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Course Notes in Typed Lambda Calculus
, 1998
"... this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of p ..."
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this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of proposition, propositional function, and variable, from those contexts in which we are dealing with completely arbitrary, logical general propositions . . . To examine this possibility more closely and to pursue it would be valuable not only from the methodological point of view that enjoins us to strive for the greatest possible conceptual uniformity but also from a certain philosophic, or if you wish, aesthetic point of view."
The ChurchTuring Thesis as an Immature Form of the ZuseFredkin Thesis (More Arguments in Support of the “Universe as a Cellular Automaton” Idea)
"... In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational model known as a ..."
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In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational model known as a "cellular automaton".
LambdaCalculus and Functional Programming
"... This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, bo ..."
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This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, booleans, and strings. It is clearly desirable to have a method of writing a piece of code that can accept the specific type as an argument. Milner developed his ideas in terms of type assignment to lambdaterms. It is based on a result due originally to Curry (Curry 1969) and Hindley (Hindley 1969) known as the principal typescheme theorem, which says that (assuming that the typing assumptions are sufficiently wellbehaved) every term has a principal typescheme, which is a typescheme such that every other typescheme which can be proved for the given term is obtained by a substitution of types for type variables. This use of type schemes allows a kind of generality over all types, which is known as polymorphism.
A symbolic and algebraic computation based LambdaBoolean reduction machine via PROLOG
, 2006
"... This paper presents a new LambdaBoolean reduction machine for LambdaBoolean and LambdaBeta Boolean reductions in the context of Lambda Calculus and introduces the role of Church–Rosser properties and functional computation model in symbolic and algebraic computation with induction. The algorithm ..."
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This paper presents a new LambdaBoolean reduction machine for LambdaBoolean and LambdaBeta Boolean reductions in the context of Lambda Calculus and introduces the role of Church–Rosser properties and functional computation model in symbolic and algebraic computation with induction. The algorithm which improved for LambdaBeta Boolean reduction is simulated by the efficient logical programming language Prolog.
Enumeration of generalized BCI lambdaterms
, 2013
"... We investigate the asymptotic number of elements of size n in a particular class of closed lambdaterms (socalled BCI(p)terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence re ..."
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We investigate the asymptotic number of elements of size n in a particular class of closed lambdaterms (socalled BCI(p)terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of BCK(p)terms and that of closed lambdaterms. Using elementary arguments we obtain upper and lowerestimates for the number of closed lambdaterms of size n. Moreover, a recurrence relation is derived which allows an efficient computation of the counting sequence. BCK(p)terms are discussed briefly. 1
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"... Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of ..."
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Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell’s reduction of functions to relations over Frege’s reduction of relations to functions. There is an interesting system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an oversimplification: one can’t assimilate predication to functional application. ∗ This paper is forthcoming in the Journal of Logic and Computation. Theauthors would like to thank Uri Nodelman for his observations on the first draft of this paper. We’d also like to thank Bernard Linsky for observations on the second draft, which led us to reconceptualize the significance of our results within a more historical context. We’d also like to acknowledge one of the referees of this journal, whose comments led us to clarify and better document the claims in the paper. Paul Oppenheimer and Edward N. Zalta 2 1.
From Search to Computation: Redundancy Criteria and Simplification at Work
"... The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of these calculi usually generate a t ..."
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The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of these calculi usually generate a tremendously growing search space. The redundancy and simplification concept is indispensable for cutting down this search space to a manageable size. For a number of subclasses of firstorder logic appropriate redundancy and simplification concepts even turn the superposition calculus into a decision procedure. Hence, the key to successfully applying firstorder theorem proving to a problem domain is to find those simplifications and redundancy criteria that fit this domain and can be effectively implemented. We present Harald Ganzinger’s work in the light of the simplification and redundancy techniques that have been developed for concrete problem areas. This includes a variant of contextual rewriting to decide a subclass of Euclidean geometry, ordered chaining techniques for ChurchRosser and priority queue proofs, contextual rewriting and historydependent complexities for the completion of conditional rewrite systems, rewriting with equivalences for theorem proving in set theory, soft typing for the exploration of sort information in the context of equations, and constraint inheritance for automated complexity analysis.