Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to AC-TRSs, and introduce new methods for effectively proving AC-termination. Since it is impossible to directly apply the notion of dependency pairs to AC-TRSs, we introduce the head parts in terms and show an analogy between the root positions in infinite reduction sequences by TRSs and the head positions in those by AC-TRSs. Indeed, this analogy is essential for the extension of dependency pairs to AC-TRSs. Based on this analogy, we define AC-dependency pairs. To simplify the task of proving termination and AC-termination, several elimination transformations such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this thesis, we show that the argument filtering method combined with the AC-dependency pair technique is essential in all the elimination transformations above. We present remarkable simple proofs for the soundness of these elimination transformations based on this observation. Moreover, we propose a new elimination transformation, called the argument filtering transformation, which is not only more powerful than all the other elimination transformations but also especially useful to make clear an essential relationship among them.

Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with super-Turing power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or super-Turing. 1

. In this paper, we develop a theory of higher-order computability suitable for comparing the expressiveness of sequential, deterministic programming languages. The theory is based on the construction of a new universal domain T and corresponding universal language KL. The domain T is universal for observably sequential domains; KL can define all the computable elements of T, including the elements corresponding to computable observably sequential functions. In addition, domain embeddings in T preserve the maximality of finite elements---preserving the termination behavior of programs over the embedded domains. 1 Background and Motivation Classic recursion theory [7, 13, 18] asserts that all conventional programming languages are equally expressive because they can define all partial recursive functions over the natural numbers. This statement, however, is misleading because real programming languages support and enforce a more abstract view of data than bitstrings. In particular, mo...

Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflection theorem), which asserts the existence of sentences that “speak about themselves”. Let T be the theory and, for each wff φ, letpφqbe the term that serves as its name. Then the theorem says that, for any wff α(v) (with one free variable), there exists a sentence β for which: T ` β ↔ α(pβq) β is sometimes called the fixed point of α(v). All that is needed for the fixed point theorem is that the diagonal function, the one that maps each φ(v) toφ(p(φ(v)q)), be representable in T. The construction of β is more transparent if we assume that the functions is represented by a term of the language, diag(x). This means that the following holds for each φ(v): T ` diag(pφ(v)q) =pφ(pφ(v)q)q (Here ‘= ’ is the equality sign of the formal language; we use it also to denote equality in our metalanguage.) In other words, we can prove in T, for each particular argument, what the

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The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alone who achieved the characterization, in the opinion of Gödel. We also explore Turing’s oracle machine and its analogous properties in analysis. Keywords: Turing a-machine, computability, Church-Turing Thesis, Kurt Gödel, Alan Turing, Turing o-machine, computable approximations,

Watson-Crick D0L systems, introduced in 1997 by V. Mihalache and A. Salomaa, arise from two major principles: the Lindenmayer rewriting and the Watson-Crick complementarity principle. Complementarity can be viewed as a purely language-theoretic operation. Majority of a certain type of symbols in a string (purines vs. pyrimidines) triggers a transition to the complementary string. The paper deals with an expressive power of deterministic interactionless Watson-Crick Lindenmayer systems. A rather surprising result is obtained: these systems, consisting of iterated morphism and a basic DNA operation, are alone able to express any Turing computable function. 1

The programming approach to computability presented in the textbook by Kfoury, Moll, and Arbib in 1982 has been embedded into a programming course following the textbook by Abelson and Sussman. This leads to a course concept teaching good programming practice and clear theoretical concepts simultaneously. Here, we explain some of the main points of this approach: the halting problem, primitive and -recursive functions and the operational counterpart of these functions, i.e., the Loop and the While programs. 1

Watson-Crick D0L systems, introduced in 1998 by Arto Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely language-theoretic operation. Majority of a certain type of symbols in a string (purines vs. pyrimidines) triggers a transition to the complementary string. This paper deals with an expressive power of deterministic interactionless Watson-Crick Lindenmayer systems. A surprising result is obtained: these systems, consisting of iterated morphism and a basic DNA operation, are alone able to express any Turing computable function. 1 Watson-Crick D0L systems For elements of formal language theory we refer to [6, 7]. Here we only briey x some notation. Let (; ) be a free monoid with the catenation operation and let the empty word be denoted : We denote jwj a the number of occurrences of a symbol a in a string w for a 2 ; w 2 : Further we denote jwj = P a2 jwj a for an a...

This paper investigates some delicate tradeoffs between the generality of an algorithmic learning device and the quality of the programs it learns successfully. There are results to the effect that, thanks to small increases in generality of a learning device, the computational complexity of some successfully learned programs is provably unalterably suboptimal. There are also results in which the complexity of successfully learned programs is asymptotically optimal and the learning device is general, but, still thanks to the generality, some of those optimal, learned programs are provably unalterably information deficient — in some cases, deficient as to safe, Preprint submitted to Elsevier Science 11 March 2007 algorithmic extractability/provability of the fact that they are even approximately optimal. For these results, the safe, algorithmic methods of information extraction will be by proofs in arbitrary, true, computably axiomatizable extensions of Peano Arithmetic. Key words: Computational learning theory; Applications of computability theory