Showing papers on "Undecidable problem published in 1986"

Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness

Abstract: Elementary translations between various kinds of recursive trees are presented. It is shown that trees of either finite or countably infinite branching can be effectively put into one-one correspondence with infinitely branching trees in such a way that the infinite paths of the latter correspond to the “P-abiding” infinite paths of the former. Here P can be any member of a very wide class of properties of infinite paths. For many properties ??, the converse holds too. Two of the applications involve (a) the formulation of large classes of highly undecidable variants of classical computational problems, and in particular, easily describable domino problems that are III11-complete, and (b) the existence of a general method for proving termination of nondeterministic or concurrent programs under any reasonable notion of fairness.

Checking Consistency of Database Constraints: a Logical Basis

Abstract: This paper addresses the problem of consistency of a set of integrity con4raints itself, independent from any state. 11 is pointed out that database constraints have not only to ,be consistent. but in addition to be finitely .+atisfiablc. Thb stronger property reflects that the constraints have to admit a finite set of [stored ps well as derivqble) facts. As opposed .tu consistency. being undecidable, linite satisfiability is semidecidable. For effickncy purposes WC investigate methods that check both linite s&i&ability as ,well as unratisfiability. Two different methods are proposed which extend two ajternative approaches to refutation. ventional databases, constraints have to admit finite models as every state consists of a finite ‘number of facts. In definile drdubtivr databases (as defined in 191) the set of deduction ryle: always has a linite minimal model, which is intended to be a model of the constraint set as well. Sqtisfiability does not necessarily imply finite sat+xbiUy, i.e., the existence of a finite model. There are ‘satisfiabk sets of for&n&s -’ called ‘axioms of, infinity’ that have ‘onI?: infinite models. Consider, e.g., a se1 of integrity constraints for a managerial database containing (among others) the following constraints: l Everybody works for somebody. . Nobody works for himself. ’ . If x works for y and F works for t, then x works for L.

The logic of distributed protocols: preliminary report

Abstract: A propositional logic of distributed protocols is introduced which includes both the logic of knowledge and temporal logic. Phenomena in distributed computing systems such as asynchronous time, incomplete knowledge by the computing agents in the system, and game-like behavior among the computing agents are all modeled in the logic. Two versions of the logic, the linear logic of protocols (LLP) and the tree logic of protocols (TLP) are investigated. The main result is that the set of valid formulas in LLP is undecidable.

On the progress of communication between two finite state machines

Abstract: We consider the following problem concerning any two finite state machines M and N that exchange messages via two 1-directional channels. “Is there a positive integer K such that the communication between M and N over K -capacity channels is guaranteed to progress indefinitely?” The problem is shown to be undecidable in general. For a practical class of communicating machines, the problem is shown to be decidable, and the decidability algorithm is polynomial. We also discuss some sufficient conditions for the problem to have a positive answer; these sufficient conditions can be checked for the given M and N in polynomial time. We apply the results to some practical protocols to show that their communications will progress indefinitely.

"Type" is not a type

Abstract: A function has a dependent type when the type of its result depends upon the value of its argument. Dependent types originated in the type theory of intuitionistic mathematics and have reappeared independently in programming languages such as CLU, Pebble, and Russell. Some of these languages make the assumption that there exists a type-of-all-types which is its own type as well as the type of all other types. Girard proved that this approach is inconsistent from the perspective of intuitionistic logic. We apply Girard's techniques to establish that the type-of-all-types assumption creates serious pathologies from a programming perspective: a system using this assumption is inherently not normalizing, term equality is undecidable, and the resulting theory fails to be a conservative extension of the theory of the underlying base types. The failure of conservative extension means that classical reasoning about programs in such a system is not sound.

Diophantine theories of free inverse semigroups

Abstract: It is known that in a free inverse semigroup I the problem of equality of words is decidable [I]. It has been established in [2] that the elementary theory of I is decidable in the case when I is monogenic and inseparable in other cases. For finitely generated nonmonogenic free inverse semigroups the undecidability of positive theories was proved in [3]. The main goal of the present note is to prove that the problem of consistency of systems of equations on a nonmonogenic free inverse semigroup is undecidable.

