5,644 research outputs found
Proof Nets for First-Order Additive Linear Logic
We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation
Proof nets for first-order additive linear logic
International audienceWe present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation. 2012 ACM Subject Classification Theory of computation → Proof theory; Theory of computation → Linear logi
Graphical representation of canonical proof: two case studies
An interesting problem in proof theory is to find representations of proof that do
not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation
of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces
artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering
of freely permutable inferences. A proof system that is free of bureaucracy is called
canonical for a logic. In this dissertation two canonical proof systems are presented,
for two logics: a notion of proof nets for additive linear logic with units, and ‘classical
proof forests’, a graphical formalism for first-order classical logic.
Additive linear logic (or sum–product logic) is the fragment of linear logic consisting
of linear implication between formulae constructed only from atomic formulae and
the additive connectives and units. Up to an equational theory over proofs, the logic
describes categories in which finite products and coproducts occur freely. A notion of
proof nets for additive linear logic is presented, providing canonical graphical representations
of the categorical morphisms and constituting a tractable decision procedure
for this equational theory. From existing proof nets for additive linear logic without
units by Hughes and Van Glabbeek (modified to include the units naively), canonical
proof nets are obtained by a simple graph rewriting algorithm called saturation. Main
technical contributions are the substantial correctness proof of the saturation algorithm,
and a correctness criterion for saturated nets.
Classical proof forests are a canonical, graphical proof formalism for first-order
classical logic. Related to Herbrand’s Theorem and backtracking games in the style
of Coquand, the forests assign witnessing information to quantifiers in a structurally
minimal way, reducing a first-order sentence to a decidable propositional one. A similar
formalism ‘expansion tree proofs’ was presented by Miller, but not given a method
of composition. The present treatment adds a notion of cut, and investigates the possibility
of composing forests via cut-elimination. Cut-reduction steps take the form
of a rewrite relation that arises from the structure of the forests in a natural way.
Yet reductions are intricate, and initially not well-behaved: from perfectly ordinary
cuts, reduction may reach unnaturally configured cuts that may not be reduced. Cutelimination
is shown using a modified version of the rewrite relation, inspired by the
game-theoretic interpretation of the forests, for which weak normalisation is shown,
and strong normalisation is conjectured. In addition, by a more intricate argument,
weak normalisation is also shown for the original reduction relation
Multiplicative-Additive Proof Equivalence is Logspace-complete, via Binary Decision Trees
Given a logic presented in a sequent calculus, a natural question is that of
equivalence of proofs: to determine whether two given proofs are equated by any
denotational semantics, ie any categorical interpretation of the logic
compatible with its cut-elimination procedure. This notion can usually be
captured syntactically by a set of rule permutations.
Very generally, proofnets can be defined as combinatorial objects which
provide canonical representatives of equivalence classes of proofs. In
particular, the existence of proof nets for a logic provides a solution to the
equivalence problem of this logic. In certain fragments of linear logic, it is
possible to give a notion of proofnet with good computational properties,
making it a suitable representation of proofs for studying the cut-elimination
procedure, among other things.
It has recently been proved that there cannot be such a notion of proofnets
for the multiplicative (with units) fragment of linear logic, due to the
equivalence problem for this logic being Pspace-complete.
We investigate the multiplicative-additive (without unit) fragment of linear
logic and show it is closely related to binary decision trees: we build a
representation of proofs based on binary decision trees, reducing proof
equivalence to decision tree equivalence, and give a converse encoding of
binary decision trees as proofs. We get as our main result that the complexity
of the proof equivalence problem of the studied fragment is Logspace-complete.Comment: arXiv admin note: text overlap with arXiv:1502.0199
Bipolar Proof Nets for MALL
In this work we present a computation paradigm based on a concurrent and
incremental construction of proof nets (de-sequentialized or graphical proofs)
of the pure multiplicative and additive fragment of Linear Logic, a resources
conscious refinement of Classical Logic. Moreover, we set a correspon- dence
between this paradigm and those more pragmatic ones inspired to transactional
or distributed systems. In particular we show that the construction of additive
proof nets can be interpreted as a model for super-ACID (or co-operative)
transactions over distributed transactional systems (typi- cally,
multi-databases).Comment: Proceedings of the "Proof, Computation, Complexity" International
Workshop, 17-18 August 2012, University of Copenhagen, Denmar
Session Types in Abelian Logic
There was a PhD student who says "I found a pair of wooden shoes. I put a
coin in the left and a key in the right. Next morning, I found those objects in
the opposite shoes." We do not claim existence of such shoes, but propose a
similar programming abstraction in the context of typed lambda calculi. The
result, which we call the Amida calculus, extends Abramsky's linear lambda
calculus LF and characterizes Abelian logic.Comment: In Proceedings PLACES 2013, arXiv:1312.221
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