modern SAT Solver’s Anatomy been shown, through extensive empirical evidence, to be critical for scalability Branching heuristics can have a signiﬁcant eﬀect on the performance of SAT solvers.
Ranging from random decision strategies to. Sep 13, · I am kinda proud of the parallel performance of CMS as it can showcase the heterogeneity of the system and the different capabilities of the solver. It’s basically doing a form of acrobatics where the solver can behave like a very agile SAT solver with one set of parameters or like a huge monolith with another set of parameters.