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COST Action IC1405
Reversible computation
extending horizons of computing
State of the art report
Working Group 2
Software and Systems
Editors:
Claudio Antares Mezzina and Rudolf Schlatte
Contributors:
Manuel Alcino Cunha, Vasileios Koutavas,
Ivan Lanese, Claudio Antares Mezzina,
Jarosław Adam Miszczak, Rudolf Schlatte,
Ulrik Pagh Schultz, Harun Siljak,
Michael Kirkedal Thomsen, German Vidal
March 17, 2017
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Contents
1 Notions of Reversibility
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2 Fully reversible languages:
2.1 Sequential Imperative languages . . . . . . .
2.2 Declarative languages . . . . . . . . . . . . .
2.3 Compiler technology, program transformation
2.4 Concurrent language . . . . . . . . . . . . . .
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3 Dependable System Abstractions
3.1 Language support for transactions: . . .
3.1.1 Database Transactions . . . . . .
3.1.2 Software Transactional Memory .
3.1.3 Non-Isolated Transactions . . . .
3.2 Checkpoint and Rollback . . . . . . . .
3.3 Library support for limited reversibility
3.4 Debugging & Program Slicing . . . . . .
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4 Execution replay
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5 Quantum Programming
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5.1 Imperative paradigm . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Functional paradigm . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Support in general purpose languages . . . . . . . . . . . . . . . 23
6 Bidirectional Transformation
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7 Robotics
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7.1 Modular self-reconfigurable robots . . . . . . . . . . . . . . . . . 27
7.2 Industrial robots . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8 Control Theory
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2
Preface
The intent of this report is to enlist all the different guises of reversibility that
can be found in software and systems. The work was initiated by Working Group
2 of the COST Action IC 1405 on reversible computations, as a continuation
of the work done by the Working Group 1 on foundations of reversibility. The
contents have been provided by the leading experts of the different research
fields that are covered: see the Contents or each section for more details.
More on the project:
• http://www.cost.eu/COST_Actions/ict/IC1405
• http://www.revcomp.eu/
3
1
Notions of Reversibility
Thinking about reversibility intuitively leads to think also about undoing. As
hinted by Bennett [12], any forward computation (or execution) can be transformed into a reversible one by just keeping an history of all the information
overwritten and hence lost (for example a variable update) by the forward computation, and then use this information to reverse (or undo) the forward computation. The ability for a system to get back to an exact past state by deleting
all its effects is what we call full reversibility. As we will see, the definition of
full reversibility varies when moving from a sequential setting to the concurrent
(possibly distributed) one.
In a distributed setting achieving full reversibility may be cumbersome if
not impossible due to the fact that some actions are by definition irreversible
(e.g. printing out a file). Several techniques used to build dependable systems
such as transactions [51], system-recovery schemes [33] and checkpoint-rollback
protocols [75], rely on some forms of undo or rollback facility.
Reversible computation is also at the core of newly-proposed programming
paradigms for developing control systems and robots with a high level of autonomy and adaptation.
Reversibility is also related to bidirectionality, which implies a rich literature
(see [26]). For the purpose and scope of this document we will just focus on
two broad classes of work: reversible computing, where the main focuses are to
provide a reversible computing model and to see what is the expressive power
or the computational cost of reversibility; and dependable systems abstractions
where dependable systems are obtained by means of some forms of reversibility.
Reversibility is also related with quantum computing, since quantum programs are always logically reversible.
The rest of the report is structured as follows: Section 2 describes fully reversible programming language both in the sequential and in the concurrent
setting. Section 3 deals with techniques used to program dependable systems
(e.g transactions, checkpoint and rollback techniques) and to debug them. Section 4 treats with replay techniques, which can be used for several aims such
as program debugging, security and simulation. Section 5 is about quantum
programming, while Section 6 is about bidirectional transformation. Section 7
is dedicated to the robotics domain, while Section 8 to control theory.
2
2.1
Fully reversible languages:
Sequential Imperative languages
Reversible programming languages can be dated back to 1986. At this time,
Lutz, after a brief meeting with Landauer, sent him a letter about some work
he did with Derby, about four years earlier, on a reversible imperative language
called Janus [87]. Their work arose from an interest to investigate if it was
possible to implement such a language and before 1986 Lutz and Derby did not
know about Landauer’s principle. The language was ‘rediscovered’ after the
turn of the century and has since then been formalized and further developed
at DIKU [144, 146, 147]. Other simple reversible imperative languages have
been developed, e.g. Frank developed R [39] to generate instruction code for
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the Pendulum processor and Matos [89] made a language for linear programs.
Listing 1: Janus code describing Fibonacci
procedure fib(int x1,int x2,int n)
if n=0
then x1 += 1
x2 += 1
else n -= 1
call fib(x1,x2,n)
x1 += x2
x1 <=> x2
fi x1=x2
procedure fib_fwd(int x1,int x2,int n)
n += 4
call fib(x1,x2,n) // forward execution
procedure fib_bwd(int x1,int x2,int n)
x1 += 5
x2 += 8
uncall fib(x1,x2,n) // backward execution
Listing 1, taken from [143], shows a Janus procedure for computing Fibonacci pairs. Given an integer n, the procedure fib computes the (n + 1)-th
and (n + 2)-th Fibonacci number. For example, the Fibonacci pair for n = 4 is
(5, 8). Returning a pair of Fibonacci numbers makes the otherwise non-injective
Fibonacci function injective. Variables n, x1, x2 are initially set to zero. Parameter passing is pass-by-reference. One key point of Janus is the use of reversible
updates + = and − = and conditional with with entry and exit guards. In the
above code the operator <=> is used to swap the value of two variables without
resorting to a third one.
A general framework for adding undo capability to a sequential programming language is presented in [82]. The paper also present a rich survey about
the undo operation and all its manifestations in different fields: such as text
editors, programming languages, function inversion, backtracking and physics.
The paper identifies two interpretations of the undo operation and motivates
them by examples. Such interpretations are: undob and undof . Let us show the
basic ideas behind them. Consider that we are editing an empty file by issuing
the following four command lines: insert w, insert x, insert y and finally insert
z. Then, the history of the effects on the file is the following sequence:
hΩ, w, wx, wxy, wxyzi
(1)
where Ω represents the empty file, and wxyz represents the state of the file
containing four lines and four characters. Now, let us introduce a time parameter
t such that in each unit of time one edit command is executed. So, if we apply
the function undob (1) to 1 we obtain the following history:
hΩ, w, wx, wxyi
(2)
if we continue by applying undob (3) to 2 we will obtain Ω. That is undob (t)
destroys the last t actions in the file history. On the other hand, by allowing
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the undo operation to move forward in the history a second interpretation is
possible. For example, if we apply undof (1) to 1 we obtain the following history:
hΩ, w, wx, wxy, wxyz, wxyi
(3)
and by applying undof (3) to 3 we obtain the following history:
hΩ, w, wx, wxy, wxyz, wxy, wxi
(4)
So the undof (t) function moves forward by just copying the state of t states
before. The main differences between the two operations are: (i) undob destroys
history while undof just moves forward preserving all the previous states; (ii)
undob moves the computation history to a point we already have seen before
while undof always creates a new history which never existed in the past. To
bring these intuitions into programming languages, a notion of undo-list is introduced. An undo-list is composed by triplets of the form ha, v, ti where a is
a variable name, v its value and t the instant in which the value has been assigned to the variable. Hence, the undo-list gives a means for restoring previous
values to variables. A computational history is then made of states, each one
represented by an undo-list. Two primitives mimicking the semantics of undob
and undof are given along with two primitives for undo-list manipulation: one
able to increase the instant t of current state variables and the second one able
to copy the value of an undo-list into another.
