Undecidability and Beyond
نویسنده
چکیده
We discuss the structure of the recursively enumerable sets under three reducibilities: Turing, truth-table and weak truth-table. Weak truth-table reducibility requires that the questions asked of the oracle be eeectively bounded. Truth-table reducibility also demands such a bound on the the length of the computations. We survey what is known about the algebraic structure and the complexity of the decision procedure for each of the associated degree structures. Each of these structures is an upper semilattice with least and greatest element. Typical algebraic questions include the existence of innma, dis-tributivity, embeddings of partial orderings or lattices and extension of embedding problems such as density. We explain how the algebraic information is used to decide fragments of the theories and then to prove their undecidability (and more). Finally, we discuss some results and open problems concerning automorphisms, deenability and the complexity of the decision problems for these degree structures.
منابع مشابه
The Boundary Between Decidability and Undecidability for Transitive-Closure Logics
To reason effectively about programs it is important to have some version of a transitive closure operator so that we can describe such notions as the set of nodes reachable from a program’s variables. On the other hand, with a few notable exceptions, adding transitive closure to even very tame logics makes them undecidable. In this paper we explore the boundary between decidability and undecid...
متن کاملUndecidability Results on Two-Variable Logics
It is a classical result of Mortimer that L 2 , rst-order logic with two variables, is decidable for satissability. We show that going beyond L 2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, viz. (i) transitive closure operations (ii) (restricted) monadic xed-point operations weak access to cardinalities, through the HH artig (or equicardinalit...
متن کاملWhat’s Decidable about Weak Memory Models? (Extended Version)
We investigate the decidability of the state reachability problem in finite-state programs running under weak memory models. In [3], we have shown that this problem is decidable for TSO and its extension with the write-to-write order relaxation, but beyond these models nothing is known to be decidable. Moreover, we have shown that relaxing the program order by allowing reads or writes to overta...
متن کاملUndecidability of a Very Simple Modal Logic with Binding
We show undecidability of the satisfiability problem of what is arguably the simplest non-sub-Boolean modal logic with an implicit notion of binding. This work enriches the series of existing results of undecidability of modal logics with binders, which started with Hybrid Logics and continued with Memory Logics. 1 Modal Logics, Names and Binders Modal Logics are languages that are able to desc...
متن کاملWhat's Decidable about Weak Memory Models?
We investigate the decidability of the state reachability problem in finite-state programs running under weak memory models. In [3], we have shown that this problem is decidable for TSO and its extension with the write-to-write order relaxation, but beyond these models nothing is known to be decidable. Moreover, we have shown that relaxing the program order by allowing reads or writes to overta...
متن کاملRelation 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 eeort 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 doi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007