Dpll algorithm. In this paper, an algorithm based on DPLL for steering the atomic time is proposed. System. ) A Horn formula is a propositional formula in conjunctive normal form that only consists of Horn clauses. Membrane computing, which is also known as a P system, is a computational model inspired by the activity of living cells. 1 Motivation M. Various sources tend to define DPLL differently: In pages 110-114 of the book Handbook of Satisfiability (Editors: Biere, A. In computer science and formal methods, a SAT solver is a computer program which aims to solve the Boolean satisfiability problem. Jan 31, 2017 · A chapter from a book on computability, logic, and mathematical foundations, reviewing the history and applications of the DPLL procedure and its variants. 1 Initialise Ato the empty assignment 2 While there is unit clause fLgin Fj A, update A7!A [L7!1] 3 If Fj Acontains no clauses, stop and output A. We would like to show you a description here but the site won’t allow us. 2 DPLL algorithms. 1 Variables de décision Une notion d’affectation de variable est utilisée dans (les versions efficaces de) DPLL. A DPLL algoritmus javítja a visszalépési algoritmust azáltal, hogy minden lépésben a következő szabályokat használja: Unit propagáció (egységterjesztés más fordításban) Ha egy tagmondat unitklóz, azaz csak egyetlen hozzá nem rendelt literált tartalmaz, akkor ez a klóz csak úgy teljesíthető, ha a literál igazzá Jan 5, 2020 · I was implementing the following DPLL algorithm in Prolog: DPLL implementation The implementation tells me if there is a solution for the given CNF, but is it also possible with DPLL algorithm to May 7, 2018 · The DPLL procedure is a recursive search algorithm. Most of these classical algorithms are limited by existing computing ability of classical computers. Among One of the well-established algorithms to solve the satisfiability problem in classical propositional logic is the DPLL procedure . Essentially, it is a (complete - \depth- rst" - backtracking) search algorithm. 1 1 The DPLL Algorithm The Davis-Putnam-Logemann-Loveland (DPLL) algorithm is a procedure that com-bines search and deduction to decide satisfiability of CNF formulas. The Davis–Putnam–Logemann–Loveland algorithm is a 1962 refinement of the propositional satisfiability step of the Davis–Putnam procedure which requires only a linear amount of memory in the worst case. DLL Algorithm: General Ideas • Iteratively set variables until – you find a satisfying assignment (done!) – you reach a conflict (backtrack and try different value) • Two main rules: – Unit Literal Rule: If an unsatisfied clause has all but 1 literal set to 0, the remaining literal must be set to 1 (a + b + c) (d’ + e) (a + c’ + d) evaluation. The DPLL procedure is a complete SAT solver that uses backtrack search and one-literal clauses to determine satisfiability or unsatisfiability of propositional formulas. The algorithm performs backtracking to find a model of a given theory or prove its unsatisfiability. The algorithm extends the original SAT -solving DPLL algorithm with the ability to reason about an arbitrary theory T. be/lRQIk8Ly5 Jul 26, 2007 · SAT-solvers have turned into essential tools in many areas of applied logic like, for example, hardware verification or satisfiability checking modulo theories. It was introduced in 1961 by Martin Davis, George DPLL SAT Solver. , Van Maaren, H. Asserting clauses. The variations include operations in the Davis-Putnam kernel (monotone literal fixing and the subsumption rule) and five branching rules in the literature (the Jeroslow-Wang rule, the two-sided DPLL algorithm. py). DPLL stands for: DPLL algorithm, for solving the boolean satisfiability problem. Unit propagation is done first because it might produce pure literals. (If you find the latter off-putting read this explanation). Show experimentally how your changes have A DPLL-Steered Timing Algorithm 3 pulsar timing residual in this paper is a simulation of future data, and the longer the prediction time, the less credible the sequence is. First, we'll implement unit propagation, which is one aspect of this formula simplification. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to treelike resolution proofs. • We model DPLL and its enhancements as transition systems instead. 