2.2
Declarative languages
Though the first reversible programming language was imperative, reversible
functional languages have lately received the most interest. This development
started in 1996 when Huelsbergen presented SECD-H [64], an evaluator for the
lambda calculus that extended Landin’s SECD evaluator [77] with a history
tape. (Kluge [71] similarly extended a machine that can reduce a program
term to normal-form and back again.) This was followed by Mu et al. who,
with applications in bidirectional computing in mind, presented a reversible
relational language [101]. This research has been further developed by Matsuda et al. [90, 92], where a reversibilization technique (including two stages:
injectivization and inversion) for a simple functional programming language
is presented. The authors mainly follow a Landauer’s embedding, although a
number of analysis are introduced in order to reduce the added information
as much as possible. More recently, work towards general purpose functional
programming languages was presented independently by Yokoyama, Axelsen,
Glück [145] (later extended [133]) and James, Sabry [67]. Another recent approach has been introduced by Nishida et al. [104], where reversibility in the
context of term rewriting is considered. Term rewriting captures the essence
of (first-order) functional computations. In [104], the authors first introduce
a reversible (but conservative) extension of term rewriting (a Landauer’s embedding). Then, a transformation for rewrite systems is also introduced, so
that the transformed systems behave with standard rewriting as the original
ones with the reversible extension (i.e., reversibility is compiled into the rewrite
rules, which opens the door to several applications). Other related approaches
include Abramsky’s [4] approach to reversible computation with pattern matching automata, which could also be represented in terms of standard notions of
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term rewriting. His approach requires a condition called biorthogonality (which,
in particular, implies injectivity), so that the considered automata are reversible
without adding any additional information. Abramsky’s work can also be seen
as a rather fundamental delineation of the boundary between reversible and irreversible computation in logical terms. Finally, let us mention other works on
reversibility for closely related languages, e.g., a probabilistic guarded command
language [151], a low level virtual machine [130], or combinatory logic [30], to
name a few.
2.3
Compiler technology, program transformation
On a related topic, transformation of reversible languages have also received
some interest lately (mainly at DIKU). Though the first compiler between two
reversible language was made by Frank [39] (between his language R and a
reversible instruction set called PISA), it was Axelsen who devised techniques for
a compiler that could perform clean translation [9]. His translation was between
Janus and PISA and was clean in the sense that the compiled program did not
have more than a constant memory overhead over the original Janus program.
It is likely that the PISA program will use more temporary memory/registers.
Lately we have seen more elaborated schemes representation of heap structure
and garbage collection [10, 99, 100].
Sometimes it is useful to have portions of code of a non-reversible program to
be reversible. This is the case of parallel discrete event simulation (PDES) [40],
where speculative simulation is used to boost the speed of the simulator. Naturally, if the speculation was made on a wrong assumption then all the computations caused by it have to be reverted. Works [117, 118] focus on automatic
generation of C++ reverse code for parallel discrete simulations.
2.4
Concurrent language
The notions of reversibility used in the sequential setting are not suitable for
concurrent and/or distributed systems. Indeed, in concurrent/distributed settings different actions can overlap their execution, hence the notion of “last
action” is not well defined. In some distributed systems there may also not
be a unique notion of time. This led to the proposal of causal-consistent reversibility [28, 78], which states that any action can be undone provided that its
consequences, if any, are undone beforehand. Notably, this avoids any reference
to the notion of time, while using causality to relate actions.
Up to now the only proposal of fully reversible concurrent language we are
aware of is the causal-consistent reversible extension of µOz [85]. The µOz
language is a kernel language of Oz [138]. µOz is a higher-order language
with thread-based concurrency and asynchronous communication via ports (i.e.,
channels). The semantics of µOz is defined using a rather standard stack-based
abstract machine. In order to make it reversible the abstract machine is extended with history information associated to each thread and each queue. This
allows one to reverse any µOz statement, but does not provide control on when
to reverse those statements. A form of control is provided in [44, 25] which
provides a causal-consistent reversible debugger for µOz.
Listing 2 describes Fibonacci function in µOz. Differently from Listing 1, in
µOz there is no need to resort to Fibonacci pairs to make the function bijective.
7
The procedure takes in input a number n and a channel res and sends on res
the n-th Fibonacci number. Executing the code in the µOz interpreter [25]
allows one to move forward and backward along the execution. While in Janus
a program can be executed both forward and backward, in µOz one can execute
a program backward only after it has been executed forward. Furthermore, in
Janus there is no need to use extra memory to store history information since
the language is naturally forward and backward deterministic, while in µOz one
has to keep extra memory to remember past actions.
Listing 2: µOz code describing Fibonacci
let fib = proc {x res}
if (x<=1) then
let one = 1 in {send res one} end
else
let z1 = port in
let z2 = port in
let w = x-1 in
let u = w-1 in
{fib w z1}; //call fib n-1
{fib u z2}; //call fib n-2
let fa = {receive z1} in
let fb = {receive z2} in
let n = fa + fb in {send res n} end
end end
end end
end end
end end
in
let ch = port in
let num = 5 in
{fib num ch}
end end
end
3
Dependable System Abstractions
A failure of a system occurs when its behavior differs from the one that has been
specified. The part of the system state leading to the failure is called an error.
Here an error is always caused by a fault, which in many cases are physical (e.g.,
hardware faults) and inevitable. Hence faults are the cause of errors that lead
to failures [135].
Fault tolerance eliminates system failures in the presence of faults, providing
high system dependability and the required level of service [8]. The two main
techniques for achieving fault tolerance are fault masking and fault treatment.
The former aims at eliminating errors before they occur by reducing the faults
leading to an error, employing for example data replication and component
redundancy.
Fault treatment, on the other hand, seeks to handle certain system errors
after they occur, stopping them from causing a failure. With this technique after
8
an error occurs the system is brought to a consistent state. If this is a previously
saved state of the system, as it happens in transactional and checkpoint recovery,
the fault tolerance mechanism is classified as backward error recovery. If the
recovery involves correcting the error without resorting to a previous state,
as in exception handling (e.g. [65]), then the mechanism is called forward error
recovery. The latter can be made more efficient in resources, whereas the former
can be more easily automated requiring little programmer intervention.
In this section we examine mechanisms for fault treatment, and in particular those employing backward recovery, as they are an instance of reversible
computation in which errors trigger reversing actions. In this framework, the
building blocks of fault treatment mechanisms can be seen as defeasible partial
agreements in a distributed setting. Reversibility provides us with a high-level
setting for examining and ultimately specifying, and verifying fault tolerant
systems with backward recovery.
3.1
Language support for transactions:
According to [49], a system implementing transactions
provides operations each of which manipulates one or more entities. The execution of an operation on an entity is called an action.
[. . . ] Associated with a database is a predicate on entities called the
consistency constraint. A database state satisfying the consistency
constraint is said to be consistent.
Transactions are used to implement the ACID properties [50] of traditional
database systems.1 Pertaining to the topic of this report, offering the consistency and isolation properties of transaction semantics requires an implementation of (limited) reversability.
Conceptually, transactions wrap a sequence of operations inside an atomic
block such that either all or none of the actions occur. A transaction is completed by either committing or aborting. Aborting reverses the effects of all
operations within the transaction, leaving the state coherent.
Systems offering transaction semantics differ in the following aspects:
• The entitites that are being protected by transactions. In software transactional memory, the entities are a number of shared variables; in database
transactions it is the state of the database.
• The operations on these entities that can be committed or aborted, usually
read/write but also create/delete.
• The semantics of aborting a transaction. After a transaction is aborted,
there might be observable changes in the protected entities since other,
concurrent transactions might have committed successfully at the same
time.
• Characteristics of the transactions themselves, e.g., whether transactions
can be long-lived or abort after some fixed time, or whether transactions
can be nested.
1 Atomicity, Consistency, Isolation, Durability – note that the Isolation property is not
explicitly listed in the introduction of [50], although very much discussed in the body.
