Robots are increasingly being studied for use in education. It is expected that robots will have the potential to facilitate children’s learning and function autonomously within real classrooms in the near future. Previous research has raised the importance of designing acceptable robots for different practices. In parallel, scholars have raised ethical concerns surrounding children interacting with robots. Drawing on a Responsible Research and Innovation perspective, our goal is to move away from research concerned with designing features that will render robots (...) more socially acceptable by end users toward a reflective dialogue whose goal is to consider the key ethical issues and long-term consequences of implementing classroom robots for teachers and children in primary education. This paper presents the results from several focus groups conducted with teachers in three European countries. Through a thematic analysis, we provide a theoretical account of teachers’ perspectives on classroom robots pertaining to privacy, robot role, effects on children and responsibility. Implications for the field of educational robotics are discussed. (shrink)

The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.

Participants were unknowingly exposed to complex regularities in a working memory task. The existence of implicit knowledge was subsequently inferred from a preference for stimuli with similar grammatical regularities. Several affective traits have been shown to influence AGL performance positively, many of which are related to a tendency for automatic responding. We therefore tested whether the mindfulness trait predicted a reduction of grammatically congruent preferences, and used emotional primes to explore the influence of affect. Mindfulness was shown to correlate negatively (...) with grammatically congruent responses. Negative primes were shown to result in faster and more negative evaluations. We conclude that grammatically congruent preference ratings rely on habitual responses, and that our findings provide empirical evidence for the non-reactive disposition of the mindfulness trait. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations (...) are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$ -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification.

In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.

A closed λ-term E is called an enumerator if M ε /gL/dg /gTn ε N E/drn/dl = β M. Here Λ° is the set of closed λ-terms, N is the set of natural numbers and the /drn/dl are the Church numerals λfx./tfnx. Such an E is called reducing if moreover M ε /gL/dg /gTn ε N E/drn/dl /a/gb M. In 1983 I conjectured that every enumerator is reducing. An ingenious recursion theoretic proof of this conjecture by Statman is presented in (...) Barendregt . The proof is not intuitionistically valid, however. Dirk van Dalen has encouraged me to find intuitionistic proofs whenever possible. In the lambda calculus this is usually not difficult. In this paper an intuitionistic version of Statmans proof will be given. It took me somewhat longer to find it than in other cases. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are (...) closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Attempts to explain behavior genetically face two major problems: the application of the concept of genetic coding and the theoretical possibility of decomposing behavior. This paper argues that using the notion of genetic coding is appropriate in explanations of protein synthesis but inadequate and even misleading in the context of explanations of behavior. Genes should be regarded as disparate components of mechanisms that account for behavior rather than as codes for behavioral phenotypes. Such mechanistic explanations, however, presuppose the possibility of (...) decomposing behavioral phenotypes, which is strongly disputed by researchers holding an interactionist view of behavior. It is argued that these researchers fail to distinguish etiological from constitutive decomposition, and that their objections apply to the former but not to the latter kind. Constitutive decomposition might identify genes as disparate components and open up the possibility of explaining behavior mechanistically by isolating causal paths from genes to behavior. Finally, research on the single gene disorder phenylketonuria is introduced to illustrate and test these views. With respect to this disorder it is demonstrated that applying the concept of genetic coding would be inappropriate and misguiding, while nonetheless the phenotype is decomposable and can be explained mechanistically by singling out a genetic causal path.t is demonstrated that applying the concept of genetic coding would be inappropriate and misguiding, while nonetheless the phenotype is decomposable and can be explained mechanistically by singling out a genetic causal path. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which (...) derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

. Evolutionary psychology and behavioural genomics are both approaches to explain human behaviour from a genetic point of view. Nonetheless, thus far the development of these disciplines is anything but interdependent. This paper examines the question whether evolutionary psychology can contribute to behavioural genomics. Firstly, a possible inconsistency between the two approaches is reviewed, viz. that evolutionary psychology focuses on the universal human nature and disregards the genetic variation studied by behavioural genomics. Secondly, we will discuss the structure of biological (...) explanations. Some philosophers rightly acknowledge that explanations do not involve laws which are exceptionless and universal. Instead, generalisations that are invariant suffice for successful explanation as long as two other stipulations are recognised: the domain within which the generalisation has no exceptions as well as the distribution of the mechanism described by the generalisation should both be specified. It is argued that evolutionary psychology can contribute to behavioural genomic explanations by accounting for these two specifications. (shrink)