1 Visualization. Mar 28, 2018 · Table 1 presents a summary of the different SLS solvers used in incomplete SAT algorithms. py to start with. Search for a sequence of proof steps (applications of inference rules) leading from to . When applied to a propositional formula in conjunctive normal form the program produces Algorithms. Propositional Logic: DPLL Algorithm Motivation 31. [1] [2] [3] At a high level, the algorithm works by transforming an SMT problem into a SAT formula where atoms are Keywords: DPLL · SAT· online Turing machines · space-bounded com-putations · sublinear space 1 Introduction DPLL(are named after the authors: Davis, Putnam, Logemann, Loveland [6,5]) algorithms are one of the most popular approachesto Boolean satisfiability prob-lem (SAT). For every possible world, if is true make sure that is true too. Given a Boolean formula, the SAT problem asks for an assignment of variables so that the entire formula evaluates to true. currentTimeMillis(); The problem is that I have to provide the test results to my problem, but compared to all the sources that I found on the internet, my algorithm has run times considerably better. Star Notifications 1. 2. The lecture covers the basics of propositional logic, the simple procedure, and the DPLL algorithm. DPLL on Horn Formulas. We have come up with an idea to make the unit-propagation faster than A personal python version of the Davis–Putnam–Logemann–Loveland (DPLL) algorithm: a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i. The recursion case then deals with a non-empty CNF formula that has no empty clauses. , Walsh, T. 5 In computer science, conflict-driven clause learning ( CDCL) is an algorithm for solving the Boolean satisfiability problem (SAT). As Million mentions, CDCL, an advancement of DPLL, is also complete. Jul 4, 2020 · Un grand merci à Diégo pour m'avoir permis de connaître cet algorithme !La vidéo de résolution de SAT par l'algo de backtracking : https://youtu. a restricted algorithm. Curate this topic Add this topic to your repo Dec 4, 2020 · Logemann-Loveland algorithm (DPLL algorithm) is a crucial representative because truth assignment is adopted to simplify CNF [2]. -- For an in-depth coverage of SA Consider, for instance, the application of the DPLL procedure to (¬p∨q)∧(¬p∨r)∧(q ∨r)∧(¬q ∨¬r). While a naive recursive backtracker would check every possible assignment until it found one that satisfies the formula, DPLL makes BCP-driven inferences to cut down on the size of the search tree – removing assignments that can’t satisfy the formula. However, we have reached a point where Dafny proofs take a lot of time (tens of minutes). 4 If Fj A3 , add new clause C to F by learning procedure. I managed to implement DPLL that doesn't count visited nodes but I can't think of any solutions to the problem of counting. The base cases are those of an empty conjunct, which is true, and a conjunct containing an empty disjunct, which is false. try making DPLL and/or WalkSAT solve more problems by varying your algorithms. A branch of the tree is no longer expanded if S i = fgor 2S i where is the empty clause. Plusieurs notions utilisées dans DP sont ajustées pour intégrer la notion d’affectation. We have depicted the basic DPLL algorithm in Fig. I have an example. The two interesting algorithms implemented for SAT solving are DPLL and unit propagation. However, most of the proposed algorithms use an exponential number of membranes, and reduction of the number of membranes must be A simple SAT solver that implements the DPLL algorithm with unit resolution License. (n = 0 is allowed. Several efficient P systems, which work in a polynomial number of steps, have been proposed for solving computationally hard problems. The notes cover the main procedure, unit propagation, conflict analysis, clause learning, and non-chronological backtracking. The authors investigated automatic code generation, but in the end the verified algorithm was manually translated into Dans l’algorithme DPLL, le backtracking est assez simple (chronologique). Directed resolution continued (DP algorithm). Digital phase-locked loop, an electronic feedback system that generates a signal. After more than 50 years the DPLL procedure still forms the basis for most efficient complete SAT solvers. Wehrle (Universit at Basel) Foundations of Arti cial Intelligence April 25, 2016 4 / 1 In logic and computer science, the Davis–Putnam–Logemann–Loveland (DPLL) algorithm is a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i. • Such calculi however cannot model meta-logical features such as backtracking, learning and restarts. The second class of algorithms used in solving the satisfiability problem is called the complete algorithm where the SAT solver based on the modern improvements or variations of a Davis–Putnam–Logemann–Loveland (DPLL) algorithm. On input a formula over Boolean variables, such as " ( x or y) and ( x or not y )", a SAT solver outputs whether the formula is satisfiable, meaning that there are possible values of x and y which make An Abstract Framework for DPLL • The DPLL procedure can be described declaratively by simple sequent-style calculi. { c } (the clause) is a unit and its literal c is positive, therefore c (the variable) is forced to be 1, then we have. 3 Related algorithms. 