The ∀∃-theory of R(≤, ∨, ∧) is undecidable

The ∀∃-theory of R(≤, ∨, ∧) is undecidable (2004)

Cached

Download Links

by Russell G. Miller , Andre O. Nies , Richard , A. Shore

Venue:	Trans. Amer. Math. Soc
Citations:	1 - 0 self

BibTeX

@ARTICLE{Miller04the∀∃-theory,
    author = {Russell G. Miller and Andre O. Nies and Richard and A. Shore},
    title = {The ∀∃-theory of R(≤, ∨, ∧) is undecidable},
    journal = {Trans. Amer. Math. Soc},
    year = {2004},
    volume = {356}
}

Bookmark

OpenURL

Abstract

Abstract. The three quantifier theory of (R, ≤T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ≤T but includes function symbols. Theorem. The two quantifier theory of (R, ≤, ∨, ∧), the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on R) is undecidable. The same result holds for various lattices of ideals of R which are natural extensions of R preserving join and infimum when it exits. 1.

Citations

411	Recursively Enumerable Sets and Degrees - Soare - 1987
259	On a problem of formal logic - Ramsey
239	Y.: The Classical Decision Problem - Börger, Grädel, et al. - 1996
79	Recursive unsolvability of Post’s problem of “Tag” and other topics in theory of Turing machines - Minsky - 1961
70	W.D.: The Decision Problem: Solvable Classes of Quantificational Formulas - Dreben, Goldfarb - 1979
66	Degrees of Unsolvability - Lerman - 1983
51	Computability of recursive functions - Shepherdson, Sturgis - 1963
44	Two recursively enumerable sets of incomparable degrees of unsolvability - Friedberg - 1957
42	1997], Logic for Applications - Nerode, Shore
31	The upper semi-lattice of degrees of recursive unsolvability - Kleene, Post - 1954
31	The recursively enumerable degrees are dense - Sacks - 1964
30	On the unsolvability of the problem of reducibility in the theory of algorithms - Muchnik - 1956
30	1998], Interpretability and definability in the recursively enumerable degrees - Nies, Shore, et al.
27	A recursively enumerable degree which will not split over all lesser ones - Lachlan
26	On the degrees of recursive unsolvability - Spector - 1956
25	Zum entscheidungsproblem der mathematischen logik - Bernays, Schonfinkel - 1928
23	Recursive enumerability and the jump operator - Sacks - 1963
21	1968], Distributive initial segments of the degrees of unsolvability - Lachlan
15	First order theory of the degrees of recursive unsolvability - Simpson
14	Decidability and invariant classes for degree structures - Lerman, Shore - 1988
14	1981],The theory of the degrees below 0 - Shore - 1981
12	Initial segments of the degrees of unsolvability - Lerman - 1971
10	1982], The undecidability of the recursively enumerable degrees (research announcement - Harrington, Shelah
10	1984], The Elementary Theory of the Recursively Enumerable Degrees is not ℵ0-categorical - Lerman, Shore, et al.
9	On the ∀∃-sentences of α-recursion theory - Shore - 1978
9	1954] The upper semi-lattice of degrees of recursive unsolvability - Kleene, Post
7	Admissible ordinals and priority arguments - Lerman
7	2001], Extension of embeddings in the computably enumerable degrees - Slaman, Soare
7	1999], The recursively enumerable degrees - Shore
6	1966], The impossibility of finding relative complements for recursively enumerable degrees - Lachlan
6	1998], The Π3-theory of the computably enumerable Turing degrees is undecidable - Lempp, Nies, et al.
6	On the 89-sentences of ff-recursion theory, in Generalized Recursion Theory - Shore
5	Incomparable prime ideals of recursively enumerable degrees - Calhoun - 1993
5	On the Σ2-theory of the upper semilattice of Turing degrees - Slaman - 1993
4	A necessary and sufficient condition for embedding principally decomposable finite lattices into the computably enumerable degrees, Annals of Pure and Applied Logic 101 - Lerman - 2000
3	On the ∀∃-sentences of α-recursion theory, in Generalized Recursion Theory - Shore
3	On the \Sigma 2-theory of the upper semilattice of Turing degrees - Slaman - 1993
3	1998], The \Pi 3-theory of the computably enumerable Turing degrees is undecidable - Lempp, Nies, et al.
2	2001], Embedding finite lattices into the ideals of computably enumerable Turing degrees - C, Lerman
2	The Recursively Enumerable Degrees’, in Handbook of Computability Theory - Shore
2	Parameter definability in the recursively enumerable degrees, to appear - Nies
1	1979], Degrees of Unsolvability: Structure and Theory, LNM 759 - Epstein
1	On the degrees of index sets, Trans - Yates
1	satisfying Nk. (zρ)[s] (zρ) correctly at the next Börger - unknown authors - 1997
1	On the unsolvability of the problem of reducibility in the theory of algorithms, Dokl - A