The partial ordering of Medvedev reducibility restricted to the family of Π0 1 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π0 1 class, which we call a ``c.e. separating class''. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes.

Abstract.The partial ordering of Medvedev reducibility restricted to the family of Π01 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π01 class, which we call a ``c.e. separating class''. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes.

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that (...) no set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 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 Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension (...) when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory △ possesses a polynomial space computable model which is exponential space decidable and thus △ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity. (shrink)

This paper is a continuation of the authors' work , where the main problem considered was whether a given recursive structure is recursively isomorphic to a polynomial-time structure. In that paper, a recursive Abelian group was constructed which is not recursively isomorphic to any polynomial-time Abelian group. We now show that if every element of a recursive Abelian group has finite order, then the group is recursively isomorphic to a polynomial-time group. Furthermore, if the orders are bounded, then the group (...) is recursively isomorphic to a polynomial-time group with universe A being the set of tally representations of natural numbers Tal = s{;1s};* or the set of binary representations of the natural numbers Bin. We also construct a recursive Abelian group with all elements of finite order but which has elements of arbitrary large finite order which is not isomorphic to any polynomial-time group with universe Tal or Bin. Similar results are obtained for structures , where f is a permutation on the set A. (shrink)

The central problem considered in this paper is whether a given recursive structure is recursively isomorphic to a polynomial-time structure. Positive results are obtained for all relational structures, for all Boolean algebras and for the natural numbers with addition, multiplication and the unary function 2x. Counterexamples are constructed for recursive structures with one unary function and for Abelian groups and also for relational structures when the universe of the structure is fixed. Results are also given which distinguish primitive recursive structures, (...) exponential-time structures and structures with honest witnesses. (shrink)

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. (...) Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)

This paper continues joint work of the authors with P. Clote, R. Soare and S. Wainer (Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145--163). An element x of the Cantor space 2 ω is said have rank α in the closed set P if x is in $D^\alpha(P)\backslash D^{\alpha + 1}(P)$ , where D α is the iterated Cantor-Bendixson derivative. The rank of x is defined to be the least α such that x has rank α in (...) some Π 0 1 set. The main result of the five-author paper is that for any recursive ordinal λ + n (where λ is a limit and n is finite), there is a point with rank λ + n which is Turing equivalent to O (λ + 2n) . All ranked points constructed in that paper are Π 0 2 singletons. We now construct a ranked point which is not a Π 0 2 singleton. In the previous paper the points of high rank were also of high hyperarithmetic degree. We now construct ▵ 0 2 points with arbitrarily high rank. We also show that every nonrecursive RE point is Turing equivalent to an RE point of rank one and that every nonrecursive ▵ 0 2 point is Turing equivalent to a hyperimmune point of rank one. We relate Clote's notion of the height of a Π 0 1 singleton in the Baire space with the notion of rank. Finally, we show that every hyperimmune point x is Turing equivalent to a point which is not ranked. (shrink)

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups ${\mathbb {Z}, Z(p^{\infty})}$ , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For (...) example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism. (shrink)

Important examples of $\Pi^0_1$ classes of functions $f \in {}^\omega\omega$ are the classes of sets (elements of ω 2) which separate a given pair of disjoint r.e. sets: ${\mathsf S}_2(A_0, A_1) := \{f \in{}^\omega2 : (\forall i < 2)(\forall x \in A_i)f(x) \neq i\}$ . A wider class consists of the classes of functions f ∈ ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: ${\mathsf S}_k(A_0,\ldots,A_k-1) := (...) \{f \in {}^\omega k : (\forall i < k) (\forall x \in A_i) f(x) \neq i\}$ . 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 ${\mathcal S}^m_k$ denote the Medvedev degrees of those ${\mathsf S}_k(A_0,\ldots,A_{k-1})$ such that no m + 1 sets among A 0,...,A k-1 have a nonempty intersection. It is shown that each ${\mathcal S}^m_k$ is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions $\mathsf{DNR}_k$ is the greatest element of ${\mathcal S}^1_k$ . If 2 ≤ l < k, then 0 M is the only degree in ${\mathcal S}^1_l$ which is below a member of ${\mathcal S}^1_k$ . Each ${\mathcal S}^m_k$ is densely ordered and has the splitting property and the same holds for the lattice ${\mathcal L}^m_k$ it generates. The elements of ${\mathcal S}^m_k$ are exactly the joins of elements of ${\mathcal S}^1_i$ for $\lceil{k \over m}\rceil \leq i \leq k$. (shrink)

The problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an effective (...) degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring. (shrink)

We examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that $\Delta _2^0 $ -categoricity is equivalent to our notion of graph-ω-c.e. categoricity.

