Elementary Formal System Research Articles

Learning of Elementary Formal Systems with Two Clauses Using Queries

An elementary formal system, EFS for short, is a kind of logic program over strings, and regarded as a set of rules to generate a language. For an EFS Γ, the language L(Γ) denotes the set of all strings generated by Γ. We consider a new form of EFS, called a restricted two-clause EFS, and denote by $r{\\cal EF\\!S}$ the set of all restricted two-clause EFSs. Then we study the learnability of $r{\\cal EF\\!S}$ in the exact learning model. The class $r{\\cal EF\\!S}$ contains the class of regular patterns, which is extensively studied in Learning Theory. Let Γ* be a target EFS in $r{\\cal EF\\!S}$ of learning. In the exact learning model, an oracle for superset queries answers “yes” for an input EFS Γ in $r{\\cal EF\\!S}$ if L(Γ) is a superset of L(Γ*), and outputs a string in L(Γ*) - L(Γ), otherwise. An oracle for membership queries answers “yes” for an input string w if w is included in L(Γ*), and answers “no”, otherwise. We show that any EFS in $r{\\cal EF\\!S}$ is exactly identifiable in polynomial time using membership and superset queries. Moreover, for other types of queries, we show that there exists no polynomial time learning algorithm for $r{\\cal EF\\!S}$ by using the queries. This result indicates the hardness of learning the class $r{\\cal EF\\!S}$ in the exact learning model, in general.

Learning elementary formal systems with queries

The elementary formal system ( EFS) is a kind of logic programs which directly manipulates strings, and the learnability of the subclass called hereditary EFSs ( HEFSs) has been investigated in the frameworks of the PAC-learning, query-learning, and inductive inference models. The hierarchy of HEFS is expressed by HEFS( m, k, t, r), where m, k, t and r denote the number of clauses, the occurrences of variables in the head, the number of atoms in the body, and the arity of predicate symbols. The present paper deals with the learnability of HEFS in the query learning model using equivalence queries and additional queries such as membership, predicate membership, entailment membership, and dependency queries. We show that the class HEFS(∗,k,t,r) is polynomial-time learnable with the equivalence and predicate membership queries and the class HEFS(∗,k,∗,r) with termination property is polynomial-time learnable with the equivalence, entailment membership, and dependency queries for the unbounded parameter ∗. A lowerbound on the number of queries is presented. We also show that the class HEFS(∗,k,t,r) is hard to learn with the equivalence and membership queries under the cryptographic assumptions. Furthermore, the learnability of the class of unions of regular pattern languages, which is a subclass of HEFSs, is investigated. The bounded unions of regular pattern languages are polynomial-time predictable with membership query. However, all unbounded unions of regular pattern languages are not polynomial-time predictable with membership queries if neither are the DNF formulas.

Advanced elementary formal systems

An elementary formal system (EFS) is a logic program such as a Prolog program, for instance, that directly manipulates strings. Arikawa and his co-workers proposed elementary formal systems as a unifying framework for formal language learning. In the present paper, we introduce advanced elementary formal systems (AEFSs), i.e., elementary formal systems which allow for the use of a certain kind of negation, which is nonmonotonic, in essence, and which is conceptually close to negation as failure. We study the expressiveness of this approach by comparing certain AEFS definable language classes to the levels in the Chomsky hierarchy and to the language classes that are definable by EFSs that meet the same syntactical constraints. Moreover, we investigate the learnability of the corresponding AEFS definable language classes in two major learning paradigms, namely in Gold's model of learning in the limit and Valiant's model of probably approximately correct learning. In particular, we show which learnability results achieved for EFSs extend to AEFSs and which do not.

Polynomial-time learning of elementary formal systems

An elementary formal system (EFS) is a logic program consisting of definite clauses whose arguments have patterns instead of first-order terms. We investigate EFSs for polynomial-time PAC-learnability. A definite clause of an EFS is hereditary if every pattern in the body is a subword of a pattern in the head. With this new notion, we show that H-EFS(m, k, t, r) is polynomial-time learnable, which is the class of languages definable by EFSs consisting of at mostm hereditary definite clauses with predicate symbols of arity at mostr, wherek andt bound the number of variable occurrences in the head and the number of atoms in the body, respectively. The class defined by all finite unions of EFSs in H-EFS(m, k, t, r) is also polynomial-time learnable. We also show an interesting series ofNC-learnable classes of EFSs. As hardness results, the class of regular pattern languages is shown not polynomial-time learnable unlessRP=NP. Furthermore, the related problem of deciding whether there is a common subsequence which is consistent with given positive and negative examples is shownNP-complete.

Rich Classes Inferrable from Positive Data: Length-Bounded Elementary Formal Systems

Inductive inference from positive data is shown to be remarkably powerful using the framework of elementary formal systems. An elementary formal system, EFS for short, is a kind of logic program on Σ+ consisting of finitely many axioms. Any context-sensitive language is definable by a restricted EFS, called a length-bounded EFS. Length-bounded EFSs with at most n axioms are considered, and it is shown that inductive inference from positive data works successfully for their models as well as for their languages. From this it follows that any class of logic programs, such as Prolog programs, corresponding to length-bounded EFSs can be inferred from positive facts.

Short note: procedural semantics and negative information of elementary formal system

We discuss procedural semantics and inference of negated ground atoms in elementary formal system (EFS). EFS is now a logic programming with associative unification. There are two problems on the SLD-resolution when we infer negated atoms. One is existence of infinitely many unifiers for two atoms, even maximally general. This prevents us finding a proper completed definitions of EFS's corresponding to the negation as failure rule. The other problem is existence of infinite derivations. They make it difficult to reject atoms not in the least Herbrand model. In the note, we give solutions for these problems. When we use the SLD-resolution to accept formal languages defined by EFS's, we assume that every refutation begins from a ground goal. Under the assumption we solve the first problem by introducing the variable-bounded EFS, which is powerful enough to define languages. The solution for the second problem is to bound the length of every SLD-derivation. We present it as an algorithm to decide whether a ground atom is in the least Herbrand model or not. We introduce the weakly reducing EFS as a class of EFS's where our algorithm is a complete realization of the closed world assumption.

Learning elementary formal system

The elementary formal systems (EFS for short) Smullyan invented to develop his recursive function theory, are proved suitable to generate languages. In this paper we first point out that EFS can also work as a logic programming language, and the resolution procedure for EFS can be used to accept languages. We give a theoretical foundation to EFS from the viewpoint of semantics of logic programs. Hence, Shapiro's theory of model inference can naturally be applied to our language learning by EFS. We introduce some subclasses of EFS's with correspond to Chomsky hierarchy and other important classes of languages. We discuss computations of unifiers between two terms. Then we give inductive inference algorithms including refinement operators for these subclasses and show their completeness.