22 research outputs found
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
MALL proof equivalence is Logspace-complete, via binary decision diagrams
Proof equivalence in a logic is the problem of deciding whether two proofs
are equivalent modulo a set of permutation of rules that reflects the
commutative conversions of its cut-elimination procedure. As such, it is
related to the question of proofnets: finding canonical representatives of
equivalence classes of proofs that have good computational properties. It can
also be seen as the word problem for the notion of free category corresponding
to the logic.
It has been recently shown that proof equivalence in MLL (the multiplicative
with units fragment of linear logic) is PSPACE-complete, which rules out any
low-complexity notion of proofnet for this particular logic.
Since it is another fragment of linear logic for which attempts to define a
fully satisfactory low-complexity notion of proofnet have not been successful
so far, we study proof equivalence in MALL- (multiplicative-additive without
units fragment of linear logic) and discover a situation that is totally
different from the MLL case. Indeed, we show that proof equivalence in MALL-
corresponds (under AC0 reductions) to equivalence of binary decision diagrams,
a data structure widely used to represent and analyze Boolean functions
efficiently.
We show these two equivalent problems to be LOGSPACE-complete. If this
technically leaves open the possibility for a complete solution to the question
of proofnets for MALL-, the established relation with binary decision diagrams
actually suggests a negative solution to this problem.Comment: in TLCA 201
Type Isomorphisms for Multiplicative-Additive Linear Logic
We characterize type isomorphisms in the multiplicative-additive fragment of linear logic (MALL), and thus for ?-autonomous categories with finite products, extending a result for the multiplicative fragment by Balat and Di Cosmo [Vincent Balat and Roberto Di Cosmo, 1999]. This yields a much richer equational theory involving distributivity and annihilation laws. The unit-free case is obtained by relying on the proof-net syntax introduced by Hughes and Van Glabbeek [Dominic Hughes and Rob van Glabbeek, 2005]. We then use the sequent calculus to extend our results to full MALL (including all units)
Normalization Without Syntax
International audienceWe present normalization for intuitionistic combinatorial proofs (ICPs) and relate it to the simplytyped lambda-calculus. We prove confluence and strong normalization. Combinatorial proofs, or "proofs without syntax", form a graphical semantics of proof in various logics that is canonical yet complexity-aware: they are a polynomial-sized representation of sequent proofs that factors out exactly the non-duplicating permutations. Our approach to normalization aligns with these characteristics: it is canonical (free of permutations) and generic (readily applied to other logics). Our reduction mechanism is a canonical representation of reduction in sequent calculus with closed cuts (no abstraction is allowed below a cut), and relates to closed reduction in lambda-calculus and supercombinators. While we will use ICPs concretely, the notion of reduction is completely abstract, and can be specialized to give a reduction mechanism for any representation of typed normal forms
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science
Extending Implicit Computational Complexity and Abstract Machines to Languages with Control
The Curry-Howard isomorphism is the idea that proofs in natural deduction can be put in correspondence with lambda terms in such a way that this correspondence is preserved by normalization. The concept can be extended from Intuitionistic Logic to other systems, such as Linear Logic. One of the nice conseguences of this isomorphism is that we can reason about functional programs with formal tools which are typical of proof systems: such analysis can also include quantitative qualities of programs, such as the number of steps it takes to terminate. Another is the possiblity to describe the execution of these programs in terms of abstract machines.
In 1990 Griffin proved that the correspondence can be extended to Classical Logic and control operators. That is, Classical Logic adds the possiblity to manipulate continuations. In this thesis we see how the things we described above work in this larger context