21,631 research outputs found
Inductive-data-type Systems
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last
two authors presented a combined language made of a (strongly normalizing)
algebraic rewrite system and a typed lambda-calculus enriched by
pattern-matching definitions following a certain format, called the "General
Schema", which generalizes the usual recursor definitions for natural numbers
and similar "basic inductive types". This combined language was shown to be
strongly normalizing. The purpose of this paper is to reformulate and extend
the General Schema in order to make it easily extensible, to capture a more
general class of inductive types, called "strictly positive", and to ease the
strong normalization proof of the resulting system. This result provides a
computation model for the combination of an algebraic specification language
based on abstract data types and of a strongly typed functional language with
strictly positive inductive types.Comment: Theoretical Computer Science (2002
Analytic Tableaux for Simple Type Theory and its First-Order Fragment
We study simple type theory with primitive equality (STT) and its first-order
fragment EFO, which restricts equality and quantification to base types but
retains lambda abstraction and higher-order variables. As deductive system we
employ a cut-free tableau calculus. We consider completeness, compactness, and
existence of countable models. We prove these properties for STT with respect
to Henkin models and for EFO with respect to standard models. We also show that
the tableau system yields a decision procedure for three EFO fragments
An interpretation of the Sigma-2 fragment of classical Analysis in System T
We show that it is possible to define a realizability interpretation for the
-fragment of classical Analysis using G\"odel's System T only. This
supplements a previous result of Schwichtenberg regarding bar recursion at
types 0 and 1 by showing how to avoid using bar recursion altogether. Our
result is proved via a conservative extension of System T with an operator for
composable continuations from the theory of programming languages due to Danvy
and Filinski. The fragment of Analysis is therefore essentially constructive,
even in presence of the full Axiom of Choice schema: Weak Church's Rule holds
of it in spite of the fact that it is strong enough to refute the formal
arithmetical version of Church's Thesis
Cut Elimination for a Logic with Induction and Co-induction
Proof search has been used to specify a wide range of computation systems. In
order to build a framework for reasoning about such specifications, we make use
of a sequent calculus involving induction and co-induction. These proof
principles are based on a proof theoretic (rather than set-theoretic) notion of
definition. Definitions are akin to logic programs, where the left and right
rules for defined atoms allow one to view theories as "closed" or defining
fixed points. The use of definitions and free equality makes it possible to
reason intentionally about syntax. We add in a consistent way rules for pre and
post fixed points, thus allowing the user to reason inductively and
co-inductively about properties of computational system making full use of
higher-order abstract syntax. Consistency is guaranteed via cut-elimination,
where we give the first, to our knowledge, cut-elimination procedure in the
presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi
Higher-Order Termination: from Kruskal to Computability
Termination is a major question in both logic and computer science. In logic,
termination is at the heart of proof theory where it is usually called strong
normalization (of cut elimination). In computer science, termination has always
been an important issue for showing programs correct. In the early days of
logic, strong normalization was usually shown by assigning ordinals to
expressions in such a way that eliminating a cut would yield an expression with
a smaller ordinal. In the early days of verification, computer scientists used
similar ideas, interpreting the arguments of a program call by a natural
number, such as their size. Showing the size of the arguments to decrease for
each recursive call gives a termination proof of the program, which is however
rather weak since it can only yield quite small ordinals. In the sixties, Tait
invented a new method for showing cut elimination of natural deduction, based
on a predicate over the set of terms, such that the membership of an expression
to the predicate implied the strong normalization property for that expression.
The predicate being defined by induction on types, or even as a fixpoint, this
method could yield much larger ordinals. Later generalized by Girard under the
name of reducibility or computability candidates, it showed very effective in
proving the strong normalization property of typed lambda-calculi..
A dependent nominal type theory
Nominal abstract syntax is an approach to representing names and binding
pioneered by Gabbay and Pitts. So far nominal techniques have mostly been
studied using classical logic or model theory, not type theory. Nominal
extensions to simple, dependent and ML-like polymorphic languages have been
studied, but decidability and normalization results have only been established
for simple nominal type theories. We present a LF-style dependent type theory
extended with name-abstraction types, prove soundness and decidability of
beta-eta-equivalence checking, discuss adequacy and canonical forms via an
example, and discuss extensions such as dependently-typed recursion and
induction principles
Building Decision Procedures in the Calculus of Inductive Constructions
It is commonly agreed that the success of future proof assistants will rely
on their ability to incorporate computations within deduction in order to mimic
the mathematician when replacing the proof of a proposition P by the proof of
an equivalent proposition P' obtained from P thanks to possibly complex
calculations. In this paper, we investigate a new version of the calculus of
inductive constructions which incorporates arbitrary decision procedures into
deduction via the conversion rule of the calculus. The novelty of the problem
in the context of the calculus of inductive constructions lies in the fact that
the computation mechanism varies along proof-checking: goals are sent to the
decision procedure together with the set of user hypotheses available from the
current context. Our main result shows that this extension of the calculus of
constructions does not compromise its main properties: confluence, subject
reduction, strong normalization and consistency are all preserved
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