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Reasoning about Object Systems in VTLoE
, 1994
"... VTLoE (Variable Type Logic of Effects) is a logic for reasoning about imperative functional programs inspired by the variable type systems of Feferman. The underlying programming language, mk , extends the callbyvalue lambda calculus with primitives for arithmetic, pairing, branching, and refere ..."
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VTLoE (Variable Type Logic of Effects) is a logic for reasoning about imperative functional programs inspired by the variable type systems of Feferman. The underlying programming language, mk , extends the callbyvalue lambda calculus with primitives for arithmetic, pairing, branching, and reference cells (mutable data). In VTLoE one can reason about program equivalence and termination, input/output relations, program contexts, and inductively (and coinductively) define data structures. In this paper we present a refinement of VTLoE. We then introduce a notion of object specification and establish formal principles for reasoning about object systems within VTLoE. Objects are selfcontained entities with local state. The local state of an object can only be changed by action of that object in response to a message. In mk objects are represented as closures with mutable data bound to local variables. A semantic principle called simulation induction was introduced in our earlier wor...
The structure of nuprl’s type theory
, 1997
"... on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) ..."
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Cited by 9 (3 self)
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on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html)
Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathemat ..."
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Cited by 8 (5 self)
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This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are prooftheoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and prooftheoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...
Why the constant `undefined'?  Logics of partial terms for strict and nonstrict functional programming languages
 Journal of Functional Programming
, 1998
"... In this article we explain two di#erent operational interpretations of functional programs by two di#erent logics. The programs are simply typed #terms with pairs, projections, ifthenelse, and least fixed point recursion. A logic for callbyvalue evaluation and a logic for callbyname evaluation ..."
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Cited by 8 (2 self)
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In this article we explain two di#erent operational interpretations of functional programs by two di#erent logics. The programs are simply typed #terms with pairs, projections, ifthenelse, and least fixed point recursion. A logic for callbyvalue evaluation and a logic for callbyname evaluation are obtained as as extensions of a system which we call the basic logic of partial terms (BPT). This logic is suitable to prove properties of programs that are valid under both strict and nonstrict evaluation. We use methods from denotational semantics to show that the two extensions of BPT are adequate for callbyvalue and callbyname evaluation. Neither the programs nor the logics contain the constant `undefined'. 1
Typical ambiguity: trying to have your cake and eat it too
 the proceedings of the conference Russell 2001
"... Would ye both eat your cake and have your cake? ..."
A type reduction from proofconditional to dynamic semantics
 Journal of Philosophical Logic
"... Abstract. Dynamic and proofconditional approaches to discourse (exemplified by Discourse Representation Theory and TypeTheoretical Grammar, respectively) are related through translations and transitions labeled by firstorder formulas with anaphoric twists. Typetheoretic contexts are defined rela ..."
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Cited by 7 (0 self)
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Abstract. Dynamic and proofconditional approaches to discourse (exemplified by Discourse Representation Theory and TypeTheoretical Grammar, respectively) are related through translations and transitions labeled by firstorder formulas with anaphoric twists. Typetheoretic contexts are defined relative to a signature and instantiated modeltheoretically, subject to change. 1
Some Theories With Positive Induction of Ordinal Strength ...
 JOURNAL OF SYMBOLIC LOGIC
, 1996
"... This paper deals with: (i) the theory ID # 1 which results from c ID 1 by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON() plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ord ..."
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Cited by 7 (3 self)
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This paper deals with: (i) the theory ID # 1 which results from c ID 1 by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON() plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are \Sigma in the ordinals. We show that these systems have prooftheoretic strength '!0.
Polynomial Time Operations in Explicit Mathematics
 Journal of Symbolic Logic
, 1997
"... In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable fu ..."
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Cited by 7 (5 self)
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In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.
A ProofTheoretic Characterization of the Basic Feasible Functionals
 Theoretical Computer Science
, 2002
"... We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27]. ..."
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Cited by 7 (6 self)
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We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].
A Theory of Classes for a Functional Language with Effects
 In Proceedings of CSL92, volume 702 of Lecture Notes in Computer Science
, 1993
"... this paper we introduce a variable typed logic of effects (i.e. a logic of effects where classes can be defined and quantified over) inspired by the variable type systems of Feferman [3, 4] for purely functional languages. A similar extension incorporating nonlocal control operations was introduced ..."
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Cited by 7 (6 self)
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this paper we introduce a variable typed logic of effects (i.e. a logic of effects where classes can be defined and quantified over) inspired by the variable type systems of Feferman [3, 4] for purely functional languages. A similar extension incorporating nonlocal control operations was introduced in [27]. The logic we present provides an expressive language for defining specifications and constraints and for studying properties and program equivalences, in a uniform framework. Thus it has an advantage over a plethora of systems in the literature that aim to capture solitary aspects of computation. The theory also allows for the construction of inductively defined sets and derivation of the corresponding induction principles. Classes can be used to express, inter alia, the nonexpansiveness of terms [29]. Other effects can also be represented within the system. These include read/write effects and various forms of interference [24]. The first order fragment is described in [16] where it is used to resolve the denotationally problematic examples of [17]. In our language atoms, references and lambda abstractions are all first class values and as such are storable. This has several consequences. Firstly, mutation and variable binding are separate and so we avoid the problems that typically arise (e.g. in Hoare's and dynamic logic) from the conflation of program variables and logical variables. Secondly, the equality and sharing of references (aliasing) is easily expressed and reasoned about. Thirdly, the combination of mutable references and lambda abstractions allows us to study object based programming within our framework. Our atomic formulas express the (operational or observational) equivalence of programs `a la Plotkin [23]. Neither Hoare's logic nor Dynamic logi...