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• Implementation aspects like optimistic versus pessimistic synchronization
and commutativity of operations; these mostly influence performance of
the system.
In the classical transactional model [14, 110] transactions are seen as sequences of read and write operations that map consistent database states to
consistent states when executed in isolation. A concurrent execution of a set
of transactions is represented as an interleaved sequence of read and write operations, and it is said to be serializable if it is equivalent to a serial (nonconcurrent) execution. A transaction is a sequence of actions that have to be
executed atomically: either it successfully completes (commits) and all its effect
are visible to the other transactions; otherwise it fails and its effects are not
visible.
Transactions, originally introduced in the field of DBMS (Data-Base Management Systems) models, provide good concurrency abstraction models in programming languages, since they ensure nice properties, such as atomicity and
isolation, di cult to obtain if manually programmed. Indeed, if a developer
were to ensure such properties, he would have to design the program relying on
low level concurrent programming primitives (e.g. critical sections, semaphores,
monitors), which is typically a difficult task, and even more, hard to debug. In
this section we will review works dealing with transactional programming languages guaranteeing atomicity, consistency, isolation and durability (or ACID)
properties. Works modelling non-classical notions of transactions (such as longrunning transactions, open-nested transactions) will be reviewed in Sections
3.1.2 and 3.1.3.
3.1.1
Database Transactions
Transactions for database systems, and the associated rollback reversibility techniques, are usually implemented as SQL statements embedded into a host programming language via libraries such as JDBC. The state of the database is protected by a transaction; the SQL SELECT, UPDATE, INSERT and DELETE
statements constitute the operations. A transaction can be aborted either by
the client or by the database system. After an aborted transaction, the database
state can be different than before the transaction was started, i.e., the reversion
of the operations within the aborted transaction leads back to a consistent state,
but not necessarily to a state identical to the starting state.
3.1.2
Software Transactional Memory
Software Transactional Memory (STM) is “a concurrency control mechanism for
executing accesses to memory shared by multiple processes. A transaction, in
this context, is a section of code that executes a series of reads and writes to the
shared memory as one atomic indivisible unit.” [125] Implementations of STM
typically have lower overhead and are less error-prone than protecting shared
memory via explicit locking. STM is used to synchronize threads running within
the same process, or processes running on the same machine.
STM implementations in various languages offer different functionality. For
example, in the Haskell implementation [88, Chapter 10], [58] the type system
guarantees that its transactional variables are only accessed inside a transaction,
while the Clojure implementation [56, Chapter 5] offers commuting operations
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and validation functions, reducing the likelihood of rollbacks and offering additional consistency checks beyond the type system of the language, but does
not statically guarantee that the transaction is side effect-free. Early research
in STM was carried out largely in functional languages, other examples include
SML/NJ: [54]; AtomCaml: [114]; and Scheme [69]. Implementations in imperative languages like Java [73] and C++ suffer from poor interoperability with
code not aware of STM, relying on load-time instrumentation and similar techniques.
Again, STM implements a specific form of reversibility that rolls back a
sequence of operations over shared state back to a consistent state.
3.1.3
Non-Isolated Transactions
Isolation in traditional ACID transactions is crucial for programming mutual
exclusion. If two transactions access the same location of the system state,
and one of them writes to this location, then the system aborts at least one
of the transactions. However, a number of works [37, 86, 29, 32, 83, 127] have
proposed dropping the Isolation principle to obtain a useful abstraction for
concurrent programming. Although non-isolated transactions cannot encode
mutual exclusion, they can be used to maintain a consistent distributed state
[37], multi-party synchronisation without the use of a coordinator [32], and in
general they provide a method to encode distributed consensus problems [128].
In [54] primitives to build composable transaction abstractions in ML are
given. Transactions are factored into four separable features: persistence, undoability, locking and threads and then each composition of these properties gives
rise to different transactional models. Following the idea of adding transactions
to Objective Caml, the language AtomCaml [114] has been proposed. The language is endowed with the new function atom(f ) of type (unit → 0 a) → 0 a
whose purpose is to execute atomically the function f. The basic idea behind
this function (and primitive) is that it tries to execute sequentially (hence not
interleaved) the entire function block, if during the execution the thread executing it has been pre-empted by the scheduler then the function rolls-back
and re-executes again. This implies that the function is executing in a monoprocessor setting, where true concurrency does not exist. Moreover there are
a few limitations about the side effects that the atomic block can have: input
operations are not allowed (since there is no way to reverse them); output operations are always buffered and flushed only if the block successfully completes,
and exceptions that escape from the atomic block boundaries force the atomic
block to complete. Changes made by an atomic block to mutable variables are
logged, that is variables are like stacks of values, and in case of failure these
changes are reverted.
Different works have approached the design of transactional languages from a
semantic point of view, formally proving some properties. Transactional Featherweight Java [66] is an object calculus with support for nested and multithreaded transactions. As usual a transaction is delimited by a special block,
in this case the onacid statement. Each time an onacid is executed a new
transaction (identified by a label) is created, and all the threads executing in it
are bound with the transaction label. Each thread is executed into a transactional environments that keeps track of all the read and write operations that
the thread performs on objects. Then by varying the semantics of operations on
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transactional environment two kinds of semantics are given: versioning semantics and strict two phase locking semantics. The first one mimics the STM (we
will discuss about STM later on) logging mechanism, since when a thread enters a new transaction, an empty log corresponding to the transaction is created.
This log keeps track of all the objects that are modified within a transaction, for
a write operation for example the old value is written in the log. A transaction
successfully commits if its log is consistent with the father’s one, otherwise it
fails. A log is consistent with the father ones, if all the values of the objects
read by the child thread have remained unchanged until the committing time.
In the strict two phase locking semantics there is no more need of a log mechanism, since before modifying an object, a transaction has to require a lock on it.
All the collected locks will be released when the transaction commits. Nested
transactions inherit father locks. By using the lock mechanism, there is no way
for a transaction to fail.
3.2
Checkpoint and Rollback
In distributed systems, checkpointing and roll-back (also known as checkpointrecovery) is a technique of backward recovery (see [8]) for creating fault tolerant
systems. The key concepts of this technique are: (i) periodic saves of system
global state; (ii) in event of a fault, the state is restored via a rollback. This
particular technique gives to a system (or to an application) the ability to save
its state and tolerate faults by simply restoring an earlier state. In fact, when
a checkpoint is executed, a snapshot of the entire system is taken and normally
it is saved into some non-volatile medium. If a fault is detected, the recovery mechanism restores the system to the last checkpointed state. There is an
abundant literature (see [8, 33] for a quick review) on protocols and techniques
on how to build a global (distributed) checkpoint, on what kind of information should represent the global state, and so on. We will address this topic
by a programming language point of view, and then not considering libraries,
middleware and operating system services.
In [19] a reversible extension of the Scala language supporting channel-based
synchronisation is presented. Two are the primitives added to Scala in order to
enact reversibility (and rollback): stable e and backtrack e to menage backtracking over speculative executions. The first one delimits the scope of the
backtrack events within e, while the second brings back the process to the dynamically closest nested block with the value of e. Backtracking has also the
effect of deleting all the communications the process has done. A backtrack
action might force neighbouring processes to also backtrack, possibly resulting in a cascade of backtracking (domino effect) for a poorly written program.
The backtrack mechanism is shown to be implementable via a point to point
algorithm avoiding thus a centralised controller.
Bringing checkpoint-recovery technique in the actor model has been tackled
by Transactor [37]: a fault tolerant programming model for composing looselycoupled distributed components. It extends the actor model with constructs
which distributed processes can use to to build globally consistent checkpoints.
Basically a transactor can decide to commit its current state to a stable one.