Complexity of Place/Transition Nets

Abstract: This is a survey of results about the complexity of decision problems for various questions about Petri nets that arise in the analysis of systems. The border between undecidable and decidable problems is discussed first and then problems are presented by decreasing complexity. As a consequence of the results presented it will follow that one has to concentrate on very restricted classes of systems in order to get practically relevant algorithms that work well for all cases, since even seemingly simple classes of Petri nets have simple problems with a provable high lower bound for the complexity of their sol’ution.

On two problems related to cancellativity

Abstract: Two decision problems that are related to the properties of right-cancellativity and left-cancellativity, respectively, of the monoidmT defined by a presentation (Σ;T), are investigated. It is shown that these problems are undecidable in general. In fact, they remain undecidable, even when they are restricted to presentations involving finite Church-Rosser Thue systems. On the other hand, if only finite presentations involving monadic Church-Rosser Thue systems are considered, then these two problems become decidable in polynomial space.

On the equivalence of some transductions involving letter to letter morphisms on regular languages

Abstract: Equivalence problems of some transductions involving letter to letter morphisms on regular languages are discussed. In particular, we deal with finite substitutions and inverses of finite substitutions. Our main results are the following: (i) The equivalence problem of inverses of finite substitutions on regular languages is undecidable, (ii) The existential equivalence problem of finite substitutions on regular languages is undecidable, and (iii) The length-equivalence problem of finite substitutions on regular languages is decidable.

The emptiness problem for indexed language is exponential‐time complete

Abstract: The class of indexed languages properly includes the class of context-free languages and is properly included in the class of context-dependent languages [1]. The emptiness problem (the problem of determining whether or not the given language is empty) is polynomial-time complete for the class of context-free languages and is undecidable for the class of context-dependent languages. The recognition problem (the problem, given a language L and word w, of determining whether or not w belongs to L) is polynomial-time complete for the class of context-free languages and is polynomialspace complete for the class of contextdependent languages. This paper shows that both the emptiness and recognition problems are exponential-time complete for the class of indexed languages. It is known in the pebble game [2] that the problem of determining whether or not the first player has the winning strategy is exponential-time complete. This paper reduces the problem to the emptiness problem for the class of indexed languages in the logarithmic space, indicating the exponential-time difficulty of the emptiness problem for the indexed language. Since Aho has shown that the problem can be answered in exponential time, the exponential-time completeness is shown. The exponential-time difficulty is also directly indicated from the fact that the emptiness problem is exponential-time complete. Consequently, the recognition problem is also exponential-time complete.

Complexity of Sufficient-Completeness

Abstract: The sufficient-completeness property of equational specifications has been found useful in providing guidelines for abstract data type specifications as well as in proving inductive properties using the inductionless-induction method. The sufficient-completeness property is known to be undecidable in general. In an earlier paper, it was shown to be decidable for constructor-preserving, complete (canonical) term rewriting systems, even when there are relations between constructors. The complexity of the sufficient-completeness property under certain conditions is discussed. It is shown that the sufficient-completeness problem for term rewriting systems without any relations on constructors is co-NP-complete. However, the problem is PSPACE-hard even for linear constructor-preserving term rewriting systems when relations on constructors are allowed.

Checking Consistency of Database Constraints

Abstract: This paper addresses the problem of consistency of a set of integrity constraints itself, independent from any state. It is pointed out that database constraints have not only to be consistent, but in addition to be finitely satisfiable. This stronger property reflects that the constraints have to admit a finite set of (stored as well as derivable) facts. As opposed to consistency, being undecidable, finite satisfiability is semidecidable. For efficiency purposes we investigate methods that check both finite satisfiability as well as unsatisfiability . Two different methods are proposed which extend two alternative approaches to refutation.