734 research outputs found
Compiling and securing cryptographic protocols
Protocol narrations are widely used in security as semi-formal notations to
specify conversations between roles. We define a translation from a protocol
narration to the sequences of operations to be performed by each role. Unlike
previous works, we reduce this compilation process to well-known decision
problems in formal protocol analysis. This allows one to define a natural
notion of prudent translation and to reuse many known results from the
literature in order to cover more crypto-primitives. In particular this work is
the first one to show how to compile protocols parameterised by the properties
of the available operations.Comment: A short version was submitted to IP
Some decidable congruences of free monoids
summary:Let be the free monoid over a finite alphabet . We prove that a congruence of generated by a finite number of pairs , where and , is always decidable
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
Strongly nonfinitely based monoids
We show that the 42-element monoid of all partial order preserving and
extensive injections on the 4-element chain is not contained in any variety
generated by a finitely based finite semigroup.Comment: 12 pages. In version 2: the proofs of two key lemmas are expanded;
two remarks are added; references are updated; a few typos and language
errors are fixe
Relation algebras with n-dimensional relational bases
Accepted versio
Hierarchical combination of intruder theories
International audienceRecently automated deduction tools have proved to be very effective for detecting attacks on cryptographic protocols. These analysis can be improved, for finding more subtle weaknesses, by a more accurate modelling of operators employed by protocols. Several works have shown how to handle a single algebraic operator (associated with a fixed intruder theory) or how to combine several operators satisfying disjoint theories. However several interesting equational theories, such as exponentiation with an abelian group law for exponents remain out of the scope of these techniques. This has motivated us to introduce a new notion of hierarchical combination for non-disjoint intruder theories and to show decidability results for the deduction problem in these theories. We have also shown that under natural hypotheses hierarchical intruder constraints can be decided. This result applies to an exponentiation theory that appears to be more general than the one considered before
Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives
Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
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