16:52 Solving satisfiability through search (DPLL algorithm). The DPLL forms the main logic of assigning variables and performing backtracking and re-assignment if conflict arise. Section 4 reviews the close connection between exhaustive DPLL algorithm and knowledge compilation by showing that the traces of exhaustive DPLL algorithms corre-spond to certain knowledge compilation languages. 1. Helmert (University of Basel) Foundations of Arti cial Intelligence. What makes a particular instance hard or easy is at most Sep 27, 2016 · 3. First DPLL is called with this formula as F and with ∅ as U (Call 1). These last decades, a promising approach has emerged for solving efficiently large size instances by using FPGA architectures. Apply (inference) rules to leaves, expanding the tree. Toggle The algorithm subsection. Sep 24, 2012 · Extracting a DPLL Algorithm Andrew Lawrence 1,2 Ulrich Berger 3 Monika Seisenberger 4 Department of Computer Science Swansea University Swansea, UK Abstract We formalize a completeness proof for the DPLL proof system and extract a DPLL SAT solver from it. In particular, show the sequence of formula manipulating operation and at the same time show how the formula it is modified. , of the form. 2 The DPLL Algorithm DPLL [6] was introduced in 1962 as a re nement of its earlier M. Another SATsolver formalized in Isabelle/HOL is by Maric´ [11]. The DPLL Algorithm 6. Non-chronological backtracking. Learn how the DPLL algorithm combines search and deduction to decide satisfiability of CNF formulas. By eliminating pure literals after unit propagation the function is assured of recursing on a variable whose value . Mar 6, 2020 · About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright O (2n) O (1) O (n) In logic and computer science, the Davis–Putnam–Logemann–Loveland ( DPLL) algorithm is a complete, backtracking -based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i. Jan 1, 1999 · Monotone literal fixing and the subsumption rule are of little use on random 3-SAT formulas, but they can speed up the DPLL algorithm significantly when solving some of the DIMACS formulas. DPLL algorithm makes our calculations a lot faster, with helping us reduce the total number of cases we need to check. It has been extended in various ways in modern imple-Fig. SAT (SATisfiability of Propositional Formula) is a well-known NP-Complete problem [1][2]. Unit propagation is an optimization that helps us prune our search space and converge to a solution if any faster. Start the DPLL algorithm by looking for any literals that can be made true without making the 3-CNF Apr 17, 2022 · 2. Feb 9, 2024 · 2 The algorithm. The DPLL Algorithm: simplify function simplify( ;v;d) Let ‘be the literal for v that is satis ed by v 7!d. :x1 _ _ :xn _ y or :x1 _ _ :xn. 2010-2019년 10년 동안 알고리즘 개선 작업은 알고리즘을 더 빠르게 만들기 위해 분기 리터럴과 새로운 데이터 구조를 선택하기 위한 더 나은 정책을 찾아냈다. After finishing this algorithm to experiment how large propositions your algorithm can solve. The pure literal rule states that if some variable appears only positively or only negatively in all clauses, then you can Iterative DPLL A decisionrefers to any time our algorithm arbitrarily assigns a variable (without being forced to do so) Selecting a literal and assigning it True is a decision Unit propagation & reassigning selected literal after backtracking are not decisions All assignments implied by the ithdecision are said to original_dpll. Conventional solvers for SAT based on traditional DPLL algorithm presents serious CPU-Times limitations, especially when addressing large size instances. , an assignment of values to some subset of the variables) and a set of undecided clauses (i. Modern SAT solvers. There are three programs: An implementation of the Davis-Putnam procedure, which takes as input a set of clauses and outputs either a satisfying valuation, or a statement that the clauses cannot be satisfied (DPLL. Une stratégie plus efficace fera l’objet de la Section 5. Now I'm calculating the running time by the standard function java api. 6 References. 