NP-SPEC: an executable specification language for solving all problems in NP

Abstract

In this paper a logic-based specification language, called NP-SPEC, is presented. The language is obtained by extending DATALOG through allowing a limited use of some second-order predicates of predefined form. NP-SPEC programs specify solutions to problems in a very abstract and concise way, and are executable. In the present prototype they are compiled to PROLOG code, which is run to construct outputs. Second-order predicates of suitable form allow to limit the size of search spaces in order to obtain reasonably efficient construction of problem solutions. NP-SPEC expressive power is precisely characterized as to express exactly the problems in the class NP. The specification of several combinatorial problems in NP-SPEC is shown, and the efficiency of the generated programs is evaluated.

Introduction

The definition and the implementation of logic-based languages allowing for specifying complex problems have recently received much attention in the research community (cf., e.g., [3], [4], [5], [6], [7]). The main features of such languages are:

• they are highly declarative, and, as such, they are easier to use and simplify the error-prone process of writing algorithms;

• they are executable, so that the defined specification can be actually run and thus constitutes a rapid prototype.

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General logic-based languages that, being Turing-complete, allow for an unrestricted specification capability are proposed in, e.g., [5], [6]. Clearly, such languages are fairly complex, and this may be an obstacle for their use (cf., e.g., [8], where the difficulties arising with handling formal specifications is pointed out as a main factor in the limited use of formal methods in software development).

In this paper we follow a different direction, and propose a language with limited expressiveness. In particular, we define NP-SPEC, which is a highly declarative executable language, which allows the user to specify exactly all problems which belong to the complexity class NP in a simple way. This restriction has two advantages:

• the language is simpler, and easier to learn;

• a more focused strategy in the search for a solution is possible, thus leading to more efficient programs.

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Like in [3], [4], we opt for a DATALOG-like, i.e., PROLOG with no function symbols, syntax. Basically, logical formulae are rules, hence they are more restricted than in general predicate logic. The main difference between NP-SPEC and the other languages relies in its semantics, which is based on the notion of model minimality. The different semantics makes negation in the body of rules not necessary—though permitted—thus allowing for simpler specifications.

We show an example of an NP-SPEC specification, which helps us to highlight the main aspects of the language. The example concerns the famous “3-coloring” problem, which is well known to be NP-complete (cf. e.g., [9, problem GT4, p. 191]), and is formally defined as follows:

In NP-SPEC, the user can make the following declarations, which specify both an instance (in the DATABASE section) and the question (in the SPECIFICATION section). In this case, the instance is a graph with six nodes and seven edges, which is 3-colorable.

In the current implementation of NP-SPEC, processing the above specification gives as output an ECLⁱPS^e, i.e., PROLOG, program. Running the ECLⁱPS^e program produces the following output, which represents the colors to be assigned to the nodes so that the input graph is 3-colored.

NP-SPEC is a DATALOG-like language. In particular:

• the clauses in the DATABASE section specify a set of facts, i.e., function-free ground atomic first-order formulas, that define the instance;

• rules in the SPECIFICATION section are universally quantified function-free definite clauses, that specify the solution of the problem.

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The main difference between NP-SPEC and DATALOG is the possibility of using second-order logic in a restricted way. Referring to the above example, the clause Partition(node,coloring,3) has the following meaning:

• The predicate coloring, which does not occur in the database section, is implicitly declared to have arity 2, i.e., the arity of domain node plus 1.

• The extension of coloring can be any set of the following kind:

$$\text{{〈n,c〉}\text{|}\text{n}\text{is}\text{in}\text{the}\text{extension}\text{of}\text{node}\text{,}\text{c∈{1,2,3}}.}$$

The intuitive meaning of the SPECIFICATION section is: the instance has no solution if for all possible extensions of coloring, there are nodes X and Y such that:

• X and Y are connected by an edge (cf. atom edge(X,Y)), and

• X and Y are colored with the same color C (cf. atoms coloring(X,C), coloring(Y,C)).

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In other words, an extension of coloring which makes fail true does not correspond to a solution.

The execution of the specification is such that as soon as the program finds an extension of coloring that does not make fail true, it outputs such an extension, or, at the request of the user, all extensions of this kind.

More precisely, in general, a specification in NP-SPEC has the following structure:

The main distinctive features of NP-SPEC are the following:

• It has a completely defined semantics, which is based on an extension of DATALOG known as DATALOG^CIRC, defined in [10].

• It has a precise connotation in terms of expressive power and computational complexity. In particular, it can specify exactly all problems in the complexity class NP.

• It offers a set of metapredicates, called tailoring predicates, which guide the system in the synthesis of algorithms that operate on search spaces of limited size so as to be reasonably efficient.

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The language NP-SPEC is meant to be simple to use and to propose intuitive and concise specifications. Our approach is meant to lift the abstraction level at which the problem must be specified for the system to be able to generate automatically the code. The user is allowed to take the decisions at such an abstract level, and her/his decisions have an impact in the generated program in a transparent way.

In the rest of the paper, we outline the main technical aspects of NP-SPEC: syntax, semantics, and computational properties. We also briefly address the current prototype and its performances, report more specification examples, and discuss related work.