When a transactor becomes stable, further communications cannot change its
state. It can be seen as a promise, to all the other transactors who communicated with it, that its state will not change. When an unstable transactor
12
decides to roll-back, or it is asked to do so, it will cause the rollback of all the
transactor whose state depended on the state of the unstable transactor. Interestingly the semantics also consider message loss. The language is proved to
be sound, that is a trace containing node failures is equivalent to a normal one
not containing failures, but possibly message losses. Moreover, checkpointing is
possible just under certain conditions, and not in general cases. Indeed, not all
the transactor programs can reach global checkpoints. A trivial program with
a transactor that sends messages introducing dependencies, but never stabilizes
or tries to checkpoint, will eliminate the ability of its dependent parties to reach
checkpoints. Hence the authors introduces the Universal Checkpoint Protocol
(UPC) that assumes a set of preconditions that will entail global checkpointing
for a set T of transactors.
Transient faults are unusual conditions that can be remedied by just reexecuting the code which raised it. These faults are usually generated by temporally unavailable resources. For example, if a server is rebooting due to an
internal error, then all the client requests issued during the rebooting time
should be re-executed. In [150] a concurrent ML language, called stabilizers,
for transient concurrent fault recovery in concurrent program is presented. The
language introduces three new primitives: stable, stabilize and cut, able to
deal with program global checkpoints. Primitive stable allows a thread to create
a new stable section, that is a new global checkpoint. Primitive stabilize, issued
by a single thread, allows the entire program to roll-back to the previous global
checkpoint and finally primitive cut discards the current global checkpoint. This
ensures that subsequent calls to stabilize will never cause the program to get
back to a state that existed logically before the cut. Even if the semantics is
proved to be safe, that is stabilization actions can never manufacture new states,
the semantics cannot avoid the domino effect, that is a stabilize operation may
unduly revert the program to a state beyond the target checkpoint.
3.3
Library support for limited reversibility
The Command Pattern [41] is a design pattern that describes how to execute,
undo and redo operations that are initiated by a user. To support the Undo
functionality, which reverses the effect of an operation, a Command objects following that pattern carries information on whether it can be reversed or not
(e.g., usually saving a document is not reversible), and instructions on how to
undo its own execution. Systems built following this pattern can undo and redo
commands issued. Many higher-level GUI toolkits offer predefined interfaces or
base classes to implement the Command pattern and get such limited reversibility (Undo/Redo) with small implementation effort. Reversible [113] is a library
for python whose aim is to provide a simple abstraction for actions that can be
reversed or rolled back and provides methods to construct, chain, and consume
them in a readable way.
Reversing the state of a system might mean going back to an earlier version
of its state. The cost of storing previous versions of large data structures can
be prohibitive. Persistent (immutable) data structures [105] offer more efficient
storing of multiple versions of a data structure, sharing structure where possible. This makes it possible to simply store previous state information without
undue memory pressure. While persistent data types can be implemented in
many languages, their usefulness is limited if existing third-party or library code
13
expects a language’s existing, non-persistent data structures. The Clojure language [57] uses persistent data structures throughout and provides support for
calling Java libraries; many purely functional languages also have persistency
built-in.
3.4
Debugging & Program Slicing
Reversible debugging has been known for the last 40 years [53, 148] and gets all
its interest and motivation from assisting the programmer in the search of possible bugs by exploring the computation both forward and backward. Retracing
back the steps is very useful when investigating a misbehavior. In a sequential
setting it is also very natural: steps are simply undone in the reverse order of
execution. Reversibility for debugging of sequential programs has been quite
extensively explored [21, 35, 72, 84], and some reversible debugging features are
available also in mainstream debuggers. For instance, GDB supports reversible
debugging since version 7.0 (released in 2009). A main limitation of sequential
reversible debuggers is the huge overhead required to store history information,
both in terms of time and in terms of space, which makes difficult the use of
reversible debuggers for large programs. State of the art techniques based on
incremental storage of information and complex fine tuning allow a consistent
reduction of the overhead making the technique applicable at the industrial
level [136].
The interplay between reversibility and concurrency makes things more complex: concurrent reversible debugging is a less explored world. All the approaches to concurrent reversible debugging we are aware of fall in the two
categories below
Non-deterministic replay debugging [7, 139, 136] : in order to go back
to a previous step, the execution is replayed non-deterministically from
the start (or from a previous checkpoint) until that step.
Deterministic replay/reverse-execute debugging [70, 24] : a log is kept
while executing, and when going back thread activities are either undone
in the exact reverse order they were executed, or the execution is replayed
from a previous checkpoint following the particular interleaving in the log.
Both approaches present drawbacks. In the first case, actions could get scheduled in a different order at every replay, and the error may not get reproduced.
So in this case, error proper to concurrency such as race conditions could not
get caught. Even if it does, one may not get any insight on the causes of the error. Following the second approach, if the error was due to one among a million
of independent parallel threads, and that one was the first one to execute, one
needs to undo all the program execution before finding the bug. Even more,
one does not understand which threads are related to a given misbehavior, since
there is no information on the relations among them. In [44] a novel approach
to concurrent reversible debugging exploiting causality information is presented,
were the notion of causally consistent reversibility is brought into a reversible
debugger.
Causality in the context of non-reversible concurrent debugging has been
addressed in different works [81, 149, 142] , which mainly rely on the Lamport’s
happens-before causality relation [76]. In all these works causality is used to
14
support determinism in replaying techniques and to define efficient dynamic slicing. In contrast, the use causality as a support for rollback primitives allowing
the programmer to find the causes of a misbehavior has been just explored are
Causeway [20] and CaReDeb[25]. Causeway is not a full-fledged debugger, but
just a post-mortem traces analyzer. It exploits a causality notion, based on
the Lamport’s happens-before, more liberal than causal consistency exploited
by CaReDeb. CaReDeb is an interpreter and debugger for µOz [85] whose key
idea is to use rollback primitives to enact reversible debugging [44]. Rollback is
done in a causally consistent fashion, that is in order to revert an action all its
causes have to be undone first.
Simple reversibility features exist in programming languages for commercial,
industrial robots from major manufacturers such as ABB, Fanuc and KUKA,
and have been shown to have significant practical value [79]. In these languages
reversibility is used when programming and debugging the robots [102]. Manufacturers however use various approaches and implementations. KUKA offers
different options, one of which is backtracking: execution is recorded during
forward motion and can be backtracked afterwards [74]. Fanuc uses a simple
implementation of program inversion based on backward interpretation [34].
This allows the user to step through programs in reverse order, but only works
on the move commands and cannot reverse control flow structures. RAPID adds
to this approach, and allows users to specify alternative reverse instructions for
backward execution of a subroutine [3]. Overall these implementations are useful for interactively reversing the movements of the robot. They are however
not suited for performing conceptually reverse executions of entire tasks.
4
Execution replay
Execution replay is a technique which finds application in several areas such as
debugging, fault-tolerant systems, security and simulation. It consists of two
phases: first a log of the execution is made (record phase), then the log is used
to control the re-execution of the program (replay phase). Checkpoints can be
used to optimize replay time (but increase the space requirements): execution
is not replayed since the very beginning, but just since the last checkpoint.
Main requirements on the log information are that it should be small enough,
so that logs of long computations fit in a reasonable storage, and easy to compute, since log production should not affect too much the performance of the
running application. Performance of replay is less critical since replay is actually performed only in case of misbehaviour, frequently only for a fragment of
the computation, and on a dedicate machine. Replay is also used for reversible
debugging [24, 115]: in order to step back in the program under debugging, its
execution is replayed but for its last action.
In a sequential setting a log only has to trace nondeterminstic events (e.g.,
user input, interrupts) thus ensuring that the replayed computation indeed coincides with the original one. Logging becomes more complex in a concurrent
scenario (and even more in a distributed one) since scheduling information needs
to be stored. There are several approaches to replay concurrent applications.
