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Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
Computability in an Introductory Course on Programming
 Bulletin of the European Association for Theoretical Computer Science 73 (2001
"... 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 simult ..."
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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
Universal Computation with WatsonCrick D0L Systems
, 2000
"... WatsonCrick D0L systems, introduced in 1997 by V. Mihalache and A. Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely languagetheoretic operation. Majority of a certain type of symbols in a s ..."
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WatsonCrick D0L systems, introduced in 1997 by V. Mihalache and A. Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely languagetheoretic 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 WatsonCrick 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
Length Of Polynomial Ascending Chains And Primitive Recursiveness
, 1992
"... In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s ..."
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In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s of such a chain depending only on n and f , which is recursive in f for every n and primitive recursive in f for n = 2. In this paper we give a better bound, expressed in a rather simple way in terms of f , which is attained when f is an increasing function. We prove that it is primitive recursive in f for all n. We also show that, on the contrary, there is no bound which is primitive recursive in n in general.
Kleene’s Amazing Second Recursion Theorem (Extended Abstract)
"... This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number ..."
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This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every nary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that
Computational Aspects of FirstOrder Logic on Finite Structures
, 1999
"... Descriptive complexity aims to classify properties of finite structures according to the logical resources that are needed to express them. The ImmermanVardi Theorem states that a property of finite ordered structures is computable in polynomialtime if and only if it can be expressed in least fixe ..."
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Descriptive complexity aims to classify properties of finite structures according to the logical resources that are needed to express them. The ImmermanVardi Theorem states that a property of finite ordered structures is computable in polynomialtime if and only if it can be expressed in least fixedpoint logic. Here, least fixedpoint logic is the extension of firstorder logic with the ability to buildin inductive definitions. It is known that firstorder logic is a fairly weak expressive language in the context of descriptive complexity. For example, least fixedpoint logic is strictly more expressive than firstorder logic on the class of all finite ordered structures. The ordered conjecture, formulated by Kolaitis and Vardi in 1992, asks whether this is also the case for any infinite class of finite ordered structures. Although some significant progress has been made since its formulation, the ordered conjecture remains open. In fact, it has been shown that any way of resolving ...
On Properties of WatsonCrick D0L Systems
"... WatsonCrick 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 languagetheoretic operation. Majority of a certain type of symbols in a string (purines v ..."
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WatsonCrick 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 languagetheoretic 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 WatsonCrick 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 WatsonCrick 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...
The WHILE Hierarchy of Program Schemes is Infinite
, 1998
"... . We exhibit a sequence Sn (n # 0) of while program schemes, i. e., while programs without interpretation, with the property that the while nesting depth of Sn is n, and prove that any while program scheme which is scheme equivalent to Sn , i. e., equivalent for all interpretations over arbitrary ..."
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. We exhibit a sequence Sn (n # 0) of while program schemes, i. e., while programs without interpretation, with the property that the while nesting depth of Sn is n, and prove that any while program scheme which is scheme equivalent to Sn , i. e., equivalent for all interpretations over arbitrary domains, has while nesting depth at least n. This shows that the while nesting depth imposes a strict hierarchy (the while hierarchy) when programs are compared with respect to scheme equivalence and contrasts with Kleene's classical result that every program is equivalent to a program of while nesting depth 1 (when interpreted over a fixed domain with arithmetic on nonnegative integers). Our proof is based on results from formal language theory; in particular, we make use of the notion of star height of regular languages. 1 Introduction When comparing programming languages, one often has a vague impression of one language being more powerful than another. However, a basic result of the ...