Satplan: Fundamentals and Applications

By Fouad Sabry

What Is Satplan

The process of computerized planning is referred to as Satplan. It transforms the instance of the planning issue into an instance of the Boolean satisfiability problem, which is then solved via a method for proving satisfiability such as the DPLL algorithm or WalkSAT. This allows the planning problem to be handled more efficiently.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Satplan

Chapter 2: Boolean satisfiability problem

Chapter 3: Constraint satisfaction problem

Chapter 4: 2-satisfiability

Chapter 5: Cook-Levin theorem

Chapter 6: Function problem

Chapter 7: DPLL algorithm

Chapter 8: WalkSAT

Chapter 9: MAX-3SAT

Chapter 10: SAT solver

(II) Answering the public top questions about satplan.

(III) Real world examples for the usage of satplan in many fields.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of satplan.

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• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 30, 2023

Book preview

Chapter 1: Satplan

Automated planning can be achieved with Satplan, also known as Planning as Satisfiability. The DPLL algorithm or WalkSAT, both of which prove satisfiability, are then used to solve the transformed instance of the planning problem.

For a particular planning problem instance, the formula is created so that it is satisfiable if and only if there is a plan with the specified horizon length, taking into account the problem instance's initial state, set of actions, goal, and horizon length. The demonstration of Cook's theorem resembles the simulation of Turing machines when dealing with the satisfiability problem. By comparing the formulas' satisfiability across a range of time horizons, a solution can be identified. Using consecutive horizon lengths (0, 1, 2, etc.) is the quickest method.

{End Chapter 1}

Chapter 2: Boolean satisfiability problem

The problem of determining whether or not there is an interpretation that satisfies a given Boolean formula is referred to as the Boolean satisfiability problem in the fields of logic and computer science. This problem is also sometimes referred to as the propositional satisfiability problem and is abbreviated as SATISFIABILITY, SAT, or B-SAT. In other words, it determines if the variables in a given Boolean formula can be consistently substituted by the values TRUE or FALSE in such a manner that the formula evaluates to TRUE. If the answer to this question is yes, then the question is considered to have been answered in the affirmative. The formula is referred to be satisfiable when this condition is met. On the other hand, in the event that there is no such assignment, the formula cannot be satisfied since the function that it expresses would evaluate to FALSE for each and every potential assignment of the variable. For instance, the formula a AND NOT b may be satisfied since it is possible to discover the values a = TRUE and b = FALSE, which together result in the expression a AND NOT b being equal to TRUE. a AND NOT a, on the other hand, is an unacceptable alternative.

The SAT problem was the first problem to be shown to be NP-complete; for more information, see the Cook–Levin theorem. This indicates that the level of difficulty of solving any issue that falls within the complexity class NP, which encompasses a broad variety of natural choice and optimization problems, is at most equal to that of SAT. However, this belief has not been mathematically proven, and finding an answer to the question of whether SAT has a polynomial-time algorithm is equivalent to the P versus NP problem, which is a famous open problem in the theory of computing. There is no known algorithm that efficiently solves each SAT problem, and it is generally believed that no such algorithm exists; however, this belief has not been mathematically proven, and finding an answer to the question of whether SAT has a polynomial-time algorithm.

In spite of this, as of the year 2007, heuristic SAT-algorithms are able to solve problem instances that include tens of thousands of variables and formulae that consist of millions of symbols. Automatic theorem proving is also possible with these algorithms.

A propositional logic formula, Boolean expression is another name for this, is composed of many parameters, combination of the operators AND (and), also denoted by ∧), OR (disjunction, ∨), NOT (negation, ¬), and parentheses.

It is argued that a formula is satisfiable if it is possible to turn it into a true statement by giving it the proper logical values (i.e.

TRUE, FALSE) to the variables that it has.

The Boolean satisfiability problem, sometimes known as the SAT, is defined as, given a formula, to determine whether it is possible to be satisfied by it.

This choice dilemma is of fundamental significance in a wide variety of subfields within computer science, Incorporating conceptual aspects of computer science, complexity theory, Either a variable, in which case it is referred to as a positive literal, or the negation of a variable is what constitutes a literal (called a negative literal).

Literals are spliced together to form a clause (or a single literal).

If a clause only has a single positive literal, then it is referred to as a Horn clause.

If a formula is a conjunction of clauses, then it is in conjunctive normal form, also known as CNF (or a single clause).

For example, x1 is a positive literal, ¬x2 is a negative literal, x1 ∨ ¬x2 is a clause.

The formula (x1 ∨ ¬x2) ∧ (¬x1 ∨ x2 ∨ x3) ∧ ¬x1 is in conjunctive normal form; Horn clauses make up the first and third sentences of this sentence, However, the second clause of it is not.

The formula may be satisfied in its entirety, by choosing x1 = FALSE, x2 = FALSE, and x3 arbitrarily, since (FALSE ∨ ¬FALSE) ∧ (¬FALSE ∨ FALSE ∨ x3) ∧ ¬FALSE evaluates to (FALSE ∨ TRUE) ∧ (TRUE ∨ FALSE ∨ x3) ∧ TRUE, and in turn to TRUE ∧ TRUE ∧ TRUE (i.e.

to TRUE).

In contrast, the CNF formula a ∧ ¬a, composed of two sentences, each of which is a literal, is unsatisfiable, since for a=TRUE or a=FALSE it evaluates to TRUE ∧ ¬TRUE (i.e, FALSE) or FALSE ∧ ¬FALSE (i.e, again FALSE), respectively.

There are a few variations of the SAT issue that, The concept of a generalized conjunctive normal form formula needs to be defined since