5 The DPLL algorithm Input: CNF formula F. , those clauses that have not yet been satisfied). I'm already provided a basic, slow working version (essentially the DPLL algorithm; furthermore, to select the next variable, it just loops an array and gets the next unused one). Local Search and Outlook M. One may design his (or her) own SAT solver for a specific applica-. Other approaches might verify, for example, functional code, and rely on a mechanism to refine the functional code or to extract it into Apr 20, 2017 · 1 Answer. Mar 10, 2015 · The DPLL algorithm has been formalized by Mari´c and Janiˇci´c [28]. This algorithm is just a simple backtracking with some pruning strategy. Jan 11, 2014 · The DPLL algorithm is an optimised search for such a solution. The DPLL algorithm works by choosing an assignment of true or false for a variable, simplifying the formula based on that choice, then recursively checking the satisfiability of the simplified formula. DPLL is an algorithm that takes the formula ϕ, uses heuristics A Of the SAT solving algorithms that you have suggested, bruteforce and DPLL are both complete algorithms that, given enough time, are guaranteed to find a satisfying assignment or prove the problem unsatisfiable. There is one example to solve following Satisfiability problem with DPLL. MIT license 43 stars 22 forks Branches Tags Activity. Complete algorithm: every that is entailed can be DPLL method DPLL procedure: Algorithm Algorithm Input: S = C 0 = fC 1; ;C k gwhere C i = L 1 _L 2 __ L r i. If C is the empty clause, stop and output UNSAT. Late 90’s and early 00’s improvements make DPLL efficient: Break-through systems: GRASP, SATO, Chaff, MiniSAT Stalmarck:1k var 1996 SATO:1k var 1996 GRASP:1k var 1996 DLL:10 var 1962 Chaff:10k var 2001 BDD:100 var 1986 DP:10 var 1960 MiniSAT:100k var 2003 The DPLL algorithm – p. A dash (-) represents the negation symbol. 4. Jan 23, 2021 · DPLL algorithm for solving the Boolean satisfiability problem (SAT) can be represented in the form of a procedure that, using heuristics A and B , select the variable x from the input formula \varphi and the value b and runs recursively on the formulas \varphi [x := b] and \varphi [x := 1 - b] . Volume 185 of Frontiers in Artificial Intelligence and Applications) it defines it as backtracking + unit The DPLL algorithm can be explained by the following pseudocode. py : solver with more branching heuristics linked_sat. In this thesis, we discuss the details of our implementation of the DPLL algorithm as well as a mathematical application of our solver. Category: Disambiguation pages. We have proposed an improved variant of the DPLL algorithm and designed an efficient data structure for it. A Horn clause is a clause with at most one positive literal, i. With DPLL algorithm, we try to check the most likely model for the proposition via setting some propositions true or false. If you are looking to discuss alternatives, I would recommend Mar 5, 2024 · Then DPLL will continue assigning sym-bols truth values until either a satisfying model is found or a symbol cannot be assigned without violating a logical constraint, at which point the algorithm will backtrack to the last working assignment. solve_dpll(cnf): while(cnf has a unit clause {X}): delete clauses contatining {X} delete { ! X} from all clauses if null clause exists: return False if CNF is null: return True select a literal {X} cnf1 = cnf + {X} cnf2 = cnf + { ! DPLL AlgorithmByC Manideepu Reddy 16MI31023 Sayan Guha 16MI31014Shourya Shashank 16MI33015 Yash Madhogaria 16MI33017 Davis-Putnam-Logemann-Loveland Algorithm. It also shows how our model counting algorithm can be used in the context of knowledge compilation (i. ) Extras: For extra credit: Try making DPLL as fast as possible. py : base solver, random selection base_sat. It has recently been extended Nov 29, 2020 · I am confused about the precise definition of the DPLL algorithm. It eschews the resolution for the splitting rule: a backtracking algorithm that chooses a literal l, and then recursively checks if a The DPLL Procedure The Davis-Putnam-Logemann-Loveland procedure is a decision procedure for CNF formulae in propositional logic. Jun 28, 2022 · DPLL algorithm using deductive verification. The other method is called the Incomplete Algorithm and it is based on random search [3]. • A transition system is a binary relation over states, M. Let ‘ be the complementary (opposite) literal to ‘. if assign of "0" to variables is prior to assign "1" to variables, Which of Unit Clause (UC) or Pure Literal (PL) is used to solve this specific example? DPLL (T) In computer science, DPLL (T) is a framework for determining the satisfiability of SMT problems. Sound algorithm: everything it claims to prove is in fact entailed. In order to go further and verify the entire DPLL algorithm, additional techniques to bring downDafny verification timeare required, asdicussed in theconclusion. for solving the CNF-SAT problem. I suggest you go read about those and try implementing them. Method 2: theorem-proving. Feb 18, 2021 · DPLL algorithm is mostly used to understand if a logical proposition is unsatisfiable or not. DPLL algorithm Aug 2, 2014 · I'm implementing the DPLL algorithm (which I've already finished). April 26, 2021. The satisfiability problem (or SAT for short) is a central problem in several fields of computer science, including theoretical computer science, artificial intelligence, hardware design, and formal Different algorithms have been proposed to solve it [17, 18, 54], with the Davis–Putnam–Logemann–Loveland (DPLL) algorithm being the one mainly used in practical applications today. tion, and in this case he (or she) has Lazy DPLLT Algorithm using a Decision Procedure Input: A formula in CNF form over theory T Output: ⊨ or UNSAT Let be the abstraction of while true do if DPLL is unsat then return UNSAT else Let be the model returned by 퐷푃 Assume is represented as a formula Aug 10, 2015 · Now, consider DPLL. Assume that the literal selected as L is p. The assignment is in lexicographical order. The one literal rule, usually known as the unit clause rule, state that if you have a unit clause (a clause containing only one literal) then you should set that literal to the value which satisfies the clause. The DPLL algorithm is a SAT solver based on recursive backtracking that makes use of BCP. 1 Pseudo-code of the basic DPLL algorithm This lab is divided into four parts, each of which contains both some tutorials and exercises. At any stage in the search there is a partial assignment (i. Wehrle (Universit at Basel) Foundations of Arti cial Intelligence April 25, 2016 3 / 1 31. Putnam (DP) algorithm. Inference (reminder) Method 1: model-checking. DPLL might then recurse on the variables associated with these literals, wasting potentially exponential time uselessly backtracking over them in the future. 5 We would like to show you a description here but the site won’t allow us. The algorithm is done when that happens (Φ is a consistent set of literals) or when it runs out of literal assignments to try and the topmost DPLL call returns false. I have been reading papers on ways to improve it, but I'm a bit Exercise 3: Implement the DPLL algorithm, you can download the file dpll. Now DPLL is called recursively with SAT solver. 1 Review - the DPLL procedure On Tuesday we talked about validity and satisfiability of formulas and the Davis-Putnam procedure for testing the satisfiability of a given formula in CNF. The atomic clock Davis-Putnam algorithm on a Maze Puzzle, which uses SAT problems to assign truth values. In computer science, the Davis–Putnam–Logemann–Loveland (DPLL) algorithm is a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i. DPLL is essentially a depth first search that alternates between three tactics. This paper follows this Add a description, image, and links to the dpll-algorithm topic page so that developers can more easily learn about it. The main problem is that as the algorithm finds satisfying valuation and returns True, the recursion rolls up and returns counter from the moment the Dec 25, 2017 · DPLL [6]) is a software, and man y of the SAT solv ers are written in C or. However, although recent implementations are able to solve problems with hundreds of thousands of variables and millions of clauses, much smaller instances remain unsolved. 0:= fC jC 2 such that ‘=2Cg 00:= fC nf‘ gjC 2 0g return 00 An improved variant of the DPLL algorithm is proposed, which performs well on most instances of the DIMACS benchmarks and it performs better than other SAT-solvers on a certain class of instances. C ++ [25]. 