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Computing with Capsules
, 2011
"... Capsules provide a clean algebraic representation of the state of a computation in higherorder functional and imperative languages. They play the same role as closures or heap or stackallocated environments but are much simpler. A capsule is essentially a finite coalgebraic representation of a reg ..."
Abstract

Cited by 7 (3 self)
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Capsules provide a clean algebraic representation of the state of a computation in higherorder functional and imperative languages. They play the same role as closures or heap or stackallocated environments but are much simpler. A capsule is essentially a finite coalgebraic representation of a regular closed λcoterm. One can give an operational semantics based on capsules for a higherorder programming language with functional and imperative features, including mutable bindings. Lexical scoping is captured purely algebraically without stacks, heaps, or closures. All operations of interest are typable with simple types, yet the language is Turing complete. Recursive functions are represented directly as capsules without the need for unnatural and untypable fixpoint combinators. 1
Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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Cited by 2 (1 self)
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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
Degrees of Undecidability in Rewriting
, 902
"... Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy cl ..."
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Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Weak and strong normalization for single terms turn out to be Σ 0 1complete, while their uniform versions as well as dependency pair problems with minimality flag are Π 0 2complete. We find that confluence is Π 0 2complete both for single terms and uniform. Unexpectedly weak confluence for ground terms turns out to be harder than weak confluence for open terms. The former property is Π 0 2complete while the latter is Σ 0 1complete (and thereby recursively enumerable). The most surprising result is on dependency pair problems without minimality flag: we prove this to be Π 1 1complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. 1
and
"... Capsules provide an algebraic representation of the state of a computation in higherorder functional and imperative languages. A capsule is essentially a finite coalgebraic representation of a regular closed λcoterm. One can give an operational semantics based on capsules for a higherorder program ..."
Abstract
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Capsules provide an algebraic representation of the state of a computation in higherorder functional and imperative languages. A capsule is essentially a finite coalgebraic representation of a regular closed λcoterm. One can give an operational semantics based on capsules for a higherorder programming language with functional and imperative features, including mutable bindings. Static (lexical) scoping is captured purely algebraically without stacks, heaps, or closures. All operations of interest are typable with simple types, yet the language is Turing complete. Recursive functions are represented directly as capsules without the need for fixpoint combinators.
Hydra Games and Tree Ordinals
"... Abstract. Hydra games were introduced by Kirby and Paris, for the formulation of a result which is independent from Peano arithmetic but depends on the transfinite structure of 0. Tree ordinals are a wellknown simple way to represent countable ordinals. In this paper we study the relation between t ..."
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Abstract. Hydra games were introduced by Kirby and Paris, for the formulation of a result which is independent from Peano arithmetic but depends on the transfinite structure of 0. Tree ordinals are a wellknown simple way to represent countable ordinals. In this paper we study the relation between these concepts; an ordinal less than 0 is canonically translated into both a hydra and a tree ordinal term, and the reduction graph of the hydra and the normal form of the term syntactically correspond to each other. 1