Showing papers on "Recursively enumerable language published in 1999"

First Order Linear Temporal Logic over Finite Time Structures

Abstract: In this work, the notion of provability for first order linear temporal logic over finite time structures, FO-LTLfin, is studied. We show that the validity problem for such a logic is not recursively enumerable, hence FO-LTLfin is not recursively axiomatizable. This negative result however does not hold in the case of bounded validity, that is truth in all temporal models where the object domain is possibly infinite, but the underlying sequence of time points does not exceed a given size. A formula is defined to be k-valid if it is true in all temporal models whose underlying time frame is not greater them k, where k is any fixed positive integer. In this work a tableau calculus is defined, that is sound and complete with respect to k-validity, when given as input the initial formula and the bound k on the size of the temporal models. The main feature of the system, extending the prepositional calculus defined in [7], is that of explicitly denoting time points and having tableau nodes labelled by either expressions intuitively stating that a formula holds in a given temporal interval, or "temporal constrsiints", i.e. linear inequalities on time points. Branch closure is reduced to unsatisfiability over the integers of the set of temporal constreiints in the branch.

Cupping the recursively enumerable degrees by d.r.e. degrees

Abstract: We prove that there are two incomplete dre\ degrees (the Turing degrees of differences of two recursively enumerable sets) such that every non-zero recursively enumerable degree cups at least one of them to , the greatest recursively enumerable (Turing) degree

New computing paradigms suggested by DNA computing: computing by carving.

Abstract: Inspired by the experiments in the emerging area of DNA computing, a somewhat unusual type of computation strategy was recently proposed by one of us: to generate a (large) set of candidate solutions of a problem, then remove the non-solutions such that what remains is the set of solutions. This has been called a computation by carving. This idea leads both to a speculation with possible important consequences—computing non-recursively enumerable languages—and to interesting theoretical computer science (formal language) questions.

Chapter 6 - The Recursively Enumerable Degrees

Abstract: Decision problems are the motivating force in the search for a formal definition of algorithm that constituted the beginnings of recursion (computability) theory. In most settings, the notion of a recursively enumerable (r.e.) set is found: the theorems of a axiomatized theory, the solvable Diophantine equations, and the true equations among words in a finitely presented group. Typically, such decision problems amount to deciding if a particular r.e. set is computable. All these sets are simply non-computable. Another view sees them as more complicated or harder to compute than the recursive sets. This is the view that leads to the notion of relative computability (reducibility) introduced by Turing. The equivalence classes under this notion of relative computability are first called the “degrees of recursive unsolvability.” The starting point for the investigation of this fundamental notion of relative computability is the r.e. degrees. The classic results of logic, such as Godel's incompleteness theorem, Church's proof of the undecidability of predicate logic, and Turing's unsolvability of the Halting problem, each proved that there is a nonrecursive r.e. degree. All the natural examples, however, of nonrecursive r.e. sets supplied by standard theories that could be proven undecidable or from other natural definitions of noncomputable r.e. sets, turned out to have the same complexity.

Iterated GSM Mappings: A Collapsing Hierarchy

Abstract: With motivations from various areas (Lindenmayer systems, iterated reading of literary works, self-generated infinite sequences, "computing by carving" as suggested in the DNA computing area), in several places mechanisms based on iterated (non-deterministic) finite state sequential transducers were considered. It is known that such mechanisms can characterize the family of recursively enumerable languages. We continue here the study of such devices, investigating the hierarchy on the number of states. We find that this hierarchy collapses: four states are enough in order to characterize the recursively enumerable languages, three states lead to non-recursive languages and cover the ET0L languages, while two states can cover the E0L (hence also context-free) languages. The case of deterministic transducers remains open.

Table-driven and context-sensitive collage languages.

