32,175 research outputs found
A first-order logic for string diagrams
Equational reasoning with string diagrams provides an intuitive means of
proving equations between morphisms in a symmetric monoidal category. This can
be extended to proofs of infinite families of equations using a simple
graphical syntax called !-box notation. While this does greatly increase the
proving power of string diagrams, previous attempts to go beyond equational
reasoning have been largely ad hoc, owing to the lack of a suitable logical
framework for diagrammatic proofs involving !-boxes. In this paper, we extend
equational reasoning with !-boxes to a fully-fledged first order logic called
with conjunction, implication, and universal quantification over !-boxes. This
logic, called !L, is then rich enough to properly formalise an induction
principle for !-boxes. We then build a standard model for !L and give an
example proof of a theorem for non-commutative bialgebras using !L, which is
unobtainable by equational reasoning alone.Comment: 15 pages + appendi
On multiple conclusion deductions in classical logic
Kneale observed that Gentzen’s calculus of natural deductions NK for classical
logic is not symmetric and has unnecessarily complicated hypothetical inference rules.
Kneale proposed inference rules with multiple conclusions as a basis for a symmetric natural
deduction calculus for classical logic. However, Kneale’s informally presented calculus is
not complete. In this paper, we define a calculus of multiple conclusion natural deductions
(MCD) for classical propositional logic based on Kneale’s multiple conclusion inference
rules. For MCD we present elementary proof search that produces proofs in normal form.
MCD proof search is motivated and explained as being a notational variant of Smullyan’s
analytic tableaux method in its initial part and a notational variant of refutation proofs
based on Robinson’s resolution in its final part. We consider MCD to have semantic motivation
of both its inference rules and its proof search. This is unusual for the natural
deduction calculi as they are syntactically motivated. Syntactic motivation is adequate for
intuitionistic logic but not a natural fit for truth-functional classical propositional logic
Graded Differential Categories and Graded Differential Linear Logic
In Linear Logic (), the exponential modality brings forth a
distinction between non-linear proofs and linear proofs, where linear means
using an argument exactly once. Differential Linear Logic () is
an extension of Linear Logic which includes additional rules for which
encode differentiation and the ability of linearizing proofs. On the other
hand, Graded Linear Logic () is a variation of Linear Logic in
such a way that is now indexed over a semiring . This -grading allows
for non-linear proofs of degree , such that the linear proofs are of
degree . There has been recent interest in combining these two
variations of together and developing Graded Differential Linear
Logic (). In this paper we present a sequent calculus for
, as well as introduce its categorical semantics, which we call
graded differential categories, using both coderelictions and deriving
transformations. We prove that symmetric powers always give graded differential
categories, and provide other examples of graded differential categories. We
also discuss graded versions of (monoidal) coalgebra modalities, additive
bialgebra modalities, and the Seely isomorphisms, as well as their
implementations in the sequent calculus of .Comment: In the proceedings of MFPS2023 (appendix included
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
Arithmetical proofs of strong normalization results for symmetric lambda calculi
International audienceWe give arithmetical proofs of the strong normalization of two symmetric -calculi corresponding to classical logic. The first one is the -calculus introduced by Curien & Herbelin. It is derived via the Curry-Howard correspondence from Gentzen's classical sequent calculus LK in order to have a symmetry on one side between ``program'' and ``context'' and on other side between ``call-by-name'' and ``call-by-value''. The second one is the symmetric -calculus. It is the -calculus introduced by Parigot in which the reduction rule , which is the symmetric of , is added. These results were already known but the previous proofs use candidates of reducibility where the interpretation of a type is defined as the fix point of some increasing operator and thus, are highly non arithmetical
A Fault Tolerant Connectionist Architecture for Construction of Logic Proofs
This chapter considers the problems of expressing logic and constructing proofs in fault tolerant connectionist networks that are based on energy minimalism. Given a first-order-logic knowledge base and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof for a given query. If a resolution-based proof of length no longer than k exists, then the global minima of the energy function that is associated with the network represent such proofs. If no proof exist then the global minima indicate the lack of a proof. The network that is generated is of size polynomial in the bound k and the knowledge size. There are no restrictions on the type of logic formulas that can be represented. Most of the chapter discusses the representation of propositional formulas and proofs; however, an extension is presented that allows the representation of unrestricted first-order logic formulas (predicate calculus). Fault tolerance is obtained using a binding technique that dynamically assigns symbolic roles to winner takes all units
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