Proof Of Undecidability Research Articles

EMPLOTTING ANALYSIS

Anita Burdman Feferman and Solomon Feferman, Alfred Tarski: Life and Logic (Cambridge: Cambridge University Press, 2004)Murray G. Murphey, C. I. Lewis: The Last Great Pragmatist (Albany, NY: State University of New York Press, 2005)The lives of analytic philosophers and logicians do not prima facie appear to be the stuff of popular intellectual biographies. Who but former students or specialists would want to know the details of lives devoted to academic pursuits that, so it would seem, all but exclude personal expression? Demonstrative cultural documents such as Also sprach Zarathustra lay claim to our attention in part because they are the product of minds in dynamic tension with the culture that surrounded them, but Gödel's undecidability proof, for example, owes nothing of its importance to its author's social or psychological life at the time he wrote it. The life of the analyst or logician would, in short, appear to be beside the point—window dressing at best, obfuscating at worst.

Robust parametric reachability for timed automata

We review the known decidability and undecidability results for reachability in parametric timed automata. Then, we present a new proof of undecidability in dense time for open timed automata that avoids equalities in clock constraints. Our result shows that the undecidability of parametric timed automata does not follow from their ability to specify punctual constraints in a dense time domain.

Folk theorems on the determinization and minimization of timed automata

Timed automata are known not to be complementable or determinizable. Natural questions are, then, could we check whether a given TA enjoys these properties? These problems are not algorithmically solvable. Minimizing the “resources” of a TA (number of clocks or size of constants) are also unsolvable problems. In this paper we provide simple undecidability proofs using a “constructive” version of the problems where we require not just a yes/no answer, but also a “witness”. Proofs are then simple reductions from the universality problem. Recent work of Finkel shows that the corresponding decision problems are also undecidable [O. Finkel, On decision problems for timed automata, Bulletin of the European Association for Theoretical Computer Science 87 (2005) 185–190].

Decidability without mathematics

The paper proposes a new definition of the conception of effectiveness (computability, general recursiveness, algoritmicity). A good name for this version of effectiveness is discernibility. The definition is based on the fact that every computation may be reduced to the operation of discerning the fundamental symbols and concatenation of formulas. This approach to effectiveness allows us to formulate the proof of undecidability in such a way that arithmetization of the syntax may be replaced by the use of concatenation in metalogic.

UNDECIDABLE EMOTIONS (WHAT IS SOCIAL THERAPY? AND HOW IS IT REVOLUTIONARY?)

Developments in the foundations of mathematics, in particular, Gödel's undecidability proof are brought to bear on group therapy, social therapy, and human emotionality. The pre-Gödelian notion that the group is nothing more than the sum total of its individual members fosters the positivistic position that what is happening in a group is decidable or determinable. In contrast, it is argued that groups, like formalized mathematics, are undecidable. This point and its therapeutic implications are expanded upon through a discussion of the group practice of social therapy. Examples of social therapy group activity are provided to illustrate the methodological and philosophical character of the therapy, and in particular, the importance of creative unresolvability to emotional growth.

Modal Logics Between Propositional and First-order

One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery produces a decidable logic, but adding it to S5 produces an undecidable one. Further, if an equality symbol is in the language, and interpreted by the equality relation, logics from K4 to S5 yield undecidable versions. (Thus transitivity is the villain here.) The proof of undecidability consists in showing that classical first-order logic can be embedded.

Special Cases and Substitutes for Rigid E -Unification

The simultaneous rigid E-unification problem arises naturally in theorem proving with equality. This problem has recently been shown to be undecidable. This raises the question whether simultaneous rigid E-unification can usefully be applied to equality theorem proving. We give some evidence in the affirmative, by presenting a number of common special cases in which a decidable version of this problem suffices for theorem proving with equality. We also present some general decidable methods of a rigid nature that can be used for equality theorem proving and discuss their complexity. Finally, we give a new proof of undecidability of simultaneous rigid E-unification which is based on Post's Correspondence Problem, and has the interesting feature that all the positive equations used are ground equations (that is, contain no variables).

Undecidability of compass logic

It is known that the tiling technique can be used to give simple proofs of undecidability of various two-dimensional modal and temporal logics. However, up until now, the simplest two-dimensional temporal logic, the compass logic of Venema, has eluded such treatment. We present a new coding of an enumeration of the tiling plane which enables us to show that the compass logic is undecidable.

Relation algebras can tile

Undecidability of the equational theory of the class RA of relation algebras can easily be proved using the undecidability of the word-problem for semigroups. With some effort and ingenuity, one can push this proof through for the larger class SA . We provide another “cause” for undecidability which works for even larger classes than SA . The reason is that we can encode the tiling problem. In doing so we will meet very simple BAO-varieties with undecidable equational theories which might be useful in other undecidability proofs. Our work is part of the research project which tries to establish the border between undecidability and decidability in relational type algebras, cf. [15,16,12,1] and the references therein. The ultimate goal of this research is to come up with versions of relational algebra which are still suitable for modern dynamic applications but whose equational theory is decidable or even tractable.

Communication errors in the [formula omitted]-calculus are undecidable

We present an undecidability proof of the notion of communication errors in the polyadic π-calculus. The demonstration follows a general pattern of undecidability proofs—reducing a well-known undecidable problem to the problem in question. We make use of an encoding of the λ-calculus into the π-calculus to show that the decidability of communication errors would solve the problem of deciding whether a lambda term has a normal form.