Abstract: In this paper, we introduce the notions of context-sensitive and ET0L collage grammars as generalizations of context-free collage grammars Both kinds of picture-generating devices are more powerful than the context-free case Nevertheless, the size of collages in an ET0L collage language can be shown to grow at most exponentially In contrast to this, there are no such bounds for context-sensitive collage languages because suitable pictorial representations of recursively enumerable sets of strings can be generated On the other hand, it is still a conjecture that ET0L collage languages exist that are not context-sensitive

Parallel functional programming on recursively defined data via data-parallel recursion

Abstract: This article proposes a new language mechanism for data-parallel processing of dynamically allocated recursively defined data. Different from the conventional array-based data- parallelism, it allows parallel processing of general recursively defined data such as lists or trees in a functional way. This is achieved by representing a recursively defined datum as a system of equations, and defining new language constructs for parallel transformation of a system of equations. By integrating them with a higher-order functional language, we obtain a functional programming language suitable for describing data-parallel algorithms on recursively defined data in a declarative way. The language has an ML style polymorphic type system and a type sound operational semantics that uniformly integrates the parallel evaluation mechanism with the semantics of a typed functional language. We also show the intended parallel execution model behind the formal semantics, assuming an idealized distributed memory multicomputer.

DNA) computing by carving

Abstract: Inspired by the experiments reported recently in the emerging area of DNA computing, we consider a somewhat unusual type of a computation strategy: generate a (large) set of candidate solutions of a problem, then remove the non-solutions such that what remains is the set of solutions We call this a computation by carving This leads both to a speculation with possible important consequences and to interesting theoretical computer science (formal language) questions The speculation is that in this way we can “compute” non-recursively enumerable languages, because the family of recursively enumerable languages is not closed under complementation The formal language theory questions concern sequences of languages with certain regularities, needed as languages to be extracted from the total language of candidate solutions of a problem Specifically, we consider sequences of languages obtained by starting from a given regular language and iteratively applying to it a given finite state sequential transducer (a gsm) Computing by carving with respect to such a sequence of languages can identify all context-sensitive languages and can also lead to non-recursively enumerable languages (but not all recursively enumerable languages can be obtained in this way) In practical circumstances, the carving process should be finite, hence, in general, approximations of the desired language are obtained We also briefly discuss this aspect

Craig’S Theorem and the Empirical Underdetermination Thesis Reassessed

Abstract: The present paper proposes to revive the twenty-year old debate on the question of whether Craig’s theorem poses a challenge to the empirical underdetermination thesis. It will be demonstrated that Quine’s account of this issue in his paper “Empirically Equivalent Systems of the World” (1975) is mathematically flawed and that Quine makes too strong a concession to the Craigian challenge. It will further be pointed out that Craig’s theorem would threaten the empirical underdetermination thesis only if the set of all relevant observation conditionals could be shown to be recursively enumerable — a condition which Quine seems to overlook —, and it will be argued that, at least within the framework of Quine’s philosophy, it is doubtful whether this condition is satisfiable.

Decidability and Undecidability Results for Recursive Petri Nets

Abstract: Recursive Petri nets (RPNs) have been introduced to model systems with dynamic structure. Whereas this model is a strict extension of Petri nets, reachability in RPNs remains decidable. Here we focus on three complementary theoretical aspects. At first, we develop decision procedures for new properties like boundedness and finiteness and we show that languages of RPNs are recursive. Then we study the expressiveness of RPNs proving that any recursively enumerable language may be obtained as the image by an homomorphism of the intersection of a regular language and a RPN language. Starting from this property, we deduce undecidability results including undecidablity for the kind of model checking which is decidable for Petri nets. At last, we compare RPNs with two other models combining Petri nets and context-free grammars features showing that these models can be simulated by RPNs.

Avoiding Coding Tricks by Hyperrobust Learning

Abstract: The present work introduces and justifies the notion of hyperrobust learning where one fixed learner has to learn all functions in a given class plus their images under primitive recursive operators. The following is shown: This notion of learnability does not change if the class of primitive recursive operators is replaced by a larger enumerable class of operators. A class is hyperrobustly Ex-learnable iff it is a subclass of a recursively enumerable family of total functions. So, the notion of hyperrobust learning overcomes a problem of the traditional definitions of robustness which either do not preserve learning by enumeration or still permit topological coding tricks for the learning criterion Ex. Hyperrobust BC-learning as well as the hyperrobust version of Ex-learning by teams are more powerful than hyperrobust Ex-learning. The notion of bounded totally reliable BC-learning is properly between hyperrobust Ex-learning and hyperrobust BC-learning. Furthermore, the bounded totally reliably BC-learnable classes are characterized in terms of infinite branches of certain enumerable families of bounded recursive trees. A class of infinite branches of a further family of trees separates hyperrobust BC-learning from totally reliable BC-learning.

