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48
A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF c ..."
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Cited by 217 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Operational Modal Logic
, 1995
"... Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact in ..."
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Cited by 65 (25 self)
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Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact interpretation as corresponding propositional operations on proofs; both S4 and Int turn out to be complete with respect to this proof realization. These results are based on operational reading of S4, where a modality is split into three operations. The logic of proofs with these operations is shown to be arithmetically complete with respect to the intended provability semantics and sufficient to realize every operation on proofs admitting propositional specification in arithmetic.
Calculating Sized Types
 HigherOrder and Symbolic Computation
, 2001
"... Many program optimizations and analyses, such as arraybounds checking, termination analysis, etc, depend on knowing the size of a function's input and output. However, size information can be dicult to compute. Firstly, accurate size computation requires detecting a size relation between different ..."
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Cited by 52 (10 self)
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Many program optimizations and analyses, such as arraybounds checking, termination analysis, etc, depend on knowing the size of a function's input and output. However, size information can be dicult to compute. Firstly, accurate size computation requires detecting a size relation between different inputs of a function. Secondly, different optimizations and analyses may require slightly different size information, and thus slightly different computation. Literature in size computation has mainly concentrated on size checking, instead of size inference. In this paper, we provide a generic framework on which di erent size variants can be expressed and computed. We also describe an effective algorithm for inferring, instead of checking, size information. Size information are expressed in terms of Presburger formulae, and our algorithm utilizes the Omega Calculator to compute as exact a size information as possible, within the linear arithmetic capability.
Choice Functions and the Scopal Semantics of Indefinites
 Linguistics and Philosophy
, 1997
"... this paper I treat conditionals using material implication, ignoring the wellknown semantic/pragmatic problems concerning their correct interpretation. Of course, one may doubt whether (7a), which is verified by any situation in which there is one woman who did not come to the party, reflects corre ..."
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Cited by 48 (12 self)
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this paper I treat conditionals using material implication, ignoring the wellknown semantic/pragmatic problems concerning their correct interpretation. Of course, one may doubt whether (7a), which is verified by any situation in which there is one woman who did not come to the party, reflects correctly the wide scope reading of the indefinite in (7). Obviously, this problem is independent of the scope problem of indefinites. For this reason and because antecedents of conditionals are one of the simplest and most striking cases of scope islands, I use such examples freely, counting on the reader to substitute her favorite theory of conditionals for material implication. This claim has been challenged in Farkas (1981), Rooth & Partee (1982:fn.6) and, more recently, in Ruys (1992) and Abusch (1994). These works all show cases where Fodor & Sag's claim is argued to be incorrect. The empirical debate will be reviewed later in this paper (subsection 3.4.2). Ruys and Abusch both conclude that Fodor & Sag's "referential" approach is inadequate. To handle the facts, Ruys proposes an indexing mechanism of indefinites within a DRTlike interpretation of LF. Abusch proposes to enrich DRT with a storage mechanism that changes the syntactic position of the N' predicate (= the restriction of the indefinite) at the representational level. Both Ruys and Abusch therefore accept the assumption of DRT about a distinct syntactic representational level for meaning. This level (sometimes called Logical Form') is additional to the syntactic level that undergoes semantic interpretation (GB's Logical Form, other theories' Surface Structure). Indefinites in Ruys and Abusch's treatments are not quantifiers. Instead, they involve the familiar treatment of DRT using free variables. I henceforth c...
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Cited by 17 (5 self)
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Ch ..."
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Cited by 11 (0 self)
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...