A λ-theory T is a consistent set of equations between λ-terms closed under derivability. The degree of T is the degree of the set of Godel numbers of its elements. H is the $\lamda$ -theory axiomatized by the set {M = N ∣ M, N unsolvable. A $\lamda$ -theory is sensible $\operatorname{iff} T \supset \mathscr{H}$ , for a motivation see [6] and [4]. In § it is proved that the theory H is ∑ 0 2 -complete. We present Wadsworth's proof (...) that its unique maximal consistent extention $\mathscr{H}^\ast (= \mathrm{T}(D_\infty))$ is Π 0 2 -complete. In $\S2$ it is proved that $\mathscr{H}_\eta(= \lambda_\eta-\text{Calculus} + \mathscr{H})$ is not closed under the ω-rule (see [1]). In $\S3$ arguments are given to conjecture that $\mathscr{H}\omega (= \lambda + \mathscr{H} + omega-rule)$ is Π 1 1 -complete. This is done by representing recursive sets of sequence numbers as λ-terms and by connecting wellfoundedness of trees with provability in Hω. In $\S4$ an infinite set of equations independent over H η will be constructed. From this it follows that there are 2^{ℵ_0 sensible theories T such that $\mathscr{H} \subset T \subset \mathscr{H}^\ast$ and 2 ℵ 0 sensible hard models of arbitrarily high degrees. In $\S5$ some nonprovability results needed in $\S\S1$ and 2 are established. For this purpose one uses the theory H η extended with a reduction relation for which the Church-Rosser theorem holds. The concept of Gross reduction is used in order to show that certain terms have no common reduct. (shrink)

The target article presents a model for schizophrenia extending four levels of abstraction: molecules, cells, cognition, and syndrome. An important notion in the model is that of coordination, applicable to both the level of cells and of cognition. The molecular level provides an “implementation” of the coordination at the cellular level, which in turn underlies the coordination at the cognitive level, giving rise to the clinical symptoms.

Abstract.Evolutionary psychology and behavioural genomics are both approaches to explain human behaviour from a genetic point of view. Nonetheless, thus far the development of these disciplines is anything but interdependent. This paper examines the question whether evolutionary psychology can contribute to behavioural genomics. Firstly, a possible inconsistency between the two approaches is reviewed, viz. that evolutionary psychology focuses on the universal human nature and disregards the genetic variation studied by behavioural genomics. Secondly, we will discuss the structure of biological explanations. (...) Some philosophers rightly acknowledge that explanations do not involve laws which are exceptionless and universal. Instead, generalisations that are invariant suffice for successful explanation as long as two other stipulations are recognised: the domain within which the generalisation has no exceptions as well as the distribution of the mechanism described by the generalisation should both be specified. It is argued that evolutionary psychology can contribute to behavioural genomic explanations by accounting for these two specifications. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both (...) translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

Barendregt gave a sound semantics of the simple type assignment system λ → by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

We present a framework for intensional reasoning in typed -calculus. In this family of calculi, called Modal Pure Type Systems (MPTSs), a propositions-as-types-interpretation can be given for normal modal logics. MPTSs are an extension of the Pure Type Systems (PTSs) of Barendregt (1992). We show that they retain the desirable meta-theoretical properties of PTSs, and briefly discuss applications in the area of knowledge representation.

Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four (...) kinds of PTSs. Most importantly PTSs are about statements of the form M:A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. (shrink)

We study the intersection type assignment system as defined by Barendregt, Coppo and Dezani. For the four essential variants of the system (with and without a universal type and with and without subtyping) we show that the emptiness (inhabitation) problem is recursively unsolvable. That is, there is no effective algorithm to decide if there is a closed term of a given type. It follows that provability in the logic of "strong conjunction" of Mints and Lopez-Escobar is also undecidable.

In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong correlation with predicativity, (...) predicative logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf''s Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russell's orders, (2) the placing of the historical system RTT underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf''s type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS. (shrink)

We study the cube of type assignment systems, as introduced in Giannini et al. 87–126), and confront it with Barendregt's typed gl-cube . The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property (...) holds only for systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one. (shrink)

A Partial Combinatory Algebra is completable if it can be extended to a total one. In [1] it is asked (question 11, posed by D. Scott, H. Barendregt, and G. Mitschke) if every PCA can be completed. A negative answer to this question was given by Klop in [12, 11]; moreover he provided a sufficient condition for completability of a PCA (M, ·, K, S) in the form of ten axioms (inequalities) on terms of M. We prove that just (...) one of these axiom (the so called Barendregt's axiom) is sufficient to guarantee (a slightly weaker notion of) completability. (shrink)

A numeral system is an infinite sequence of different closed normal -terms intended to code the integers in -calculus. Barendregt has shown that if we can represent, for a numeral system, the functions Successor, Predecessor, and Zero Test, then all total recursive functions can be represented. In this paper we prove the independancy of these three particular functions. We give at the end a conjecture on the number of unary functions necessary to represent all total recursive functions.

We work in λH, the untyped λ-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing a λ-term Ξ such that ΞM = ΞΙ ⟺ ΞM ≠ ΞΩ. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories.