14,455 research outputs found
A new graphical calculus of proofs
We offer a simple graphical representation for proofs of intuitionistic
logic, which is inspired by proof nets and interaction nets (two formalisms
originating in linear logic). This graphical calculus of proofs inherits good
features from each, but is not constrained by them. By the Curry-Howard
isomorphism, the representation applies equally to the lambda calculus,
offering an alternative diagrammatic representation of functional computations.Comment: In Proceedings TERMGRAPH 2011, arXiv:1102.226
Expansion Trees with Cut
Herbrand's theorem is one of the most fundamental insights in logic. From the
syntactic point of view it suggests a compact representation of proofs in
classical first- and higher-order logic by recording the information which
instances have been chosen for which quantifiers, known in the literature as
expansion trees.
Such a representation is inherently analytic and hence corresponds to a
cut-free sequent calculus proof. Recently several extensions of such proof
representations to proofs with cut have been proposed. These extensions are
based on graphical formalisms similar to proof nets and are limited to prenex
formulas.
In this paper we present a new approach that directly extends expansion trees
by cuts and covers also non-prenex formulas. We describe a cut-elimination
procedure for our expansion trees with cut that is based on the natural
reduction steps. We prove that it is weakly normalizing using methods from the
epsilon-calculus
ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
We present a new graphical calculus that is sound and complete for a
universal family of quantum circuits, which can be seen as the natural
string-diagrammatic extension of the approximately (real-valued) universal
family of Hadamard+CCZ circuits. The diagrammatic language is generated by two
kinds of nodes: the so-called 'spider' associated with the computational basis,
as well as a new arity-N generalisation of the Hadamard gate, which satisfies a
variation of the spider fusion law. Unlike previous graphical calculi, this
admits compact encodings of non-linear classical functions. For example, the
AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in
the ZX-calculus. Consequently, N-controlled gates, hypergraph states,
Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the
Clifford hierarchy also enjoy encodings with low constant overhead. This
suggests that this calculus will be significantly more convenient for reasoning
about the interplay between classical non-linear behaviour (e.g. in an oracle)
and purely quantum operations. After presenting the calculus, we will prove it
is sound and complete for universal quantum computation by demonstrating the
reduction of any diagram to an easily describable normal form.Comment: In Proceedings QPL 2018, arXiv:1901.0947
A complete graphical calculus for Spekkens' toy bit theory
While quantum theory cannot be described by a local hidden variable model, it
is nevertheless possible to construct such models that exhibit features
commonly associated with quantum mechanics. These models are also used to
explore the question of {\psi}-ontic versus {\psi}-epistemic theories for
quantum mechanics. Spekkens' toy theory is one such model. It arises from
classical probabilistic mechanics via a limit on the knowledge an observer may
have about the state of a system. The toy theory for the simplest possible
underlying system closely resembles stabilizer quantum mechanics, a fragment of
quantum theory which is efficiently classically simulable but also non-local.
Further analysis of the similarities and differences between those two theories
can thus yield new insights into what distinguishes quantum theory from
classical theories, and {\psi}-ontic from {\psi}-epistemic theories.
In this paper, we develop a graphical language for Spekkens' toy theory.
Graphical languages offer intuitive and rigorous formalisms for the analysis of
quantum mechanics and similar theories. To compare quantum mechanics and a toy
model, it is useful to have similar formalisms for both. We show that our
language fully describes Spekkens' toy theory and in particular, that it is
complete: meaning any equality that can be derived using other formalisms can
also be derived entirely graphically. Our language is inspired by a similar
graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'
toy bit theory and stabilizer quantum mechanics can be analysed and compared
using analogous graphical formalisms.Comment: Major revisions for v2. 22+7 page
Advanced Proof Viewing in ProofTool
Sequent calculus is widely used for formalizing proofs. However, due to the
proliferation of data, understanding the proofs of even simple mathematical
arguments soon becomes impossible. Graphical user interfaces help in this
matter, but since they normally utilize Gentzen's original notation, some of
the problems persist. In this paper, we introduce a number of criteria for
proof visualization which we have found out to be crucial for analyzing proofs.
We then evaluate recent developments in tree visualization with regard to these
criteria and propose the Sunburst Tree layout as a complement to the
traditional tree structure. This layout constructs inferences as concentric
circle arcs around the root inference, allowing the user to focus on the
proof's structural content. Finally, we describe its integration into ProofTool
and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785
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