Divergence of F⩽ type checking

System F ⩽ is an extension of second-order typed lambda calculus, where a subtype hierarchy among types is defined, and bounded second-order lambda abstraction is allowed. This language is a basis for much of the current research on integration of typed functional languages with subtypes and inheritance. An algorithm to perform type checking for F ⩽ expressions has been known since the language Fun was defined. The algorithm has been proved complete, by the author and Curien, which means that it is a semi-decision procedure for the type-checking problem. In this paper we show that this algorithm is not a decision procedure, by exhibiting a term which makes it diverge. This result was the basis of Pierce's proof of undecidability of typing for F ⩽. We study the behavior of the algorithm to show that our diverging judgement is in some sense contained in any judgement which makes the algorithm diverge. On the basis of this result, and of other results in the paper, we claim that the chances that the algorithm will loop while type-checking a “real program” are negligible. Hence, the undecidability of F ⩽ type-checking should not be considered as a reason to prevent the adoption of F ⩽ as a basis for defining programming languages of practical interest. Finally, we show the undecidability of an important subsystem of F ⩽.

Termination of term rewriting: interpretation and type elimination

We investigate proving termination of term rewriting systems by interpretation of terms in a well-founded monotone algebra. The well-known polynomial interpretations can be considered as a particular case in this framework. A classification of types of termination, including simple termination, is proposed based on properties in the semantic level. A transformation on term rewriting systems eliminating distributive rules is introduced. Using this distribution elimination a new termination proof of the system SUBST of Hardin and Laville (1986) is given. This system describes explicit substitution in λ-calculus.Another tool for proving termination is based on introduction and removal of type restrictions. A property of many-sorted term rewriting systems is called persistent if it is not affected by removing the corresponding typing restriction. Persistence turns out to be a generalization of direct sum modularity, but is more powerful for both proving confluence and termination. Termination is proved to be persistent for the class of term rewriting systems for which not both duplicating rules and collapsing rules occur, generalizing a similar result of Rusinowitch for modularity. This result has nice applications, in particular in undecidability proofs.

A new method for undecidability proofs of first order theories

We claim that the reduction of Post's Correspondence Problem to the decision problem of a theory provides a useful tool for proving undecidability of first order theories given by some interpretation. The goal of this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering.

An undecidable problem in correspondence theory

In this paper we prove undecidability of first-order definability of propositional formulas. The main result is proved for intuitionistic formulas, but it remains valid for other kinds of propositional formulas by analogous arguments or with the help of various translations.For general background on correspondence theory the reader is referred to van Benthem [1], [2] (see [3] for a survey of recent results).The method for the proofs of undecidability in this paper will be to simulate calculations of a Minsky machine by intuitionistic formulas. §3 concerns this simulation. Effective procedures for the construction of simulating modal formulas can be found in the literature (cf. [4]).The principal results of the paper are in §4. §5 gives some further undecidability results, that will be proved in another paper by modification of the method of this paper.I am indebted to the referee for drawing my attention to some errors in an earlier version of this paper.

A new proof of undecidability of safety of logic queries

A new proof of undecidability of safety of logic queries

Theory of Boolean algebras with a locally finite group of automorphisms

In this note we consider the class of model K(G)= { ,] B is a Boolean algebra in the signature , ~ is an automorphism of B from the group G}. It has been shown in [i] that if G is a finite solvable group, then Th(K(G)) is decidable. It has been independently proved in [2, 3] that if G is an arbitrary finite group, then Th(K(G)) is decidable. It has been proved in [3, 4] that if G is not a locally finite group, then Th(K(G)) is undecidable. In [3] a conjecture is proposed that if G is a locally finite recursive group, then Th(K(G)) is decidable. In the present article a family of locally finite recursive groups F is constructed such that Th(K(G)) is hereditarily undecidable for all G.~ F. The proof of undecidability relies on [5].

Concise survey of mathematical logic

AbstractThis paper contains proofs of the compactness, completeness and Löwenheim-Skolem theorems for predicate logic, together with their application to nonstandard numbers; proofs of undecidability in predicate logic and number theory, and the Gödel incompleteness theorem.

Michael O. Rabin. A simple method for undecidability proofs and some applications. Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, edited by Yehoshua Bar-Hillel, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 38–68.

Michael O. Rabin. A simple method for undecidability proofs and some applications. Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, edited by Yehoshua Bar-Hillel, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 38–68.

What makes some language theory problems undecidable

In the theory of automata and formal languages, the undecidability of various properties has been studied for specific classes of languages. Here we abstract the essence of various proofs of undecidability and find wide classes of properties and general conditions on families of languages such that these proofs of undecidability hold. The paper also illustrates the manner in which the degree of undecidability of a property changes as we consider more and more complicated families of languages.

Decision Problems of Phrase-Structure Grammars

Phrase-structure grammars were first introduced and studied by Chomsky as devices for generating the sentences of a language. By means of increasingly heavy restrictions on the productions (rewriting rules), four types of grammars were singled out by Chomsky: type 0 (unrestricted), type 1 (context-dependent), type 2 (context-free), and type 3 (finite state). In this paper, a number of decision problems are resolved for these classes of grammars and the languages they generate, largely in the negative. A table of decision problems for grammars of the four different types is presented. This table indicates the problems which have been found to be decidable or undecidable. The ambiguity problem for type 3 grammars and the emptiness and infiniteness problems for type 2 grammars are shown to be decidable. A known unsolvable problem, the Post correspondence problem, is the key to the undecidability proofs which are given. For type 2 grammars, the ambiguity and equivalence problems are proved undecidable; the emptiness and infiniteness problems for type 1 grammars are shown to be undecidable. There is no algorithm to decide whether a language of a given type can be generated by a grammar of a more restricted type. The results on type 2 grammars were first obtained by Bar-Hillel, Perles, and Shamir.