Showing papers on "Undecidable problem published in 1981"

Embedded implicational dependencies and their inference problem

Abstract: It is shown that the general inference problem for embedded implicational dependencies (EIDs) is undecidable. For the more important case of finite inference (i.e., inference for finite data bases), the problem is not even recursively enumerable (r.e.); rather, it is complete in co-r.e. These results hold even for typed EIDs without equality, as well as for (untyped) template dependencies. The case for typed template dependencies remains open. The complexity of the inference problem for full dependencies has also been characterized - it is complete in exponential time for full implicational dependencies, and even for full typed template dependencies.

Equations between regular terms and an application to process logic

Abstract: Regular terms with the Kleene operations ∪,;, and * can be thought of as operators on languages, generating other languages An equation r1 = r2 between two such terms is said to be satisfiable just in case languages exist which make this equation true We show that the satisfiability problem even for *-free regular terms is undecidable Similar techniques are used to show that a very natural extension of the Process Logic of Harel, Kozen and Parikh is undecidable

Finite and infinite regular thue systems

Abstract: This dissertation studies certain classes of Thue system, concentrating on the Church-Rosser property. The following new results are obtained about infinite regular Thue systems S: (1) if S is Church-Rosser, the word problem is solvable in linear time; (2) if S is monadic Church-Rosser, it defines a nontrivial boolean algebra of DCFLs; (3) if S is monadic Church-Rosser and so is another system T, equivalence of S and T is decidable; (4) if S is monadic, it is decidable if S is Church-Rosser; (5) if S is not monadic it is undecidable if S is Church-Rosser. The following new results are obtained about finite Thue systems S: (1) it is undecidable if there exists another finite Thue system T which is equivalent to S and is Church-Rosser (respectively: almost confluent, preperfect); (2) it is undecidable if S generates a Church-Rosser congruence. Some of these results generalise results about finite Thue systems, and some answer what had previously been open questions. The results are obtained using the theories of automata and formal languages, of Turing machines, and of finitely presented groups.

How to Gödel a Frege-Russell: Gödel's incompleteness theorems and logicism

Abstract: Any viable philosophy of mathematics must square with incontrovertible metamathematical results. Thus, as is well known, any version of formalism committed to identifying mathematical truth with provability in some formal system is to be rejected in light of Godel's first incompleteness theorem, according to which any consistent formal system capable of representing the primitive recursive numbertheoretic functions has an undecidable sentence. In fact, Godel showed how, given any such system, one could construct such a sentence which, on the standard interpretation of its symbols, makes a definite claim about natural numbers. Thus, either such a Godel sentence or its negation must be an arithmetical truth, but neither is provable in the system for which it was constructed. Any formalist philosophy which respects "arithmetical truth" runs afoul of the first incompleteness theorem.' The impact of Godel's incompleteness theorems on logicism, however, is not so straightforwardly assessed. While there are independent grounds for regarding logicism (as conceived by Frege and Russell) as a lost cause, it remains instructive to examine the force of G6del's theorems in this quarter. In fact, a negative result (i.e., incompatibility of logicism with a metatheorem) could be quite powerful, especially if it does not depend on a particularly restrictive way of drawing the controversial line between logic and non-logic. The purpose of this note is to present precisely this kind of an argument: so long as logic is taken to be formalizable (i.e., logically valid formulas form a recursively enumerable set), a natural thesis of epistemological logicism is, modulo quite elementary assumptions, incompatible with Godel's second incompleteness theorem.2 However, for this argument, it needs to be assumed that the logicist system is to be finitely axiomatized. Extending the argument to the infinite case turns out to depend on a crucial step whose

Two-way counter machines and Diophantine equations

Abstract: Let Q be the class of deterministic two-way one-counter machines accepting only bounded languages. Each machine in Q has the property that in every accepting computation, the counter makes at most a fixed number of reversals. We show that the emptiness problem for Q is decidable. When the counter is unrestricted or when the machine is provided with two reversal-bounded counters, the emptiness problem becomes undecidable. The decidability of the emptiness problem for Q is useful in proving the solvability of some numbertheoretic problems. It can also be used to prove that the language L = {u1iu2i2|i≥0} cannot be accepted by any machine in Q (u1 and u2 are distinct symbols). The proof technique is new in that it does not employ the usual "pumping", "counting", or "diagonal" argument. Note that L can be accepted by a deterministic two-way machine with two counters, each of which makes exactly one reversal.

Decision problems of some statistically motivated monadic modal calculi

Abstract: Logical calculi corresponding to the theoretical level of statistical inference (as understood, for example, in foundations of GUHA-style hypothesis formation) may be described as some generalized monadic modal predicate calculi. It is shown that various such calculi are undecidable when endowed with general semantics (arbitrary probabilistic structures) but, roughly, all reasonable such calculi become decidable when semantics is restricted to identically independently distributed structures.

Program logic without binding is decidable

Abstract: When the "binding mechanisms" of assignment, quantification, and procedure definition are removed from a conventional first order total correctness logic of programs, the remaining logical system is decidable in time approximately one exponential in the length of the input. This system is maximal in the sense that the presence of any one of the three binding mechanisms would make it undecidable. Such a decision procedure can play a central role in the construction of program verifiers based on decision methods.

On restricted one-counter machines

Abstract: Let ℂ be the class of real-time nondeterministic one-counter machines whose counters make at mostone reversal. Let ℂ1 (respectively, ℂ2) be the subclass consisting of machines whose only nondeterministic move is in the choice of when to reverse the counter (respectively, when to start using the counter). ℂ1 and ℂ2 are among the simplest known classes of machines for which the universe problem has been shown undecidable. (The universe problem for a class of machines is the problem of deciding if an arbitrary machine in the class accepts all its inputs.) Here, we show that the classes of languages accepted by machines in ℂ1 and ℂ2 are incomparable. Moreover, the union of the language classes is properly contained in the class defined by ℂ. We also, briefly, look at the closure properties of these machines.

Undecidable Problems Associated with Combinatiorial Systems and Their One‐One Degrees of Unsolvability

Abstract: The genesis of the problems investigated in this paper are the results on Turing degrees and many-one degrees of undecidable problems of J. C. SHEPHERDSON [9] and R. OVERBEEK [7] and the results of J. P. CLEAVE 111, [2], [3] on one-one degrees. As was proved in [I] each of the following: halting problem, derivability problem and confulence problem of a deterministic combinatorial system is cylindrical. Thus, it is clear that OVERBEEK'S result version on many-one degrees was the best possible improvement of SHEPHERDSON'S representation theorem. The aim of this paper is to show that in case of nondeterministic systems these problems need not be cylindrical. Besides, results concerning the place of problems in DERKER-MYHILL classification of r.e. sets are obtained (Chap. 4) and home improvement of &EAvE's paper [2] is given (Chap. 3). Some results are partly annouced in [a] but the essential proofs are given in this paper (Theorems 2 and 4). The author wishes to express his sincere gratitude to Prof. A. W. MOSTOWSKI for his help and encouragement.

Computability Abstract Theory

Abstract: There are some abstract axiomatic theories which imitate certain features of arithmetical computability. For instance lambda-calculus (see “Combinatory logic”), though originated for other purposes, can be interpreted as a theory of this kind. But the most known one is due to Wagner (69) and was developed by Strong (70), Gross and Venturini Zilli (71) and others. It is an extension of lambda-calculus (LaC). The primitive notions are: (..,..) is a function of application as in LaC, and three individual constants denoted usually by *, Ψ, α.