Let A and B be subsets of the space 2 N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2 N into 2 N such that A = Φ -1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, $a \subset b$ implies $\Phi (a) \subset \Phi(b)$ . The set A is said to be monotone (...) if a ∈ A and $a \subset b$ imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The ▵ 0 1 sets are all reducible to the (Σ 0 1 but not ▵ 0 1 ) sets, which are in turn all reducible to the strictly ▵ 0 2 sets, which are all in turn reducible to the strictly Σ 0 2 sets. In addition, the nontrivial Σ 0 n sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly Π 0 2 monotone sets which have different monotone degrees. We show that every Σ 0 2 monotone set is actually Σ 0 2 positive. We also consider reducibility for subsets of the space of compact subsets of 2 N . This leads to the result that the finitely iterated Cantor-Bendixson derivative D n is a Borel map of class exactly 2n, which answers a question of Kuratowski. (shrink)

We show that in the lattice E Π of Π 0 1 classes there are initial segments [ $\emptyset$ , P] = L(P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a Π 0 1 class P such that L is isomorphic to the lattice L(P)*, which is L(P), modulo finite differences. For the 2-element lattice, we obtain (...) a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new Π 0 1 class P constructed, P has a single, non-computable limit point and L(P)* has three elements, corresponding to $\emptyset$ , P and a minimal class P $_0 \subset$ P. The element corresponding to P 0 has no complement in the lattice. On the other hand, the theory of L(P) is shown to be decidable. A Π 0 1 class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then L(P)* is always a Boolean algebra. We show that if P is a decidable Π 0 1 class and L(P) is not a Boolean algebra, then the theory of L(P)interprets the theory of arithmetic and is therefore undecidable. (shrink)

We study computability theoretic properties of and equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between equivalence structures and between equivalence structures.

The notion of superhigh computably enumerable (c.e.) degrees was first introduced by (Mohrherr in Z Math Logik Grundlag Math 32: 5–12, 1986) where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in (Proc Amer Math Soc 94:123–128, 1985) that the diamond lattice can be embedded into the c.e. tt-degrees preserving 0 (...) and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh. (shrink)

We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π01 class P is a subshift if and only if there exists a computable function F mapping 2ℕ to 2ℕ such that P is the set of itineraries of elements of 2ℕ. Π01 subshifts are constructed in 2ℕ and in 2ℤ which have no computable elements. We also consider the symbolic dynamics of maps on the unit interval.

A computable graph is constructed which is not computably isomorphic to any polynomial-time graph with a standard universe . Conditions are given under which a computable graph is computably isomorphic to a polynomial-time graph with a standard universe — for example, if every vertex has finite degree. Two special types of graphs are studied. It is shown that any computable tree is recursively isomorphic to a p-time tree with standard universe and that any computable equivalence relation is computably isomorphic to (...) a p-time equivalence relation with standard universe. (shrink)

Index sets are used to measure the complexity of properties associated with the differentiability of real functions and the existence of solutions to certain classic differential equations. The new notion of a locally computable real function is introduced and provides several examples of Σ04 complete sets.

We consider certain linear orders with a function on them, and discuss for which types of functions the resulting structure is or is not computably categorical. Particularly, we consider computable copies of the rationals with a fixed-point free automorphism, and also ω with a non-decreasing function.

A minimal extension of a Π01 class P is a Π01 class Q such that P ⊂ Q, Q – P is infinite, and for any Π01 class R, if P ⊂ R ⊂ Q, then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q′ have the same Cantor-Bendixson derivative. We show that for any class P which has a single limit point A, (...) and that point of degree ≤ 0, P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q. (shrink)

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb N}}$ , (...) there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes and (...) for the subclasses of decidable and of homogeneous ${\Pi^{0}_{1}}$ classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. (shrink)

We investigate effective categoricity of computable equivalence structures . We show that is computably categorical if and only if has only finitely many finite equivalence classes, or has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively categorical, (...) we further investigate when they are categorical. We also obtain results on the index sets of computable equivalence structures. (shrink)

We investigate effective categoricity of computable Abelian p-groups . We prove that all computably categorical Abelian p-groups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian p-groups are categorical and relatively categorical.

