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The Symbol Grounding Problem
, 1990
"... There has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling. This paper describes the "symbol grounding problem": How can the semantic interpretation of a formal symbol system be made intrin ..."
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Cited by 806 (14 self)
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There has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling. This paper describes the "symbol grounding problem": How can the semantic interpretation of a formal symbol system be made intrinsic to the system, rather than just parasitic on the meanings in our heads? How can the meanings of the meaningless symbol tokens, manipulated solely on the basis of their (arbitrary) shapes, be grounded in anything but other meaningless symbols? The problem is analogous to trying to learn Chinese from a Chinese/Chinese dictionary alone. A candidate solution is sketched: Symbolic representations must be grounded bottomup in nonsymbolic representations of two kinds: (1) "iconic representations" , which are analogs of the proximal sensory projections of distal objects and events, and (2) "categorical representations" , which are learned and innate featuredetectors that pick out the invariant features of object and event categories from their sensory projections. Elementary symbols are the names of these object and event categories, assigned on the basis of their (nonsymbolic) categorical representations. Higherorder (3) "symbolic representations" , grounded in these elementary symbols, consist of symbol strings describing category membership relations (e.g., "An X is a Y that is Z"). Connectionism is one natural candidate for the mechanism that learns the invariant features underlying categorical representations, thereby connecting names to the proximal projections of the distal objects they stand for. In this way connectionism can be seen as a complementary component in a hybrid nonsymbolic/symbolic model of the mind, rather than a rival to purely symbolic modeling. Such ...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Classical And Constructive Hierarchies In Extended Intuitionistic Analysis
"... This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with t ..."
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Cited by 4 (3 self)
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This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with the property that every constructive partial functional defined on {# : R(#)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(#) is equivalent in #) for some stable A(#, #) (which belongs to the classical analytical hierarchy). The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems.
Metamathematical Properties of Intuitionistic Set Theories with Choice Principles
"... This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive ZermeloFraenkel Set T ..."
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This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive ZermeloFraenkel Set Theory and full Intuitionistic ZermeloFraenkel augmented by any combination of the principles of Countable Choice, Dependent Choices and the Presentation Axiom. Also Markov’s principle may be added. Moreover, these properties hold effectively. For instance from a proof of a statement ∀n ∈ ω ∃m ∈ ω ϕ(n, m) one can effectively construct an index e of a recursive function such that ∀n ∈ ω ϕ(n, {e}(n)) is provable. Thus we have an explicit method of witness and program extraction from proofs involving choice principles. As for the proof technique, this paper is a continuation of [32]. [32] introduced a selfvalidating semantics for CZF that combines realizability for extensional set theory and truth.
UNAVOIDABLE SEQUENCES IN CONSTRUCTIVE ANALYSIS
"... Kleene’s formalization FIM of intuitionistic analysis ([3] and [2]) includes bar induction, countable and continuous choice, but is consistent with the statement that there are no nonrecursive functions ([5]). Veldman ([12]) showed that in FIM the constructive analytical hierarchy collapses at Σ 1 ..."
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Kleene’s formalization FIM of intuitionistic analysis ([3] and [2]) includes bar induction, countable and continuous choice, but is consistent with the statement that there are no nonrecursive functions ([5]). Veldman ([12]) showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions
Oberwolfach Proof Theory and Constructive Math
"... Vesley [1965], as extended by Kleene [1969]) includes bar induction, countable and continuous choice, but cannot prove that the constructive arithmetical hierarchy is proper. Veldman showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions to interp ..."
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Vesley [1965], as extended by Kleene [1969]) includes bar induction, countable and continuous choice, but cannot prove that the constructive arithmetical hierarchy is proper. Veldman showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions to interpreting the constructive content of classical analysis, just as the collapse of the arithmetical hierarchy at Σ 0 3 in HA + MP0 + ECT0 limits the scope and effectiveness of recursive analysis. Question: Can we do better by working within classical extensions of nonclassical theories, or within classically correct theories obeying e.g. Church’s Rule or Brouwer’s Rule? We work in a twosorted language L with variables over numbers and oneplace numbertheoretic functions (choice sequences). Our base theory M – the minimal theory used by Kleene [1969] to formalize the theory of recursive partial functionals, function
Note on Π 0 n+1LEM, Σ0 n+1LEM and Σ0 n+1DNE⋆
"... Abstract. In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ 0 n+1DNE from Π 0 n+1LEM over HA, and hence the independence of Σ 0 n+1LEM from Π 0 n+1LEM over HA, for all n ≥ 0. We show that the same relat ..."
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Abstract. In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ 0 n+1DNE from Π 0 n+1LEM over HA, and hence the independence of Σ 0 n+1LEM from Π 0 n+1LEM over HA, for all n ≥ 0. We show that the same relative independence results hold for these arithmetical principles over Kleene and Vesley’s system FIM of intuitionistic analysis [3], which extends HA and is consistent with PA but not with classical analysis. 1 The double negations of the closures of Σ 0 n+1LEM, Σ 0 n+1DNE and Π 0 n+1LEM are also considered, and shown to behave differently with respect to HA and FIM. Various elementary questions remain to be answered.
Global semantic typing for inductive and coinductive computing
"... Common datatypes, such as N, can be identified with term algebras. Thus each type can be construed as a global set; e.g. for N this global set is instantiated in each structure S to the denotations in S of the unary numerals. We can then consider each declarative program as an axiomatic theory, and ..."
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Common datatypes, such as N, can be identified with term algebras. Thus each type can be construed as a global set; e.g. for N this global set is instantiated in each structure S to the denotations in S of the unary numerals. We can then consider each declarative program as an axiomatic theory, and assigns to it a semantic (Currystyle) type in each structure. This leads to the intrinsic theories of [18], which provide a purely logical framework for reasoning about programs and their types. The framework is of interest because of its close fit with syntactic, semantic, and proof theoretic fundamentals of formal logic. This paper extends the framework to data given by coinductive as well as inductive declarations. We prove a Canonicity Theorem, stating that the denotational semantics of an equational program P, understood operationally, has type τ over the canonical model iff P, understood as a formula has type τ in every “datacorrect ” structure. In addition we show that every intrinsic theory is interpretable in a conservative extension of firstorder arithmetic. 1998 ACM Subject Classification F.3 Logics and meanings of programs