Some of them [36] aim at perfectly replaying the order of execution of actions,
by completely logging the scheduling. Perfect replay allows one to reproduce
the logged execution. The information stored during the logging phase depends
15
also on the kind of inter-process communication mechanism used by the system:
shared memory [36, 81, 22] or message passing [43, 103].
5
Quantum Programming
Quantum algorithms and communication protocols are described using a language of quantum circuits, describing reversible logical circuits in the language
of unitary operations. While this method is convenient in the case of simple
algorithms, it is very hard to operate on compound or abstract data types like
arrays or integers using this notation. This lack of abstraction motivated the
development in the field of quantum programming languages.
One should note that it is now always possible to identify a direct reference
to the reversibility in quantum programming languages. They provide some
syntax elements – like reverse invocation using exlamation mark (!) – typical
for the reversible languages. However, in most cases this is the only explicit
element related to reversibility. Of course quantum programming languages are
reversible because they use unitary operations as basic building blocks. Thus it
is always possible to execute a block which is a reverse of a given block – for a
given unitary operator U this is simply its conjugate transpose, U † = (U T )∗ . As
such quantum programs are always logicaly reversible and quantum programming languages must ensure this. However this is usually stated implicitly by
allowing unitary operations only.
5.1
Imperative paradigm
The first familly of quantum programming languages consists of languages based
on the imperative paradigm. Those languages provide syntax known from programming languages like Pascal and C, and extend it with the elemenets necessary to operate on quantum memeory.
QCL QCL (Quantum Computation Language) [107, 108, 109] is one of the
most advanced quantum programming language with working interpreter. Its
syntax resembles the syntax of C programming language and classical data types
are similar to data types in C or Pascal.
The programmes written in QCL can be executed using the available interpreter [106]. The interpreter can be executed in a batch mode or in as an
interactive programme. The interpreter is built on a top of libqc simulation
library written in C++ and offers an excellent speed of execution of simulated
programmes. As the simulation of quantum computing requires a considerable
amount of computing resources, there were also some attempts to provide a
paralellized version of libqc library [45].
The basic built-in quantum data type in QCL is qureg (quantum register).
It can be interpreted as the array of qubits (quantum bits).
qureg x1[2]; // 2-qubit quantum
qureg x2[2]; // 2-qubit quantum
H(x1); // Hadamard operation on
H(x2[1]); // Hadamard operation
register x1
register x2
x1
on the second qubit of the x2
Listing 3: Basic operations on quantum registers and subregisters in QCL.
16
QCL standard library provides standard quantum operators used in quantum
algorithms, such as:
• Hadamard H and Not operations on many qubits,
• controlled not CNot with many target qubits and Swap gate,
• rotations: RotX, RotY and RotZ,
• phase Phase and controlled phase CPhase.
QCL supports user-defined operators and functions known from languages
like C or Pascal. Classical subroutines are defined using procedure keyword.
Also standard elements, known from C programming language, like looping (e.g.
for i=1 to n { ... }) and conditional structures (e.g. if x==0 { ... }),
can be used to control the execution of quantum and classical elements. In
addition to this, it provides two types of quantum subroutines, which for the
core of the reversible part of the lanbguage.
The first type is used for unitary operators. By using it one can define new
operations, which in turn can be used to manipulate quantum register. For
example operator diffuse, defined in Listing 4, defines inverse about the mean
operator used in Grover’s algorithm. Such syntax enabled the higher level of
abstraction and extend the library of functions available for a programmer.
operator diffuse(qureg
H(q);
//
Not(q);
//
CPhase(pi,q);
//
!Not(q);
//
!H(q);
//
}
q) {
Hadamard Transform
Invert q
Rotate if q=1111..
undo inversion
undo Hadamard Transform
Listing 4: The implementation of the inverse about the mean operation in QCL
[109]. Constant pi represents number π. Exclamation mark ! is used to indicate that the interpreter should use the inverse of a given operator. Operation
diffuse is used in the quantum search algorithm.
QCL introduces special syntax – exclamation mark ! – to indicate the inverse
of a given operator. Subprocedures defined using operator keyword can called
in the reveres version using !diffuse syntax.
QCL utilize different types of quantum memeory to control the operations on
quantum registers and to perform optimization of the quantum circuit generation. Quantum memory can be declared using quantum types qureg, quconst,
quvoid and quscratch. Type qureg is used as a base type for general quantum registers. Other types allow for the optimisation of a generated quantum
circuit. The summary of the types defined in QCL is presented in Table 1.
LanQ The second important example of quantum programming language
based on the imperative paradigm is LanQ [97]. LanQ was developed to address
the problems arising from the lack of elements supporting quantum communication. Additionally, LanQ is the first quantum programming language with
full operation semantics specified [98]. This allows for the formal reasoning
17
Table 1: Types of quantum registers used for memory management in QCL.
Type
Description
Usage
qureg
general quantum register
basic type
quvoid
register which has to be empty when target register
operator is called
quconst
register which must be invariant for quantum conditions
all operators used in quantum conditions
quscratch
register which has to be empty be- temporary registers
fore and after the operator is called
about the correctness of programmes written in LanQ and for the further development of the language. Semantics is also crucial for the optimization of the
programmes written in LanQ.
The programmes written in LanQ can be executed and tested using an
available interpreter [96]. The interpreter was developed as a part of a PhD
thesis [97], but its development stopped in 2007.
Its main feature is the support for creating multipartite quantum protocols.
LanQ, as well as cQPL presented in the next section, are built with quantum
communication in mind. Thus, in contrast to QCL, they provide the features
for facilitating the simulation of quantum communication.
The syntax of the LanQ programming language is very similar to the syntax
of C programming language. In particular it supports:
• Classical data types: int and void.
• Conditional statements of the form if ( cond ) { . . . } else { . . . }
• Looping with while keyword while ( cond ) { . . . }
• User defined functions, int fun( int i) { int res; . . . return res; }
LanQ is built around the concepts of process and interprocess communication, known for example from UNIX operating system. It provides the support
for controlling quantum communication between many parties. The implementation of teleportation protocol presented in Listing 6 provides an example of
LanQ features, which can be used to describe quantum communication.
Function main() in Listing 6 is responsible for controlling quantum computation. The execution of protocol is divided into the following steps:
1. Creation of the classical channel for communicating the results of measurement:
channel[int] c withends [c0,c1];.
2. Creation of Bell state used as a quantum channel for teleporting a quantum state
(psiEPR aliasfor [psi1, psi2]); this is accomplished by calling external function createEPR() creating an entangled state.
3. Instruction fork executes alice() function, which is used to implement
a sender; original process continues to run.
18
void alice(channelEnd[int] c0, qbit auxTeleportState) {
int i;
qbit phi;
// prepare state to be teleported
phi = computeSomething();
// Bell measurement
i = measure (BellBasis, phi, auxTeleportState);
send (c0, i);
}
void bob(channelEnd[int] c1, qbit stateToTeleportOn) {
int i;
i = recv(c1);
// execute one of the Pauli gates according to the protocol
if (i == 1) {
Sigma_z(stateToTeleportOn);
} else if (i == 2) {
Sigma_x(stateToTeleportOn);
} else if (i == 3) {
Sigma_x(stateToTeleportOn);
Sigma_z(stateToTeleportOn);
}
dump_q(stateToTeleportOn);
}
Listing 5: Modules used in the quantum teleportation programme implemented
in LanQ (see: Listing 6).
void main() {
channel[int] c withends [c0,c1];
qbit psi1, psi2;
psiEPR aliasfor [psi1, psi2];
psiEPR = createEPR();
c = new channel[int]();
fork alice(c0, psi1);
bob(c1, psi2);
}
Listing 6:
Teleportation protocol implemented in LanQ [97].
Functions
Sigma_x(), Sigma_y() and Sigma_z() are responsible for implementing Pauli
matrices. Function createEPR() (not defined in the listing) creates maximally
entangled state between parties — Alice and Bob. Quantum communication is
possible by using the state, which is stored in a global variable psiEPR. Function
computeSomething() (not defined in the listing) is responsible for preparing a
state to be teleported by Alice.
4. In the last step function bob() implementing a receiver is called.
Types in LanQ are used to control the separation between classical and quantum computation. In particular they are used to prohibit copying of quantum
19
registers. The language distinguishes two groups of variables [97, Chapter 5]:
• Duplicable or non-linear types for representing classical values, e.g. bit,
int, boolean. The value of a duplicable type can be exactly copied.
• Non-duplicable or linear types for controlling quantum memory and quantum resources, e.g. qbit, qtrit channels and channel ends (see example
in Listing 6). Types from this group do not allow for cloning, ie. it is
impossible to make a copy of such variable.
One should note that quantum types defined in LanQ are mainly used to
check validity of the programme before its run. However, such types do not help
to define abstract operations. As a result, even simple arithmetic operations
have to implemented using elementary quantum gates.
5.2
Functional paradigm
Second familly of quantum programming languages consists of languages based
on the functional paradigm. Most of the laguages from this familly are constructed with the focus on the formal properties of the
cQPL Classical elements of cQPL are very similar to classical elements implemented in imperative programming languages. The syntax resembles that of
classical programming languages based on C programming language.
In particular cQPL provides conditional structures using if ... then ... else
block and loops are introduced with while keyword.
Quantum memory can be accessed in cQPL using the variables of type qbit
or qint. Basic operations on quantum registers are presented in Listing 7. In
particular, the execution of quantum gates is performed by using *= operator.
new qbit q1 := 0;
new qbit q2 := 1;
// execute CNOT gate on both qubits
q1, q2 *= CNot;
// execute phase gate on the first qubit
q1 *= Phase 0.5;
Listing 7: State initialisation and basic gates in cQPL. Data type qbit represents a single qubit.
It should be pointed out that qint data type provides only a short-cut for
accessing the table of qubits.
Only a few elementary quantum gates are built into the language:
• Single qubit gates H, Phase and NOT implementing elementary one-qubits
gates.
• CNot operator implementing controlled negation,
• FT(n) operator for n-qubit quantum Fourier transform.
This set of operations allows to simulate an arbitrary quantum computation.
Besides, it is possible to define new gates by specifying their matrix elements
directly.
Measurement is performed using measure/then keywords and print command allows to display the value of a variable.
20
measure a then {
print "a is |0>";
} else {
print "a is |1>";
};
In similar manner like in QCL, it is also possible to inspect the value of a
state vector using dump command.
The main feature of cQPL is its ability to build and test quantum communication protocols easily. Communicating parties are described using modules. In
analogy to LanQ, cQPL introduces channels which can be used to send quantum
data.
Once again we stress out that the notion of channels used in cQPL and
LanQ is different from that used in quantum theory. In quantum mechanics
channels, sometimes referred to as operations, are used to describe allowed physical transformations, while in quantum programming they are used to describe
communication links.
Communicating parties are described by modules, introduced using module
keyword. Modules can exchange quantum data (states). This process is accomplished using send and receive keywords.
To compare cQPL and LanQ one can use the implementation of the teleportation protocol. The implementation of teleportation protocol in cQPL is
presented in Listing 8, while the implementation in LanQ is provided in Listing 6.
module Alice {
proc createEPR: a:qbit, b:qbit {
a *= H;
b,a *= CNot; /* b: Control, a: Target */
} in {
new qbit teleport := 0;
new qbit epr1 := 0;
new qbit epr2 := 0;
call createEPR(epr1, epr2);
send epr2 to Bob;
/* teleport: Control, epr1: Target
teleport, epr1 *= CNot;
new bit m1 :=
new bit m2 :=
m1 := measure
m2 := measure
*/
0;
0;
teleport;
epr1;
/* Transmit the classical measurement results to Bob */
send m1, m2 to Bob;
};
module Bob {
receive q:qbit from Alice;
receive m1:bit, m2:bit from Bob;
if (m1 = 1) then { q *= [[ 0,1,1,0 ]];
21
/* Apply sigma_x */ };
if (m2 = 1) then { q *= [[ 1,0,0,-1 ]];
/* Apply sigma_z */};
/* The state is now teleported */
dump q;
};
Listing 8: Teleportation protocol implemented in cQPL (from [93]). Two parties
– Alice and Bob – are described by modules. Modules in cQPL are introduced
using module keyword and can exchange quantum data using send/receive
structure.
One can note tha cQPL modules resemble to some extent the objects used
in object-oriented languages.
QML Another quantum programming language following a functional paradigm
is QML developed by Altenkirch and Grattage [5, 47]. The QML compiler was
described in [48] and can be downloaded from the project web page [46].
The name suggest that QML was designed as a quantum version of ML
language [95]. The language, however, is implemented in Haskell and follows
some syntactic conventions used in Haskell.
The QML compiler requires GHC (Glasgow Haskell Compiler) [2] in version
6 in order to run QML programmes. In order to run a program written in QML
one needs to load the definitions in qml.hs into the interactive environment
ghci and use one of the functions described in Table 2.
Table 2: Possible methods of evaluation of QML programmes [48].
Function
Evaluation method
runM
Unitary matrix representing a reversible part of the programme.
runI
Isometry providing a full description of the programme for terms
that produce no garbage.
runS
Superoperator initializing the heap
and tracing-out the garbage.
Programme structure A programme written in QML consists of a sequence of function definitions. Each definition is of the form
funName (var1,type1) ... (varN,typeN) |- funBody :: retType
For example, the classical not function (Cnot) is defined as
Cnot (q,qb) |- if q then qtrue
else qfalse :: qb
Using the same syntax the user can define constants, which in QML are
equivalent to functions. For example, to use a constant representing a superpo√
sition |0i+|1i
one can declare it as
2
-- one-qubit superposition
Qsup |- hF*qtrue + hF*qfalse :: qb
22
Here the term hF is defined as √12 . The * operation allows to associate the
probability amplitude with a term.
The state of compound systems can be represented using the () operation.
For example, the constant representing the EPR pair (ie. one of the Bell states)
is defined as
Epr |- hF * (qtrue,qtrue) + hF * (qfalse,qfalse) :: qb*qb;
One should note that in the above example the resulting type is described as
qb*qb and here * operator is used to describe a type of two-qubit state.
Subroutines As in any functional programming language, programmes
written in QML are composed of small functions. This makes the written code
more readable and easier to maintain.
Subroutines in QML can operate on an arbitrary number of arguments. A
subroutine is introduced by the following syntax
ProcName (arg) -|
code
code
One should note that procedure names have to start with the upper case
letter. Moreover, similarly as in Haskell, the indention is important and denotes
the continuation of the code block.
Cnot (b,qb) |- if b then qfalse else qtrue :: qb;
CNot (s,qb) (b,qb) |- if s then Cnot (b) else b :: qb;
-- classically-controlled quantum Not
CQnot (s,qb) (t,qb) |- if s then Qnot (t) else t :: qb;
The quantum CNot operation can be defined in terms of the above by using
a quantum conditional operation
QCNot (s,qb) (t,qb) |- ifo s then (qtrue,Qnot (t))
else (qfalse,t) :: qb * qb;
where Qnot is defined as
Qnot (b,qb) |- ifo b then qfalse else qtrue :: qb;
5.3
Support in general purpose languages
The last method for implementing high-level quantum programming concepts
is to extend standard programming languages using specialized library of functions.
Libquantum and Q Language Along with QCL several other imperative
quantum programming languages were proposed. Notably Q Language developed by Betteli [15, 16] and libquantum [141] have the ability to simulate noisy
environment. Thus, they can be used to study decoherence and analyze the
impact of imperfections in quantum systems on the accuracy of quantum algorithms.
