394 research outputs found
Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems
We describe the first results of a project of analyzing in which theories
formal proofs can be ex- pressed. We use this analysis as the basis of
interoperability between proof systems.Comment: In Proceedings PxTP 2017, arXiv:1712.0089
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Models and termination of proof reduction in the -calculus modulo theory
We define a notion of model for the -calculus modulo theory and
prove a soundness theorem. We then define a notion of super-consistency and
prove that proof reduction terminates in the -calculus modulo any
super-consistent theory. We prove this way the termination of proof reduction
in several theories including Simple type theory and the Calculus of
constructions
Rules and derivations in an elementary logic course
When teaching an elementary logic course to students who have a general
scientific background but have never been exposed to logic, we have to face the
problem that the notions of deduction rule and of derivation are completely new
to them, and are related to nothing they already know, unlike, for instance,
the notion of model, that can be seen as a generalization of the notion of
algebraic structure. In this note, we defend the idea that one strategy to
introduce these notions is to start with the notion of inductive definition
[1]. Then, the notion of derivation comes naturally. We also defend the idea
that derivations are pervasive in logic and that defining precisely this notion
at an early stage is a good investment to later define other notions in proof
theory, computability theory, automata theory, ... Finally, we defend the idea
that to define the notion of derivation precisely, we need to distinguish two
notions of derivation: labeled with elements and labeled with rule names. This
approach has been taken in [2]
On the definition of the classical connectives and quantifiers
Classical logic is embedded into constructive logic, through a definition of
the classical connectives and quantifiers in terms of the constructive ones.Comment: Why is this a Proof?, Festschrift for Luiz Carlos Pereira , 201
Free fall and cellular automata
Three reasonable hypotheses lead to the thesis that physical phenomena can be
described and simulated with cellular automata. In this work, we attempt to
describe the motion of a particle upon which a constant force is applied, with
a cellular automaton, in Newtonian physics, in Special Relativity, and in
General Relativity. The results are very different for these three theories.Comment: In Proceedings DCM 2015, arXiv:1603.0053
Linear-algebraic lambda-calculus
With a view towards models of quantum computation and/or the interpretation
of linear logic, we define a functional language where all functions are linear
operators by construction. A small step operational semantic (and hence an
interpreter/simulator) is provided for this language in the form of a term
rewrite system. The linear-algebraic lambda-calculus hereby constructed is
linear in a different (yet related) sense to that, say, of the linear
lambda-calculus. These various notions of linearity are discussed in the
context of quantum programming languages. KEYWORDS: quantum lambda-calculus,
linear lambda-calculus, -calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language
interpreter/simulator file (see "other formats"). See the more recent
arXiv:quant-ph/061219
The physical Church-Turing thesis and the principles of quantum theory
Notoriously, quantum computation shatters complexity theory, but is innocuous
to computability theory. Yet several works have shown how quantum theory as it
stands could breach the physical Church-Turing thesis. We draw a clear line as
to when this is the case, in a way that is inspired by Gandy. Gandy formulates
postulates about physics, such as homogeneity of space and time, bounded
density and velocity of information --- and proves that the physical
Church-Turing thesis is a consequence of these postulates. We provide a quantum
version of the theorem. Thus this approach exhibits a formal non-trivial
interplay between theoretical physics symmetries and computability assumptions.Comment: 14 pages, LaTe
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
The probability of non-confluent systems
We show how to provide a structure of probability space to the set of
execution traces on a non-confluent abstract rewrite system, by defining a
variant of a Lebesgue measure on the space of traces. Then, we show how to use
this probability space to transform a non-deterministic calculus into a
probabilistic one. We use as example Lambda+, a recently introduced calculus
defined through type isomorphisms.Comment: In Proceedings DCM 2013, arXiv:1403.768
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