4,192 research outputs found
Implicit complexity for coinductive data: a characterization of corecurrence
We propose a framework for reasoning about programs that manipulate
coinductive data as well as inductive data. Our approach is based on using
equational programs, which support a seamless combination of computation and
reasoning, and using productivity (fairness) as the fundamental assertion,
rather than bi-simulation. The latter is expressible in terms of the former. As
an application to this framework, we give an implicit characterization of
corecurrence: a function is definable using corecurrence iff its productivity
is provable using coinduction for formulas in which data-predicates do not
occur negatively. This is an analog, albeit in weaker form, of a
characterization of recurrence (i.e. primitive recursion) in [Leivant, Unipolar
induction, TCS 318, 2004].Comment: In Proceedings DICE 2011, arXiv:1201.034
Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes.
In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class C? For concrete languages L (such as TAUT or SAT) and concrete promise classes C (such as NP∩coNP, UP, BPP, disjoint NP-pairs etc.), these ques-tions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new character-izations for Q1 and Q2 that apply to almost all promise classes C and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount on advice is avail-able in the underlying machine model. This continues a recent line of research on proof systems with advice started by Cook and Kraj́ıček [6]
On Modal Logics of Partial Recursive Functions
The classical propositional logic is known to be sound and complete with
respect to the set semantics that interprets connectives as set operations. The
paper extends propositional language by a new binary modality that corresponds
to partial recursive function type constructor under the above interpretation.
The cases of deterministic and non-deterministic functions are considered and
for both of them semantically complete modal logics are described and
decidability of these logics is established
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
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
Satisfaction classes in nonstandard models of first-order arithmetic
A satisfaction class is a set of nonstandard sentences respecting Tarski's
truth definition. We are mainly interested in full satisfaction classes, i.e.,
satisfaction classes which decides all nonstandard sentences. Kotlarski,
Krajewski and Lachlan proved in 1981 that a countable model of PA admits a
satisfaction class if and only if it is recursively saturated. A proof of this
fact is presented in detail in such a way that it is adaptable to a language
with function symbols. The idea that a satisfaction class can only see finitely
deep in a formula is extended to terms. The definition gives rise to new
notions of valuations of nonstandard terms; these are investigated. The notion
of a free satisfaction class is introduced, it is a satisfaction class free of
existential assumptions on nonstandard terms.
It is well known that pathologies arise in some satisfaction classes. Ideas
of how to remove those are presented in the last chapter. This is done mainly
by adding inference rules to M-logic. The consistency of many of these
extensions is left as an open question.Comment: Thesis for the degree of licentiate of philosophy, 74 pages, 4
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