23
Q Language is implemented as a class library for C++ programming language and libquantum is implemented as a C programming language library.
Q Language provides classes for basic quantum operations like QHadamard,
QFourier, QNot, QSwap, which are derived from the base class Qop. New operators can be defined using C++ class mechanism. Both Q Language and
libquantum share some limitations with QCL, since it is possible to operate on
single qubits or quantum registers (ie. arrays of qubits) only.
Quantum-Octave In a similar fashion the basic high-level structures used for
developing quantum programming languages were developed as a set of functions for the general purpose scientific computing system. The structures introduced in [42] are similar to the elements used in QCL and Q Language and
were described in quantum pseudo-code based on QCL quantum programming
language. They were implemented in GNU Octave language for scientific computing. The procedures used in the implementation are available as a package
quantum-octave providing a library of functions, which facilitates the simulation of quantum computing. This package allows also to incorporate high-level
programming concepts into the simulation in GNU Octave.
GNU Octave environment is to the large extent compatible with Matlab,
thus the described package can be also used with Matlab. As such it connects
the features unique for high-level quantum programming languages, with the
full palette of efficient computational routines commonly available in modern
scientific computing systems.
Quipper Quipper provides an implementation of programming language for
quantum computing based on the functional paradigm. The language hasb been
describe in [52]. It also gaines a considerable attnsion from the general scientific
community [60, 59, 137]. Moreover, some popular quantum algorithms havce
been implemented using Quipper [126].
The interpreter is implemented as library for Haskell programming language
and, effectively, Quipper programes are written in Haskell. It can be donwloaded
from [1]
6
Bidirectional Transformation
A bidirectional transformation (BX) is a “mechanism for maintaining the consistency of two (or more) related sources of information” [27]. To be more precise,
given a relation R ⊆ A × B specifying when two artifacts a ∈ A and b ∈ B
are consistent, denoted R(a, b), the goal of a BX framework is to derive trans→
−
←
−
formations R : ∆A × B → ∆B and R : ∆B × A → ∆A that can be used to
restore consistency when one of the artifacts is updated: given a ∈ A and b ∈ B
such that R(a, b), if a suffers an update δ ∈ ∆A that may break the consistency
→
−
→
−
between a and b, then R can be applied to obtain an update R (δ, b) ∈ ∆B that,
←
−
when applied to b, restores the desired consistency; likewise for R . Notice that
a BX framework cannot directly handle a synchronization scenario, where both
a ∈ A and b ∈ B may be concurrently updated.
BX frameworks differ on the kind of updates they handle: most are statebased and only consider the outcome of the update, i.e., ∆A is just a value of
24
type A; some are delta-based [31] and may handle a more detailed characterization of the updates - ∆A can, for example, include a correspondence relation
between elements of A before and after the update, making clear which where
deleted, modified or inserted. BX frameworks also differ on the kind of consistency relation R ∈ A×B that can be specified: some allow R to be, in principle,
an arbitrary relation - a value a ∈ A can be consistent with with many different
value of type B, and vice-versa; some only allow R to be deterministic (or functional) - for every value a ∈ A there is at most a value b ∈ A such that R(a, b);
some impose further restructions, for example, forcing R to be a bijection.
The BX frameworks that are more related with reversible computation are
precisely those where R must be deterministic and updates are state-based. In
this case, R can be specified by a transformation (or function) from A to B,
→
−
and also fulfills the role of R , being used to (trivially) recover consistency when
an update is performed to a value a ∈ A2 . This BX scenario is also known
as view-update, since R can be seen as a function that extracts a view b ∈ B
←
−
from a ∈ A, and R is responsible to propagate updates to the view back to the
→
−
source. In this scenario it is also common to denote R (and R ) by get : A → B
←
−
and R by put : B × A → A (known as putback ).
There are two basic properties that a BX view-update framework is usually
required to satisfy in order to be well-behaved [38]:
∀a ∈ A, b ∈ B · get(put(b, a)) = b
(PutGet)
∀a ∈ A · put(get(a), a) = a
(GetPut)
To simplify, here we assume get and put to be total functions, but the laws can
be trivially generalized to the partial setting. Law PutGet ensures that put can
indeed be used to restore consistency, i.e., put(b, a) must return an a0 of which
b is a view. Law GetPut forces put to return a null update when the input
values are already consistent, i.e., if get(a) = b then put(b, a) must return the
original a. Notice that GetPut sets a very lax lower-bound on what constitutes
a reasonable behavior for put: when the view is updated there is considerable
freedom on how to propagate it back into the source. Stronger alternatives
to this law have been proposed, for example variants of the principle of least
change [94], that requires put(b, a) to return a consistent a0 ∈ A that is as close
as possible to the original a ∈ A.
Several approaches have been proposed to obtain well-behaved view-update
frameworks. One of the most popular is the combinatorial approach first introduced in the so-called lenses [38]: a total well-behaved lens l from A to B,
denoted l ∈ A B, comprises both getl : A → B and putl : B × A → A,
satisfying PutGet and GetPut. A lens framework consists of a set of simple
primitive lenses and a set of combinators that build more complex lenses out
of simpler ones: these combinators ensure “by-construction” that the resulting
lenses are well-behaved whenever their parameter lenses are also well-behaved.
A simple example of a primitive lens could be fst : A × B A, to extract the
first component of a pair, defined as follows:
getfst (a, b) = a
putfst (b0 , (a, b)) = (a, b0 )
−
→
that R no longer needs to receive the original b ∈ B as input, as the updated
0
consistent b ∈ B is now uniquely determined by the updated a0 ∈ A.
2 Notice
25
The most fundamental lens combinator is sequential composition that, given
lenses l ∈ A B and k ∈ B C yields a lens l; k : A C, defined as follows:
getl;k (a) = getk (getl (a))
putl;k (c0 , a) = putl (putk (c0 , getl (a)), a)
Several domain-specific lens frameworks have been proposed, for example, for
manipulating string data [17] or to solve the classic database view-update problem [18]. Popular lens libraries have also been implemented for general purpose
languages such as Haskell3 or Scala4 .
Another popular technique to obtain an well-behaved view-update framework is bidirectionalization: given the definition of get somehow bidirectionalize
it in order derive a suitable put. The constant complement approach [11], first
proposed in the database community to solve the classic view-update problem, is a standard way to perform syntactic bidirectionalization. This approach
builds on the fact that an injective function f : A → B can easily be inverted on order to obtain a (possibly partial) function f −1 : B → A. Given
get : A → B, getc : A → C is a view complement function if the tupled function
get M getc : A → B × C is injective5 . Essentially, the goal of the view complement function getc is to keep all the information that get discards. Given getc ,
we can define put as follows:
put(b, a) = (get M getc )−1 (b, getc (a))
Notice that is always possible to obtain a view complement function (the identity function is a possibility) – the key issue is to obtain one that produces
a complement C as small as possible, so that the above put is as defined as
possible, i.e., it can propagate back more updates. [91] is a good example of
applying this approach to a non-database domain, namely to bidirectionalize
get functions defined in a functional language with support for algebraic data
types. In this concrete framework, an updatability checker is also derived, that
can be used to determine if a given update can be propagated back.
For particular view functions, namely polymorphic ones, it is also possible
to perform semantic bidirectionalization [140], that is, define a generic higherorder put function that receives get as a parameter, and executes it in a kind
of “simulation mode” to determine how to propagate back an update. The
main advantage of this technique is that, since it does not need to impose
any syntactic restrictions on how get is defined, it can easily be deployed in a
general purpose language (Haskell was the language of choice in [140]). The
main drawback (appart from the restriction to polymorphic functions) is poor
updatability, namely when handling updates that change the shape of the view.
Several extensions have been proposed to overcome these drawbacks, namely
combining it with syntactic bidirectionalization [91].
