3,415 research outputs found
Proof Outlines as Proof Certificates: A System Description
We apply the foundational proof certificate (FPC) framework to the problem of
designing high-level outlines of proofs. The FPC framework provides a means to
formally define and check a wide range of proof evidence. A focused proof
system is central to this framework and such a proof system provides an
interesting approach to proof reconstruction during the process of proof
checking (relying on an underlying logic programming implementation). Here, we
illustrate how the FPC framework can be used to design proof outlines and then
to exploit proof checkers as a means for expanding outlines into fully detailed
proofs. In order to validate this approach to proof outlines, we have built the
ACheck system that allows us to take a sequence of theorems and apply the proof
outline "do the obvious induction and close the proof using previously proved
lemmas".Comment: In Proceedings WoF'15, arXiv:1511.0252
A System of Interaction and Structure
This paper introduces a logical system, called BV, which extends
multiplicative linear logic by a non-commutative self-dual logical operator.
This extension is particularly challenging for the sequent calculus, and so far
it is not achieved therein. It becomes very natural in a new formalism, called
the calculus of structures, which is the main contribution of this work.
Structures are formulae submitted to certain equational laws typical of
sequents. The calculus of structures is obtained by generalising the sequent
calculus in such a way that a new top-down symmetry of derivations is observed,
and it employs inference rules that rewrite inside structures at any depth.
These properties, in addition to allow the design of BV, yield a modular proof
of cut elimination.Comment: This is the authoritative version of the article, with readable
pictures, in colour, also available at
. (The published version contains
errors introduced by the editorial processing.) Web site for Deep Inference
and the Calculus of Structures at <http://alessio.guglielmi.name/res/cos
A Cut-Free Sequent Calculus for Defeasible Erotetic Inferences
In recent years, the effort to formalize erotetic inferences (i.e., inferences
to and from questions) has become a central concern for those working
in erotetic logic. However, few have sought to formulate a proof theory
for these inferences. To fill this lacuna, we construct a calculus for (classes
of) sequents that are sound and complete for two species of erotetic inferences
studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made to axiomatize the former in a sequent
system, there is currently no proof theory for the latter. Moreover, the extant
axiomatization of erotetic evocation fails to capture its defeasible character
and provides no rules for introducing or eliminating question-forming operators.
In contrast, our calculus encodes defeasibility conditions on sequents and
provides rules governing the introduction and elimination of erotetic formulas.
We demonstrate that an elimination theorem holds for a version of the cut
rule that applies to both declarative and erotetic formulas and that the rules
for the axiomatic account of question evocation in IEL are admissible in our
system
Normalisation Control in Deep Inference via Atomic Flows
We introduce `atomic flows': they are graphs obtained from derivations by
tracing atom occurrences and forgetting the logical structure. We study simple
manipulations of atomic flows that correspond to complex reductions on
derivations. This allows us to prove, for propositional logic, a new and very
general normalisation theorem, which contains cut elimination as a special
case. We operate in deep inference, which is more general than other syntactic
paradigms, and where normalisation is more difficult to control. We argue that
atomic flows are a significant technical advance for normalisation theory,
because 1) the technique they support is largely independent of syntax; 2)
indeed, it is largely independent of logical inference rules; 3) they
constitute a powerful geometric formalism, which is more intuitive than syntax
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