167 research outputs found
Improving legibility of natural deduction proofs is not trivial
In formal proof checking environments such as Mizar it is not merely the
validity of mathematical formulas that is evaluated in the process of adoption
to the body of accepted formalizations, but also the readability of the proofs
that witness validity. As in case of computer programs, such proof scripts may
sometimes be more and sometimes be less readable. To better understand the
notion of readability of formal proofs, and to assess and improve their
readability, we propose in this paper a method of improving proof readability
based on Behaghel's First Law of sentence structure. Our method maximizes the
number of local references to the directly preceding statement in a proof
linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page
Multiplication-Related Classes of Complex Numbers
The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced.Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandMarco B. Caminati and Giuseppe Rosolini. Custom automations in Mizar. Journal of Automated Reasoning, 50(2):147â160, 2013.Artur KorniĆowicz. On rewriting rules in Mizar. Journal of Automated Reasoning, 50(2): 203â210, 2013.Karol PÄ
k. Improving legibility of natural deduction proofs is not trivial. Logical Methods in Computer Science, 10, 2014.28219721
Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq
We mechanise the undecidability of various frst-order axiom systems in Coq, employing
the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments
of Peano arithmetic (PA) as well as ZF and related fnitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e.
Hilbertâs tenth problem (H10), and the Post correspondence problem (PCP), respectively.
In the synthetic setting based on the computability of all functions defnable in a constructive foundation, such as Coqâs type theory, it sufces to defne these reductions as metalevel functions with no need for further encoding in a formalised model of computation.
The concrete cases of PA and the considered set theories are supplemented by a general
synthetic theory of undecidable axiomatisations, focusing on well-known connections to
consistency and incompleteness. Specifcally, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic
extensions still justifed by such standard models are shown incomplete. As a by-product of
the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of frst-order logic with a single binary relation
A proof calculus which reduces syntactic bureaucracy
International audienceIn usual proof systems, like the sequent calculus, only a very limited way of combining proofs is available through the tree structure. We present in this paper a logic-independent proof calculus, where proofs can be freely composed by connectives, and prove its basic properties. The main advantage of this proof calculus is that it allows to avoid certain types of syntactic bureaucracy inherent to all usual proof systems, in particular the sequent calculus. Proofs in this system closely reflect their atomic flow, which traces the behaviour of atoms through structural rules. The general definition is illustrated by the standard deep-inference system for propositional logic, for which there are known rewriting techniques that achieve cut elimination based only on the information in atomic flows
Hammering towards QED
This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong âone-strokeâ tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistantâs logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QED-like efforts.Blanchetteâs Sledgehammer research was supported by the Deutsche Forschungs-
gemeinschaft projects Quis Custodiet (grants NI 491/11-1 and NI 491/11-2) and
Hardening the Hammer (grant NI 491/14-1). Kaliszyk is supported by the Austrian
Science Fund (FWF) grant P26201. Sledgehammer was originally supported by the
UKâs Engineering and Physical Sciences Research Council (grant GR/S57198/01).
Urbanâs work was supported by the Marie-Curie Outgoing International Fellowship
project AUTOKNOMATH (grant MOIF-CT-2005-21875) and by the Netherlands
Organisation for Scientific Research (NWO) project Knowledge-based Automated
Reasoning (grant 612.001.208).This is the final published version. It first appeared at http://jfr.unibo.it/article/view/4593/5730?acceptCookies=1
Metalevel and reflexive extension in mechanical theorem proving
In spite of many years of research into mechanical assistance for mathematics
it is still much more difficult to construct a proof on a machine than on
paper. Of course this is partly because, unlike a proof on paper, a machine
checked proof must be formal in the strictest sense of that word, but it is
also because usually the ways of going about building proofs on a machine
are limited compared to what a mathematician is used to. This thesis looks
at some possible extensions to the range of tools available on a machine
that might lend a user more flexibility in proving theorems, complementing
whatever is already available.In particular, it examines what is possible in a framework theorem
prover. Such a system, if it is configured to prove theorems in a particular
logic T, must have a formal description of the proof theory of T written
in the framework theory F of the system. So it should be possible to use
whatever facilities are available in F not only to prove theorems of T, but
also theorems about T that can then be used in their turn to aid the user
in building theorems of T.The thesis is divided into three parts. The first describes the theory
FSâ, which has been suggested by Feferman as a candidate for a framework
theory suitable for doing meta-theory. The second describes some experiments with FSâ, proving meta-theorems. The third describes an experiment
in extending the theory PRA, declared in FSâ, with a reflection facility.More precisely, in the second section three theories are formalised:
propositional logic, sorted predicate logic, and the lambda calculus (with
a deBruijn style binding). For the first two the deduction theorem and
the prenex normal form theorem are respectively proven. For the third, a
relational definition of beta-reduction is replaced with an explicit function.In the third section, a method is proposed for avoiding the work involved
in building a full Godel style proof predicate for a theory. It is suggested
that the language be extended with quotation and substitution facilities directly, instead of providing them as definitional extensions. With this, it
is possible to exploit an observation of Solovay's that the Lob derivability
conditions are sufficient to capture the schematic behaviour of a proof
predicate. Combining this with a reflection schema is enough to produce
a non-conservative extension of PRA, and this is demonstrated by some
experiments
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