114 research outputs found
An applicative theory for FPH
In this paper we introduce an applicative theory which characterizes the
polynomial hierarchy of time.Comment: In Proceedings CL&C 2010, arXiv:1101.520
Global semantic typing for inductive and coinductive computing
Inductive and coinductive types are commonly construed as ontological
(Church-style) types, denoting canonical data-sets such as natural numbers,
lists, and streams. For various purposes, notably the study of programs in the
context of global semantics, it is preferable to think of types as semantical
properties (Curry-style). Intrinsic theories were introduced in the late 1990s
to provide a purely logical framework for reasoning about programs and their
semantic types. We extend them here to data given by any combination of
inductive and coinductive definitions. This approach is of interest because it
fits tightly with syntactic, semantic, and proof theoretic fundamentals of
formal logic, with potential applications in implicit computational complexity
as well as extraction of programs from proofs. We prove a Canonicity Theorem,
showing that the global definition of program typing, via the usual (Tarskian)
semantics of first-order logic, agrees with their operational semantics in the
intended model. Finally, we show that every intrinsic theory is interpretable
in a conservative extension of first-order arithmetic. This means that
quantification over infinite data objects does not lead, on its own, to
proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories
are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were
used to characterize major computational complexity classes Their extensions
described here have similar potential which has already been applied
Extracting Programs from Constructive HOL Proofs via IZF Set-Theoretic<br> Semantics
Church's Higher Order Logic is a basis for influential proof assistants --
HOL and PVS. Church's logic has a simple set-theoretic semantics, making it
trustworthy and extensible. We factor HOL into a constructive core plus axioms
of excluded middle and choice. We similarly factor standard set theory, ZFC,
into a constructive core, IZF, and axioms of excluded middle and choice. Then
we provide the standard set-theoretic semantics in such a way that the
constructive core of HOL is mapped into IZF. We use the disjunction, numerical
existence and term existence properties of IZF to provide a program extraction
capability from proofs in the constructive core.
We can implement the disjunction and numerical existence properties in two
different ways: one using Rathjen's realizability for IZF and the other using a
new direct weak normalization result for IZF by Moczydlowski. The latter can
also be used for the term existence property.Comment: 17 page
Metalinguistic views of quantum mechanics and its formalizability
Much like the way we distinguish between formalism and experimentalism, we distinguish between ascertainment by proof and ascertainment by measurement. We argue that quantum mechanics, which characteristically encompasses both kinds of ascertainment, is too complex to be fully captured by formalism alone, and needs relativization to language in its complementaristic conception. In particular, we argue that there is a partial tie between the two ascertainments. Although, at higher levels, inferences or proofs may well be accepted as less constructive than direct measurements, they are tied at a basic level of constructivity. An inference is here of the same constructive nature as that of a direct measurement. The levelled approach is helpful, e.g., for understanding Bohr´s wave-particle complementarity and its recent challenge by the double-prism experiment (as well as, e.g., for understanding a thesis of a programmable experimentability within "quantum computation")
Step-Indexed Normalization for a Language with General Recursion
The Trellys project has produced several designs for practical dependently
typed languages. These languages are broken into two
fragments-a_logical_fragment where every term normalizes and which is
consistent when interpreted as a logic, and a_programmatic_fragment with
general recursion and other convenient but unsound features. In this paper, we
present a small example language in this style. Our design allows the
programmer to explicitly mention and pass information between the two
fragments. We show that this feature substantially complicates the metatheory
and present a new technique, combining the traditional Girard-Tait method with
step-indexed logical relations, which we use to show normalization for the
logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240
On the Semantics of Intensionality and Intensional Recursion
Intensionality is a phenomenon that occurs in logic and computation. In the
most general sense, a function is intensional if it operates at a level finer
than (extensional) equality. This is a familiar setting for computer
scientists, who often study different programs or processes that are
interchangeable, i.e. extensionally equal, even though they are not implemented
in the same way, so intensionally distinct. Concomitant with intensionality is
the phenomenon of intensional recursion, which refers to the ability of a
program to have access to its own code. In computability theory, intensional
recursion is enabled by Kleene's Second Recursion Theorem. This thesis is
concerned with the crafting of a logical toolkit through which these phenomena
can be studied. Our main contribution is a framework in which mathematical and
computational constructions can be considered either extensionally, i.e. as
abstract values, or intensionally, i.e. as fine-grained descriptions of their
construction. Once this is achieved, it may be used to analyse intensional
recursion.Comment: DPhil thesis, Department of Computer Science & St John's College,
University of Oxfor
Realizability and recursive mathematics
Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the
foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures.
Uealizability applies recursion-theoretic concepts to give interpretations of constructivism
along lines suggested originally by Heyting and Kleene. The research reported in the
dissertation revives the original insights of Kleene—by which realizability structures are
viewed as models rather than proof-theoretic interpretations—to solve a major problem of
classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization.
Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped
"constructivities," approaches to the mathematics of the calculable which range
from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic:
to sort through the jungle, set standards for classification and determine those
features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in
any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies
on a complete constructivization of the basic mathematical objects and logical operations.
The other is classical recursive mathematics, as represented by the work of Dekker, Myhill,
and Nerode. Classical constructivists use standard logic in a mathematical universe
restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for
intuitionism and classical constructivism. Between these realms arc connected semantically
through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses
all of the intuitionistic mathematics that does not involve choice sequences. (This includes
all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure,
V(A7), based on Kleene realizability. Since realizability takes set variables to range over
"effective" objects, large parts of classical constructivism appear over the model as inter¬
preted subsystems of intuitionistic set theory. For example, the entire first-order classical
theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals
and ordinals under realizability. In brief, we prove that a satisfactory partial solution to
the classification problem exists; theories in classical recursive constructivism are identical,
under a natural interpretation, to intuitionistic theories. The interpretation is especially
satisfactory because it is not a Godel-style translation; the interpretation can be developed
so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way
mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical
theory of effective structures, leaving pure set theory and a bit of model theory. Not only
are the theorems of classical effective mathematics faithfully represented in intuitionistic
set theory, but also the arguments that provide proofs of those theorems. Via realizability,
one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are
often more straightforward than their recursion-theoretic counterparts. The new proofs
are also more transparent, because they involve, rather than recursion theory plus set
theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results
from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on
the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be
applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer
science. The classical theory of effectively given computational domains a la Scott can
be subsumed into the Kleene realizability universe as a species of countable noneffective
domains. In this way, the theory of effective domains becomes a chapter (under interpre¬
tation) in an intuitionistic study of denotational semantics. We then show how the "extra
information" captured in the logical signs under realizability can be used to give proofs of
classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles
a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a
number of open problems in the metamathematics of constructivity. First, there is the
perennial problem of finding and delimiting in the wide constructive universe those features
that correspond to structures familiar from classical mathematics. In the realizability
model, it is easy to locate the collection of classical ordinals and to show that they form,
intuitionistically, a set rather than a proper class. Also, one interprets an argument of
Dekker and Myhill to prove that the classical powerset of the natural numbers contains at
least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including
the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be
accomplished. Every set over the model with decidable equality and every metric space is
enumerated by a collection of natural numbers
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