579 research outputs found
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
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
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
On Berry's conjectures about the stable order in PCF
PCF is a sequential simply typed lambda calculus language. There is a unique
order-extensional fully abstract cpo model of PCF, built up from equivalence
classes of terms. In 1979, G\'erard Berry defined the stable order in this
model and proved that the extensional and the stable order together form a
bicpo. He made the following two conjectures: 1) "Extensional and stable order
form not only a bicpo, but a bidomain." We refute this conjecture by showing
that the stable order is not bounded complete, already for finitary PCF of
second-order types. 2) "The stable order of the model has the syntactic order
as its image: If a is less than b in the stable order of the model, for finite
a and b, then there are normal form terms A and B with the semantics a, resp.
b, such that A is less than B in the syntactic order." We give counter-examples
to this conjecture, again in finitary PCF of second-order types, and also
refute an improved conjecture: There seems to be no simple syntactic
characterization of the stable order. But we show that Berry's conjecture is
true for unary PCF. For the preliminaries, we explain the basic fully abstract
semantics of PCF in the general setting of (not-necessarily complete) partial
order models (f-models.) And we restrict the syntax to "game terms", with a
graphical representation.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main
changes for this version: 4.1: proof of game term theorem corrected, 7.: the
improved chain conjecture is made precise, more references adde
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
A Strong Call-By-Need Calculus
We present a call-by-need ?-calculus that enables strong reduction (that is, reduction inside the body of abstractions) and guarantees that arguments are only evaluated if needed and at most once. This calculus uses explicit substitutions and subsumes the existing strong-call-by-need strategy, but allows for more reduction sequences, and often shorter ones, while preserving the neededness.
The calculus is shown to be normalizing in a strong sense: Whenever a ?-term t admits a normal form n in the ?-calculus, then any reduction sequence from t in the calculus eventually reaches a representative of the normal form n. We also exhibit a restriction of this calculus that has the diamond property and that only performs reduction sequences of minimal length, which makes it systematically better than the existing strategy. We have used the Abella proof assistant to formalize part of this calculus, and discuss how this experiment affected its design
Canonical Proof nets for Classical Logic
Proof nets provide abstract counterparts to sequent proofs modulo rule
permutations; the idea being that if two proofs have the same underlying
proof-net, they are in essence the same proof. Providing a convincing proof-net
counterpart to proofs in the classical sequent calculus is thus an important
step in understanding classical sequent calculus proofs. By convincing, we mean
that (a) there should be a canonical function from sequent proofs to proof
nets, (b) it should be possible to check the correctness of a net in polynomial
time, (c) every correct net should be obtainable from a sequent calculus proof,
and (d) there should be a cut-elimination procedure which preserves
correctness. Previous attempts to give proof-net-like objects for propositional
classical logic have failed at least one of the above conditions. In [23], the
author presented a calculus of proof nets (expansion nets) satisfying (a) and
(b); the paper defined a sequent calculus corresponding to expansion nets but
gave no explicit demonstration of (c). That sequent calculus, called LK\ast in
this paper, is a novel one-sided sequent calculus with both additively and
multiplicatively formulated disjunction rules. In this paper (a self-contained
extended version of [23]), we give a full proof of (c) for expansion nets with
respect to LK\ast, and in addition give a cut-elimination procedure internal to
expansion nets - this makes expansion nets the first notion of proof-net for
classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and
Computation
- …