III. GEOLOGÍA
3.3. COBERTRA VOLCANICA-SEDIMENTARIA
3.3.2. Geología del cuaternario
The notion of bisimulation equivalence was developed by Park [1981] and is based upon the concept of simulation due to Milner [1982]. If one process, Sim,simulates
another, Sys, then it means that Sim is able to perform any action that Sys can, and that on completion of that action, the remaining behaviour of Sim can simulate the remaining behaviour of Sys. Bisimulation takes this concept a step further by requiring the simulation relation to besymmetric. If two process simulate one another then they are not necessarily bisimilar, since the two simulation relations may not be the symmetric forms of each other.
Strong Bisimilarity
There are many variants of bisimulation, of whichstrong bisimulation is the proto- typical and most natural. Strong bisimulation requires every action to be matched exactly, regardless of whether they are internal or external actions.
Definition 2.1(Strong Bisimulation). A relationRis astrong bisimulationif when- ever (P, Q)∈ Rthen
• ifP −→α P0 thenQ−→α Q0 and (P0, Q0)∈ R, • ifQ−→α Q0 thenP −→α P0 and (P0, Q0)∈ R.
This definition is often shortened by requiring the relation to be symmetric. This results in the following equivalent definition.
Definition 2.2(Strong Bisimulation). A symmetric relationRis astrong bisimula- tion if whenever (P, Q)∈ Rthen ifP −→α P0 thenQ−→α Q0 and (P0, Q0)∈ R.
Strong bisimilarity is the union of all strong bisimulations. In other words,P and
Qare strong bisimilar (denotedP ∼Q) if and only if there exists a strong bisimulation Rsuch that (P, Q)∈ R. β β α β α β α P ∼ Q β γ α β α γ α R S
Figure 2.4. Strong bisimilarity.
Figure 2.4 gives examples of processes that are strong bisimilar (P ∼ Q) and processes that are not strong bisimilar (RS). IfR makes the transition−→α , then the resulting process can perform either action β or γ. On the other hand, after making a transition −→α , the process S is in one of two states; one of which can perform actionβ, and the other which can perform actionγ. Neither of these states are capable of bothβ andγ.
Weak Bisimilarity
One of the most useful properties of process calculus, particularly as an approach to verification, is the ability to abstract from the internal behaviour of a system. Strong bisimilarity requires processes to match execution on a step-by-step basis. This is too discriminating when comparing design and high-level specification because their respective internal behaviours are often very different. Weak bisimilarity identifies processes that exhibit the same external behaviour but allows internal actions to be matched byzero or more internal (τ) actions.
We introduce the notation =⇒to be zero or more τtransitions, and=α⇒is short- hand for =⇒−→α .
Definition 2.3(Weak Bisimulation). A symmetric relationRis aweak bisimulation
if (P, Q)∈ Rimplies that wheneverP −→α P0 then there existsQ0such thatQ=α⇒Q0
Processes P and Q are weak bisimilar, denoted P ≈ Q, if there exists a weak bisimulationRsuch that (P, Q)∈ R.
α β β τ α β τ P ≈ Q α β α τ β τ R 6≈ S
Figure 2.5. Weak bisimilarity.
Figure 2.5 shows examples of processes that are weak bisimilar (P ≈ Q) and processes that are not (R 6≈ S). After making an internal transition, S loses one capability (either αor β), while Rstill has a choice between αandβ. On the other hand, internal transitions by P and Qresult in the inability to perform actionαin both cases.
Branching Bisimilarity
Weak bisimilarity isn’t the only equivalence that treats internal actions abstractly.
Branching bisimilarity [van Glabbeek and Weijland 1996] is similar to weak bisimi- larity, however it also matches the branching structure more accurately.
Definition 2.4 (Branching Bisimilarity). A symmetric relation R is a branching bisimulation if (P, Q)∈ Rimplies that wheneverP −→α P0 then there existsQ0 such thatQ=τ⇒Q0 −→α Q00and (P, Q0)∈ Rand (P0, Q00)∈ R.