Blame is multifarious. It can be passionate or dispassionate. It can be expressed or kept private. We blame both the living and the dead. And we blame ourselves as well as others. What’s more, we blame ourselves, not only for our moral failings, but also for our non-moral failings: for our aesthetic bad taste, gustatory self-indulgence, or poor athletic performance. And we blame ourselves both for things over which we exerted agential control (e.g., our voluntary acts) and for things over (...) which we lacked such control (e.g., our desires, beliefs, and intentions). I argue that, despite this manifest diversity in our blaming practices, it’s possible to provide comprehensive account of blame. Indeed, I propose a set of necessary and sufficient conditions that aims to specify blame’s extension in terms of its constitution as opposed to its function. And I argue that this proposal has a number of advantages beyond accounting for blame in all its disparate forms. For one, it can account for the fact that one’s having had control over whether one was to φ is a necessary condition for one’s being fittingly blamed for having φ-ed. For another, it can account for why, unlike fitting shame, fitting blame is always deserved, which in turn explains why there is something morally problematic about ridding oneself of one’s fitting self-blame (e.g., one’s fitting guilt). (shrink)

The idea of basic action is a fixed point in the contemporary investigation of the nature of action. And while there are arguments aimed at putting the idea in place, it is meant to be closer to a gift of common sense than to a hard-won achievement of philosophical reflection. It first appears at the stage of innocuous description and before the announcement of philosophical positions. And yet, as any decent magician knows, the real work so often gets done in (...) the set-up. I argue that the seemingly innocent idea of basic action is, in fact, bound up with a wide-spread conception of the nature of bodily or physical action (section 3). Its legitimacy is vital to the intelligibility of the causal theory of action, according to which physical action consists of a mere event and a condition of mind joined (in the right way) by the bond of causality. Left unchecked, means-end reason threatens to permeate physical action, and thus threatens the sovereignty of the sphere of material events at the center of the causal theory: such events, including the movements of one’s body when one intentionally moves it, are thought to be constitutively independent of the subject’s rational capacities. Basic action is a necessary countermeasure, a sort of metaphysical containment wall needed to preserve the separate jurisdictions of the mind of the acting subject and what merely happens. I argue that so long as action theory proceeds under the assumption that there must be basic action, it must regard our relation to the progress of our own deeds as not different in principle from Marx’s understanding of the relation of the non-worker (Nichtarbeiter) to the material processes that realize his own ideas, namely the work done on the factory floor. Each is alienated from the progress, or getting done, of his deeds. In each the process of doing something intentionally turns out to be a case of delegating tasks to another power. How the process comes to completion is not willed, and at best watched: the causal work is not the agent’s work, his knowledge not self-knowledge (section 5). We should find this unacceptable. And if this is right, it should be a pressing question whether there must be basic action. I argue that we have no good reason to think so: basic action is not forced on us by argument (sections 2 and 4). It is open to us to consider an alternative, one on which action is not, in the fundamental case, barren of means- end structure, but instead permeated by it: we register as a force in nature not at the limit of the means-end order but precisely in its constitution. [from article introduction]. (shrink)

This chapter examines the views of seventeenth-century British philosophers on the notion of logic and demonstrative knowledge, particularly Francis Bacon, Thomas Hobbes, and John Locke, offering an overview of traditional Aristotelianism in relation to logic and describing Bacon's approach to demonstration and logic. It also analyzes the contribution of the Cambridge Platonists and evaluates the influence of Cartesianism. The chapter concludes that theorizing about logic and demonstrative knowledge followed an arc familiar from other branches of philosophy such as metaphysics or (...) the philosophy of science. (shrink)

Argumentation, which can be abstractly defined as the interaction of different arguments for and against some conclusion, is an important skill to learn for everyday life, law, science, politics and business. The best way to learn it is to try it out on real instances of arguments found in everyday conversational exchanges and legal argumentation. The introductory chapter of this book gives a clear general idea of what the methods of argumentation are and how they work as tools that can (...) be used to analyze arguments. Each subsequent chapter then applies these methods to a leading problem of argumentation. Today the field of computing has embraced argumentation as a paradigm for research in artificial intelligence and multi-agent systems. Another purpose of this book is to present and refine tools and techniques from computing as components of the methods that can be handily used by scholars in other fields. (shrink)

Shows how analogy-making pervades human thought at all levels, influencing the choice of words and phrases in speech, providing guidance in unfamiliar situations, and giving rise to great acts of imagination.

A brief exploration of the nature of, and motivations for, contemporary forms of metaethical quietism. Also outlines some of the prominent objections to such positions and discusses some of the limitations of these objections from the quietist's perspective.

Does criminal liability require an act? -- Motive and criminal liability -- The costs to criminal theory of supposing that intentions are irrelevant to permissibility -- Transferred intent -- The nature and justifiability of nonconsummate offenses -- Strict liability, justice, and proportionality -- The sequential principle of relative culpability -- Willful ignorance, knowledge, and the equal culpability thesis : a study of the significance of the principle of legality -- Rapes without rapists : consent and reasonable mistake -- Mistake of (...) law and culpability -- On the supposed priority of justification to excuse -- Partial defenses -- The "but everybody does that!" defense -- The de minimis "defense" to criminal liability -- Why punish the deserving -- Malum prohibitum and retributivism -- Already punished enough. (shrink)