48,144 research outputs found
Proofs Without Syntax
"[M]athematicians care no more for logic than logicians for mathematics."
Augustus de Morgan, 1868.
Proofs are traditionally syntactic, inductively generated objects. This paper
presents an abstract mathematical formulation of propositional calculus
(propositional logic) in which proofs are combinatorial (graph-theoretic),
rather than syntactic. It defines a *combinatorial proof* of a proposition P as
a graph homomorphism h : C -> G(P), where G(P) is a graph associated with P and
C is a coloured graph. The main theorem is soundness and completeness: P is
true iff there exists a combinatorial proof h : C -> G(P).Comment: Appears in Annals of Mathematics, 2006. 5 pages + references. Version
1 is submitted version; v3 is final published version (in two-column format
rather than Annals style). Changes for v2: dualised definition of
combinatorial truth, thereby shortening some subsequent proofs; added
references; corrected typos; minor reworking of some sentences/paragraphs;
added comments on polynomial-time correctness (referee request). Changes for
v3: corrected two typos, reworded one sentence, repeated a citation in Notes
sectio
A Provably Correct Translation of the λ-Calculus into a Mathematical Model of C++
We introduce a translation of the simply typed λ-calculus into C++, and give a mathematical proof of the correctness of this translation. For this purpose we develop a suitable fragment of C++ together with a denotational semantics. We introduce a formal translation of the λ-calculus into this fragment, and show that this translation is correct with respect to the denotational semantics. We show as well a completeness result, namely that by translating λ-terms we obtain essentially all C++ terms in this fragment. We introduce a mathematical model for the evaluation of programs of this fragment, and show that the evaluation computes the correct result with respect to this semantics.
On the Completeness of Interpolation Algorithms
Craig interpolation is a fundamental property of classical and non-classic
logics with a plethora of applications from philosophical logic to
computer-aided verification. The question of which interpolants can be obtained
from an interpolation algorithm is of profound importance. Motivated by this
question, we initiate the study of completeness properties of interpolation
algorithms. An interpolation algorithm is \emph{complete} if, for
every semantically possible interpolant of an implication , there
is a proof of such that is logically equivalent to
. We establish incompleteness and different kinds of
completeness results for several standard algorithms for resolution and the
sequent calculus for propositional, modal, and first-order logic
On the equivalence between MV-algebras and -groups with strong unit
In "A new proof of the completeness of the Lukasiewicz axioms"} (Transactions
of the American Mathematical Society, 88) C.C. Chang proved that any totally
ordered -algebra was isomorphic to the segment
of a totally ordered -group with strong unit . This was done by the
simple intuitive idea of putting denumerable copies of on top of each other
(indexed by the integers). Moreover, he also show that any such group can
be recovered from its segment since , establishing an
equivalence of categories. In "Interpretation of AF -algebras in
Lukasiewicz sentential calculus" (J. Funct. Anal. Vol. 65) D. Mundici extended
this result to arbitrary -algebras and -groups with strong unit. He
takes the representation of as a sub-direct product of chains , and
observes that where . Then he let be the -subgroup generated by inside . He proves that this idea works, and establish an equivalence of
categories in a rather elaborate way by means of his concept of good sequences
and its complicated arithmetics. In this note, essentially self-contained
except for Chang's result, we give a simple proof of this equivalence taking
advantage directly of the arithmetics of the the product -group , avoiding entirely the notion of good sequence.Comment: 6 page
Completeness of the ZX-Calculus
The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum
mechanics and quantum information theory. It comes equipped with an equational
presentation. We focus here on a very important property of the language:
completeness, which roughly ensures the equational theory captures all of
quantum mechanics. We first improve on the known-to-be-complete presentation
for the so-called Clifford fragment of the language - a restriction that is not
universal - by adding some axioms. Thanks to a system of back-and-forth
translation between the ZX-Calculus and a third-party complete graphical
language, we prove that the provided axiomatisation is complete for the first
approximately universal fragment of the language, namely Clifford+T.
We then prove that the expressive power of this presentation, though aimed at
achieving completeness for the aforementioned restriction, extends beyond
Clifford+T, to a class of diagrams that we call linear with Clifford+T
constants. We use another version of the third-party language - and an adapted
system of back-and-forth translation - to complete the language for the
ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the
added axioms, and finally, we provide a complete axiomatisation for an altered
version of the language which involves an additional generator, making the
presentation simpler
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
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