P and Q are branching bisimilar (denoted P - Q) if there exists a branching bisimulationRsuch that (P, Q)∈ R.
The relationship of branching bisimilar processes is illustrated in Figure 2.6. The difference between this and weak bisimilarity, is the additional requirement thatP - Q0.
2.3.3
Quantum Process Calculus
In addition toCommunicating Quantum Processes (CQP)which is used in this thesis, there are two other process calculi that have been designed to model quantum sys- tems; Quantum Process Algebra (QPAlg)[Lalire 2006; Lalire and Jorrand 2004] and
P P0 Q Q0 Q00 α τ∗ α
Figure 2.6. Branching bisimilarity.
significant differences between the languages and semantics, they each have common features not found in classical process calculi:
• Quantum state. The quantum state is considered as a global resource in order to represent entanglement.
• Quantum communication. It is possible for processes to send and receive both quantum and classical information.
• Quantum operators. Further primitives are included to model quantum op- erations such as unitary operators and measurements.
The transitions of quantum processes are dependent on the quantum state, and for this reason, the transition relations are defined usingconfigurations. For example, a CQP configuration (σ;ω;P) consists of a quantum stateσand qubit listωalongside a process P. A transition takes the form (σ;ω;P)−→α (σ0;ω0;P0). These transition relations must also deal with the probabilistic outcomes arising from quantum mea- surements; in CQP, there are probabilistic transitions which select one configuration from a probabilistic distribution:
p1•(σ1;ω1;P1)· · · pn•(σn;ωn;Pn) pi
(σi;ωi;Pi).
A similar method is used in QPAlg, however in qCCS, probabilistic distributions are used throughout. The full syntax and semantics of CQP will be detailed in the following chapters. In particular, the semantics of measurement will be discussed in Chapters 3 and 4.
3
Behavioural Equivalence for
Communicating Quantum
Processes
This chapter presents a first attempt at using CQP for the verification of quantum protocols. We investigate behavioural equivalence for quantum processes, specificallyprobabilistic branching bisimilarity, in order to determine whether two systems act in the same way. Behavioural equivalence requires the ability to model the interaction of a process with the environment because it is these interactions that constitute the observational properties of the process. The existing reduction semantics of CQP describes only internal interactions, hence a core part of this chapter is dedicated to the definition of the operational semantics in terms of a labelled transition system.
There are a number of challenges involved in designing the labelled transition system when it comes to the representation of quantum information due to its non- local nature, and the related work by Feng et al. [2006]; Lalire [2005, 2006]; Ying et al. [2007, 2009] proves extremely valuable in this respect. It is important to note that the labelled transition system is not intended as a replacement for the reduction semantics, but provides a complementary semantics to use when modelling external interactions is required. We focus on the quantum teleportation protocol because it has a very simple high-level specification, yet the protocol features many aspects of the language, including measurement, unitary transformations and communication. With a specific protocol in mind, it is possible to critically consider the properties that an equivalence must possess, and the range of features used in teleportation significantly adds to the understanding we gain.
T ::= Int | Unit | Qbit | b[ ˜T] | Op(1) | Op(2) | · · · v ::= 0 | 1 | · · · | unit | H | · · · e ::= v | x | measuree˜ | e˜∗=ee | e+e P ::= 0 | (P kP) | P+P | e?[˜x: ˜T].P | e![˜e].P | {e}.P | (qbit x)P | (νx:b[T])P Figure 3.1. Syntax of CQP.
advantages to consider equivalence with CQP, including the use of the type system for the formal analysis of congruence properties. The implementation of the type system in CQP not only guarantees that qubits are only used by a single process, but also provides a clear and structured way to prove other results.
As we shall find, the equivalence we define in this chapter is not a congruence, however for a small class of quantum protocols this equivalence is preserved by all process constructs. This class of protocols includes quantum teleporatation, thereby allowing us to show the correctness of teleportation as a component in a larger system. Arguably the most important contribution from this chapter is the deeper understand- ing we gain about the observable aspects of quantum measurement. It is through the application to practical quantum processes that we are able to advance to the results in Chapter 4.