this paper. Here, drawing on all the research mentioned above for inspiration, we present a coherent unified theory of nonmonotonic formal systems. At the level of abstraction we achieve, we are finally able to see that nonmonotone systems pervade ordinary mathematical practice. There is no sign of any realization of the existence of such mathematical examples in the previous nonmonotonic logic literature. Perhaps these connections can only be seen by having a common abstract notion. What this commonality does for us is to make available known mathematical techniques from other areas of conventional mathematics for constructing and classifying belief sets (extensions) and, simultaneously, to make evident a common thread among disparate parts of mathematics and disparate nonmonotonic systems from artificial intelligence and computer science. On the level of Mathematical Philosophy there is a connection worth stating as well. Non-monotone reasoning takes place during the process of discovery of mathematical theorems, when one posits temporarily some proposition on the basis of no evidence against it, and explores the consequences of such a belief until new mathematical facts force their abandonment. These nonmonotone belief sets have their traces eradicated when final belief sets are achieved and demonstrative proofs are finished and published. The only hint of provisional belief sets left in mathematical papers is in the motivational remarks explaining what obstacles were overcome and by what changes in viewpoint the proof was achieved. Here is the main definition. A nonmonotone rule system consists of a set U and a set of triples (ff; fi; fl) called rules. Here ff = (ff 1 ; : : : ; ff n ) is a finite sequence of elements of U , called premises, and fi = (fi 1 ; : : : ...

Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with non-empty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally non-recursive functions, and rn is the weak degree of the n-random reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is one-to-one, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.

A number of nonmonotonic reasoning formalism have been introduced to model the set of beliefs of an agent. For example, Reiter [Rei80] introduced default logic where the set of beliefs of an agent reasoning with incomplete information corresponded to a extension of a default theory < D,W>. In the realm of logic programming,

Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive Cantor-Bendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is shown that, for all ordinals α>1, no set of degree ≥ 0 ′ ′ can be a member of any thin Π0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π0 1 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper non-principal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree 0 ′ ′. Introduction.

Abstract. We study the reverse mathematics and computability-theoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2-coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT 2 2 does not imply RT 2 2. 1.

Cellular automata have rich computational properties and, at the same time, provide plausible models of physics-like computation. We study decidability issues in the phasespace of these automata, construed as automatic structures over infinite words. In dimension one, slightly more than the first order theory is decidable but the addition of an orbit predicate results in undecidability. We comment on connections between this “what you see is what you get” model and the lack of natural intermediate degrees.

Important examples of Π0 1 classes of functions f ∈ ωω are the classes of sets (elements of ω2) which separate a given pair of disjoint r.e. sets: S2(A0, A1): = { f ∈ ω2: (∀i &lt; 2)(∀x ∈ Ai)f(x) � = i}. A wider class consists of the classes of functions f ∈ ωk which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: Sk(A0,..., Ak−1): = { f ∈ ωk: (∀i &lt; k)(∀x ∈ Ai)f(x) � = i}. We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let Sm k denote the Medvedev degrees of those Sk(A0,..., Ak−1) such that no m + 1 sets among A0,..., Ak−1 have a nonempty intersection. It is shown that each Sm k is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions DNRk is the greatest element of S1 k. If 2 ≤ l &lt; k, then 0M is the only degree in S1 l which is below a member of S1 k. Each Sm k is densely ordered and has the splitting property and the same holds for the lattice Lm k it generates. The elements of Sm k are exactly the joins of elements of S1 i for ⌈ k ⌉ ≤ i ≤ k. m

Abstract. Consider a Martin-Löf random ∆ 0 2 set Z. We give lower bounds for the number of changes of Zs ↾n for computable approximations of Z. We show that each nonempty Π 0 1 class has a low member Z with a computable approximation that changes only o(2 n) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↾n changes more than c2 n times. 1

The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics and construct a finite set of axioms that are strong enough to prove all proper theorems, but no more. Thus a proof of consistency and a proof of completeness were required. These proofs should be carried only by strictly finitary means so as to be beyond any reasonable criticism. As Hilbert pointed out [19], to carry out this project one needs to develop a better understanding of proofs as objects of mathematical discourse: To reach our goal, we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves. Furthermore, Hilbert hoped to find a single, mechanical procedure that would, at least in principle, provide correct answers to all well-defined questions

We introduce a formal definition of Wolfram’s notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process. 1 Computational Processes Degrees of unsolvability were introduced in two important papers by Post [21] and Kleene and Post [12]. The object of these papers was the study of the complexity of decision problems and in particular their relative complexity: how does a solution to one problem contribute to the solution of another, a notion that can be formalized in terms of Turing reducibility and Turing degrees. Post was particularly interested in the degrees of recursively enumerable (r.e.) degrees. The Turing degrees of r.e. sets together with Turing reducibility form a partial order and in fact an upper semi-lattice R. It is easy to see that R has least element /0, the degree of decidable sets, and a largest element /0 ′ , the degree of the halting set. Post asked whether there are any other r.e. degrees and embarked on a program to establish the existence of such an intermediate degree by constructing a suitable r.e. set. Post’s efforts produced a number of interesting ideas such as simple, hypersimple and hyperhypersimple sets but failed to produce