The cupping theorem in r/m

Abstract: It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] ≺ [b] ≺ [a] there exists [c] ∈ R/M such that [c] ≺ [a] and [a] = [b] ∨ [c].

An Overview of the Computably Enumerable Sets

Abstract: This chapter summarizes some of the results of the algebraic structure of the computably enumerable (c.e.) sets since 1987, when the subject was covered in Soare. In addition to defining computable functions, there was in interest in defining computable generated sets. Church and Kleene defined a set of positive integers to be “recursively enumerable,” if it is the range of a recursive function. A little more was done with these sets until Post proposed a formal system for generating sets rather than computing their characteristic functions. Post showed that the normal sets are exactly the recursively enumerable (r.e.) sets, providing the empty set is added as an r.e. set. Post, however, thought not so much in formal systems as in informal terms and described the corresponding informal concept of effectively enumerable set or generated set. The chapter considers the various properties of Є: namely, definable properties, automorphisms, invariant properties, decidability and undecidability results, and miscellaneous results.

On a Class of Recursively Enumerable Sets

Abstract: We define a class of so-called ∑(n)-sets as a natural closure of recursively enumerable sets Wn under the relation “∈” and study its properties.

Descriptional complexity of multi-continuous grammars

Abstract: The present paper discusses multi-continuous grammars and their descriptional complexity with respect to the number of nonterminals. It proves that six-nonterminal multi-continuous grammars characterize the family of recursively enumerable languages. In addition, this paper formulates an open problem area closely related to this characterization.

Binary enumerability of real numbers

Abstract: A real number x is called binary enumerable, if there is an effective way to enumerate all "1"-positions in the binary expansion of x. If at most k corrections for any position are allowed in the above enumerations, then x is called binary k-enumerable. Furthermore, if the number of the corrections is bounded by some computable function, then x is called binary ω-enumerable. This paper discusses some basic properties of binary enumerable real numbers. Especially, we show that there are two binary enumerable real numbers x and y such that their difference x - y is not binary ω-enumerable (in fact we have shown that it is even of no "ω-r.e. Turing degree").

There is no complete axiom system for shuffle expressions

Abstract: In this paper we show that neither the set of all valid equations between shuffle expressions nor the set of schemas of valid equations is recursively enumerable. Thus, neither of the sets can be recursively generated by any axiom system.

Learning from Random Text

Abstract: Learning in the limit deals mainly with the question of what can be learned, but not very often with the question of how fast. The purpose of this paper is to develop a learning model that stays very close to Gold's model, but enables questions on the speed of convergence to be answered. In order to do this, we have to assume that positive examples are generated by some stochastic model. If the stochastic model is fixed (measure one learning), then all recursively enumerable sets are identifiable, while straying greatly from Gold's model. In contrast, we define learning from random text as identifying a class of languages for every stochastic model where examples are generated independently and identically distributed. As it turns out, this model stays close to learning in the limit. We compare both models keeping several aspects in mind, particularly when restricted to several strategies and to the existence of locking sequences. Lastly, we present some results on the speed of convergence: In general, convergence can be arbitrarily slow, but for recursive learners, it cannot be slower than some magic function. Every language can be learned with exponentially small tail bounds, which are also the best possible. All results apply fully to Gold-style learners, since his model is a proper subset of learning from random text.

Multiset Processing by Means of Systems of Finite State Transducers

Abstract: We introduce a computing mechanism of a biochemical inspiration (similar to a P system from the area of Computing with Membranes) which consists of a multiset of symbol-objects and a set of finite state transducers. The transducers process symbols in the current multiset in the usual manner. A computation starts in an initial configuration and ends in a halting configuration. The power of these mechanisms is investigated, as well as the closure properties of the obtained family. The main results say that (1) systems with two components and an unbounded number of states in each component generate all gsm images of all permutation closures of recursively enumerable languages, while (2) systems with two states in each component but an unbounded number of components can generate the permutation closures of all recursively enumerable languages, and (3) the obtained family is a full AFL. Result (2) is related to a possible (speculative) implementation of our systems in biochemical media.

