7,098 research outputs found
Deep Inference and Symmetry in Classical Proofs
In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems
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
Cirquent calculus deepened
Cirquent calculus is a new proof-theoretic and semantic framework, whose main
distinguishing feature is being based on circuits, as opposed to the more
traditional approaches that deal with tree-like objects such as formulas or
sequents. Among its advantages are greater efficiency, flexibility and
expressiveness. This paper presents a detailed elaboration of a deep-inference
cirquent logic, which is naturally and inherently resource conscious. It shows
that classical logic, both syntactically and semantically, is just a special,
conservative fragment of this more general and, in a sense, more basic logic --
the logic of resources in the form of cirquent calculus. The reader will find
various arguments in favor of switching to the new framework, such as arguments
showing the insufficiency of the expressive power of linear logic or other
formula-based approaches to developing resource logics, exponential
improvements over the traditional approaches in both representational and proof
complexities offered by cirquent calculus, and more. Among the main purposes of
this paper is to provide an introductory-style starting point for what, as the
author wishes to hope, might have a chance to become a new line of research in
proof theory -- a proof theory based on circuits instead of formulas.Comment: Significant improvements over the previous version
Importing SMT and Connection proofs as expansion trees
Different automated theorem provers reason in various deductive systems and,
thus, produce proof objects which are in general not compatible. To understand
and analyze these objects, one needs to study the corresponding proof theory,
and then study the language used to represent proofs, on a prover by prover
basis. In this work we present an implementation that takes SMT and Connection
proof objects from two different provers and imports them both as expansion
trees. By representing the proofs in the same framework, all the algorithms and
tools available for expansion trees (compression, visualization, sequent
calculus proof construction, proof checking, etc.) can be employed uniformly.
The expansion proofs can also be used as a validation tool for the proof
objects produced.Comment: In Proceedings PxTP 2015, arXiv:1507.0837
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
On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics
We consider two styles of proof calculi for a family of tense logics,
presented in a formalism based on nested sequents. A nested sequent can be seen
as a tree of traditional single-sided sequents. Our first style of calculi is
what we call "shallow calculi", where inference rules are only applied at the
root node in a nested sequent. Our shallow calculi are extensions of Kashima's
calculus for tense logic and share an essential characteristic with display
calculi, namely, the presence of structural rules called "display postulates".
Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable
for proof search due to the presence of display postulates and other structural
rules. The second style of calculi uses deep-inference, whereby inference rules
can be applied at any node in a nested sequent. We show that, for a range of
extensions of tense logic, the two styles of calculi are equivalent, and there
is a natural proof theoretic correspondence between display postulates and deep
inference. The deep inference calculi enjoy the subformula property and have no
display postulates or other structural rules, making them a better framework
for proof search
A System of Interaction and Structure II: The Need for Deep Inference
This paper studies properties of the logic BV, which is an extension of
multiplicative linear logic (MLL) with a self-dual non-commutative operator. BV
is presented in the calculus of structures, a proof theoretic formalism that
supports deep inference, in which inference rules can be applied anywhere
inside logical expressions. The use of deep inference results in a simple
logical system for MLL extended with the self-dual non-commutative operator,
which has been to date not known to be expressible in sequent calculus. In this
paper, deep inference is shown to be crucial for the logic BV, that is, any
restriction on the ``depth'' of the inference rules of BV would result in a
strictly less expressive logical system
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