5 See also. Clause learning. However, DPLL makes three improvements over simple backtracking search: 1. To prove logical consequence take the axioms and negated conjecture in CNF, and check for unsatisfiability. Enter in the box below a series of clauses (one for each line), using alphanumeric characters to represent the variables, separating it using spaces. Besides, DPLL simplifies $\phi$ along the backtracking, instead of doing Talk about the characteristics of each algorithm, and compare the two algorithms to explain any differences in the results. The first part is the SAT encoding of the basic propositions; the second part covers validity checking; part three covers the DPLL algorithm implementation; and the fourth part covers some SAT applications. 4 Relation to other notions. As part of a college class, I'm asked to improve the performance of a basic DPLL sat solver. In this dissertation, we examine variations of the DPLL algorithm, a popular algorithm for solving the satisfiability problem, which is NP-complete. After the call to Unit-propagate, the values of F and U remain the same. Simple Java implementation of the Davis–Putnam–Logemann–Loveland (DPLL) algorithm Topics java algorithm math algorithms mathematics discrete-mathematics sat-solver dpll dpll-algorithm Feb 20, 2016 · We know DPLL algorithm is backtracking + unit propagation + pure literal rule. Implication graphs. Therefore, lower bounds for treelike resolution (known since the 1960s The DPLL Algorithm Print version of the lecture in Logic and Proof presented on 31 January 2017 by Dr Christoph Haase 7. Set C 0 as the root of the tree. The DPLL Procedure The Davis-Putnam-Logemann-Loveland procedure is a decision procedure for CNF formulae in propositional logic. Nov 20, 2020 · DPLL algorithm. Davis, H. Learn how to use decision procedures for propositional logic, also known as SAT solvers, to automatically prove the validity of formulas. The basic step consists of picking a literal and assigning to it the value of either true or false if possible DPLL은 DPLL(T) 알고리즘을 통해 1차 로직의 fragment를 증명하는 자동 정리를 위해 확장되었습니다. , in the Feb 26, 2018 · $\begingroup$ In this case, if you read the Wikipedia articles on the DPLL algorithm and on Boolean satisfiability, they describe the main techniques: conflict analysis and clause learning, non-chronological backtracking, and two-watched-literals unit propagation. This algorithm underlies most modern SAT solvers. py : solver with linked list structure (only with JW branching heuristic) Aug 15, 2006 · DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. Mar 13, 2017 · I'm implementing DPLL algorithm that counts the number of visited nodes. For instance, you can generate some very large propositions using this generator and feed the generated propositions to your solver. (Suggested Length: 2 - 3 pages. DPLL(f) •If f is empty, return true (embrace truth) •If there is an empty clause in f, return false (reject falsity) •If there is a unit clause U in f, return DPLL(f(U)) (accept the inevitable) •If there is a pure literal U in f, return DPLL(f(U)) (go with the flow) •For some variable v (take a guess) –If DPLL(f(v)) then return true Aug 27, 2020 · An algorithm for SAT solving DPLL Algorithm: Outline DPLL:Davis-Putnam-Logemann-Loveland [4] Algorithmic improvements over DP algorithm Basic idea:case splitting and simplification Simplification:unit propagation and pure literal deletion Unit propagation:1-clauses (unit clauses) fix variable values:if f xg2 S, in order to satisfy , variable must be set to 1. , Heule, M. Let me briefly review it in a somewhat more precise formulation Late 90’s and early 00’s improvements make DPLL efficient: Break-through systems: GRASP, SATO, Chaff, MiniSAT Stalmarck:1k var 1996 SATO:1k var 1996 GRASP:1k var 1996 DLL:10 var 1962 Chaff:10k var 2001 BDD:100 var 1986 DP:10 var 1960 MiniSAT:100k var 2003 The DPLL algorithm – p. If S i = fgthen S is satis able and we can stop the procedure. This version of DPLL implements unit clause and non-chronological backtrack. The internal workings of CDCL SAT solvers were inspired by DPLL solvers. That step wouldn't have happened that way, because there were unit clauses in the input which would have been resolved first. Exponential lower bounds on the running time of Nov 2, 2017 · The DPLL algorithm, through the pure and unit rules, gradually converts the list of clauses into a list of literals that satisfy all the original clauses. Here’s the best way to solve it. C2 : {d, a} C3: {b, !d, !a} Apply the DPLL algorithm to prove if the following 3-CNF formula is satisfaiable. The beauty of that algorithm is that it is both simple and efficient. This approach was extended to formalize a SAT solver including optimizations such as clause learning and the lazy two-watched-literal data structure [27]. Feb 2009. DPLL Algorithm I32. e. gv ii yf wt ev gv tx or fm zw