The Complexity of Universal Text-Learners

Abstract: The present work deals with language learning from text. It considers universal learners for classes of languages in models of additional information and analyzes their complexity in terms of Turing degrees. The following is shown: If the additional information is given by a set containing at least one index for each language from the class to be learned but no index for any language outside the class, then there is a universal learner having the same Turing degree as the inclusion problem for recursively enumerable sets. This result is optimal in the sense that any other successful learner has the same or higher Turing degree. If the additional information is given by the index set of the class of languages to be learned then there is a computable universal learner. Furthermore, if the additional information is presented as an upper bound on the size of some grammar that generates the language, then a high oracle is necessary and sufficient. Finally, it is shown that for the concepts of finite learning and learning from good examples, the index set of the class to be learned gives insufficient information due to the restrictive convergence constraints, these criteria need the jump of the index set instead of the index set itself. So, they have infinite access to the information of the index set in finite time.

Grammar Systems as Language Analyzers and Recursively Enumerable Languages

Abstract: We consider parallel communicating grammar systems which consist of several grammars and perform derivation steps, where each of the grammars works in a parallel and synchronized manner on its own sentential form, and communication steps, where a transfer of sentential forms is done. We discuss accepting and analyzing versions of such grammar systems with context-free productions and present characterizations of the family of recursively enumerable languages by them. In accepting parallel communicating grammar systems rules of the form α → A with a word α and a nonterminal A are applied as in the generating case, and the language consists of all terminal words which can derive the axiom. We prove that all types of these accepting grammar systems describe the family of recursively enumerable languages, even if λ-rules are forbidden. Moreover, we study analyzing parallel communicating grammar systems, the derivations of which perform the generating counterparts backwards. This requires a modification of the generating derivation concept to strong-returning parallel communicating grammar systems which also generate the family of recursively enumerable languages.

Prefix pushdown automata and their simplification

Abstract: A prefix pushdown automatonM, accepts a wordx, with respect to a languageZ, if and only if M makes a sequence of moves so it reads xy, for some y∊Z, and enters a final state. This paper demonstrates that for every recursively enumerable languageL, there exist a linear languageZ, and a prefix pushdown automatonM, so that L equals the prefix language that M accepts with respect to Z. Besides the acceptance by final state, this result is established in terms of acceptance by empty pushdown and acceptance by final state and empty pushdown. In addition, the present paper demonstrates this result for some simplified versions of prefix pushdown automata. Finally, it discusses the descriptional complexity of these automata

Selected logic papers

Abstract: Recursive enumerability and the jump operator on the degrees less than 0' a simple set which is not effectively simple the recursively enumerable degrees are dense metarecursive sets (with G. Kreisel) Post's problem, admissable ordinals and regularity on a theorem of Lachlan and Martin a minimal hyperdegree (with R.O. Gandy) measure-theoretic uniformity in recursion theory and set theory forcing with perfect closed sets recursion in objects of finite type the a-finite injury methods (with S.G. Simpson) remarks against foundational activity countable admissible ordinals and hyperdegrees the 1-section of a type 'n' object the k-section of a type 'n' object Post's problem, absoluteness and recursion in finite types effective bonds on Morley rank on the number of countable models Post's problem in E-recursion the limits of E-recursive enumerability effective versus proper forcing.

Middle quotients of linear languages

Abstract: For two languagesX and Y, the middle quotient of X with respect to Y is denoted by and defined as and In addition, if coincides with where and u is the reversal of v}, then is a stable middle quotient, denoted by This paper proves that for every recursively enumerable language and , where A B C, and D are a linear language, a deterministic linear binary language, a linear language, and a minimal deterministic linear ternary language, respectively Consequently and hold, too

Some reducibilities and splittings of recursively enumerable sets

Abstract: The existence of a recursively enumerable (RE)T-degreea that does not contain an RE semirecursive setA ∈a with theQ-universal splitting property is proved. Each nonrecursive RE contiguous degree contains an RE setA with the universalT-Q-reduction property, butA is notT-Q-maximal. Each nonrecursive REW-degree contains an RE setA with the universalW-sQ-reduction property, butA is notW-sQ-maximal. Each creative set is partially semimaximal.