More information about BX can be found in an (early) state of the art
survey [27], in the proceedings of the ongoing workshop series on BX, and in
the respective community wiki6 .
3 https://github.com/ekmett/lens
4 https://github.com/julien-truffaut/Monocle
5 The
tupled function f M g is defined as λx.(f (x), g(x))
6 http://bx-community.wikidot.com
26
7
Robotics
Robots can be defined as generic mechatronic devices controlled by a computer.
This definition naturally raises the question of what it means for the computer
to be running a reversible program. Given the massive diversity in robotic
machinery, the meaning of robotic reversibility is likely to depend on the type
of the robot, and is not necessarily meaningful for all kinds of robots.
Industrial robots are segmented tool-manipulating robots often used in production; they are normally controlled by a single host computer. As described
in Section 3.4, limited forms of reversibility for debugging and interactive programming has traditionally been a feature found in many major commercial
industrial robot systems. An explicit connection between reversibility in robot
programming languages and reversible computing was however first made in
the context of modular robotics. We describe the use of reversible languages to
control modular self-reconfigurable robots and to provide a more general form
of control for industrial robots.
7.1
Modular self-reconfigurable robots
Modular robotics is an approach to the design, construction and operation of
robotic devices aiming to achieve flexibility and reliability by using a reconfigurable assembly of simple subsystems [131]. Robots built from modular components can potentially overcome the limitations of traditional fixed-morphology
systems because they are able to rearrange modules automatically on a by-need
basis, a process known as self-reconfiguration. Although self-reconfiguration
tends to be a reversible process, physical constraints such as changes in the environment or motion limitations due to gravity may impact the reversibility of
a given self-reconfiguration sequence.
Schultz et al investigated the distributed execution of a pre-specified selfreconfiguration sequence in the ATRON modular robot [121, 122]. A sequence is
specified using a simple, centralized scripting language named DynaRole. DynaRole programs could be manually written ro automatically generated by a planner. The distributed controller generated from this language allows for parallel
self-reconfiguration steps and is highly robust to communication errors and loss
of local state due to software failures. Furthermore, the self-reconfiguration sequence can automatically be reversed: any self-reconfiguration process described
in the language is reversible, subject to physical constraints. The distributed
scripting facility is however limited to specifying straightforward sequences of
actions, there is no support for specifying when sequences should execute, nor
is there support for e.g. conditionals and local state.
As a concrete example, consider the DynaRole program shown in Listing 9,
which is an excerpt of a 65-line program describing a self-reconfiguration sequence for a 7-module ATRON robot changing its shape between a flat shape
resembling the number 8 and a car (the 8-shape is an intermediate shape used
when changing from a snake to a car). The sequence consists of an unconditional series of steps performed either in sequentially or in parallel. Each step
can manipulate a gripper (“retract” or “extend”) or rotate the main joint (“rotateFromToBy”). The language is designed to allow statement-level inversion:
the inverse of retracting a given gripper is to extend it (and vice versa), and
the inverse of rotating from a given position to a target position is to rotate
27
sequence eight2car {
M0.Connector[$CONNECTOR_0].retract() &
M3.Connector[$CONNECTOR_4].retract();
M3.rotateFromToBy(0,324,false,150) ;
M4.rotateFromToBy(0,108,true,150);
M4.Connector[$CONNECTOR_0].extend() &
// ....
M3.Connector[$CONNECTOR_2].retract() ;
M1.rotateFromToBy(108,216,true,150) ;
}
sequence car2eight = reverse eight2car;
Listing 9: DynaRole sequence describing the “8 to car” self-reconfiguration sequence
back again. Note that rotation commands often do not carry a “from” argument
since this is normally given by the current position of the controller. The special keyword “reverse” shown in the last line of the listing provides direct access
to the reverse semantics. In addition to defining and reversing sequences, the
language also allows sequences to be composed.
In an effort to provide a more general reversible language for controlling
modular robots, the language µrRoCE (micro reversible robust collaborative ensembles) was devised as a minimalistic, general-purpose and reversible language
for programming modular robot systems [124]. The language was prototyped as
an embedded DSL in Scheme integrated with the USSR simulator for modular
robots [23], based on code ideas of the RoCE language [123, 120]. The language
integrates a minimal set of operations and control flow operators inspired from
Janus with a distributed execution model based on state-machines. Essentially,
behaviors can be either continuous or reactive based on events, and behaviors
are evaluted using either a forwards or a reverse execution semantics.
(define DockingCar
(@Front (if (and (@LeftWheel (module-nearby 1))
(@RightWheel (module-nearby 3)))
(begin (@LeftWheel (connector-extend 1))
(@RightWheel (connector-extend 3)))
’nop
(and (@LeftWheel (is-connected 1))
(@RightWheel (is-connected 3))))))
Listing 10: Docking of cars in µrRoCE (details omitted)
As a concrete example, consider the µRoCE program shown in Listing 10.
The program describes a docking sequence between to small robot cars made
from a “front” module and two “wheel” modules [132]. The “DockingCar” definition triggers on any module playing the role “Front” (which is the otherwise
passive module placed between two wheel modules). The definitions consist of
a Janus-style reversible conditional that tests if another module is close to the
wheel modules, and if so extends the connectors (and otherwise does nothing).
28
The post-assertion for the conditional states that the wheels should now be
connected. Reversing this behavior provides an undocking behavior.
7.2
Industrial robots
Industrial robots are increasingly being used for complex operations like assembly of products. The likelihood of errors however increases with the complexity
of the assembly operation. As an extension to the standard use of reversibility
for debugging (see also Section 3.4), it has been proposed that certain classes of
errors during assembly operations can be addressed using reverse execution, allowing the robot to temporarily back out of an erroneous situation, after which
the assembly operation can be automatically retried [80, 119]. The assembly
sequence is programmed in a reversible DSL named RASQ, which makes it is
possible to automatically derive a disassembly sequence from a given assembly
sequence, or vice versa. If an error is encountered because the assembly becomes stuck, the execution direction changes immediately, trying to undo the
operation and by getting the assembly “unstuck.”
object nut, bolt
sequence attach_nut_bolt {
moveto (...pos above table...)
pickup (nut, fixed_gripper, (...pos of nut...))
moveto (...)
try 3 (force<1) {
moveto (...pos on bolt...)
call apply_and_turn_nut
}
release (nut, fixed_gripper, (...))
moveto (...pos above table...)
}
sequence apply_and_turn_nut { ...commands... }
reverse { ...commands that undo apply_and_turn_nut...
}
grip fixed_gripper (nut) { ...commands for gripping a nut... }
Listing 11: Sample RASQ program, vector constants and a few other details are
omitted for clarity, only the body of “attach_nut_bolt” is shown.
The RASQ language is based on semantic modeling of the actions that the
robot makes during the assembly, reversing an assembly sequence is done by
reversing the actions using program inversion. A key contribution of RASQ is
however that it allows the programmer to explicitly override the definition of
the reverse of any given action. For example, the reverse of a simple push to an
object is not normally achieved by moving the gripper in the reverse direction.
Rather, “unpushing” could be achieved by pulling the object after gripping it,
or perhaps pushing it from a different angle.
As a concrete example, consider the RASQ program in Listing 11. A RASQ
program consists of declarations of objects, sequences and grips. Objects are to
be manipulated by sequences of commands, using grips to pick up and release
objects (an object is moved by moving the robot while the object is gripped.)
29
Key to reversibility is the declaration of a “sequence” as a number of series of
steps defined in terms of operations (such as moving the gripper), calls to other
sequences, and try-blocks that cause the body to be evaluated a number of
times.
8
Control Theory
Control theory and the foundations of reversibility are closely related, beginning
from Maxwell’s demon and Landauer’s principle [13], since Maxwell’s demon is a
control-based concept and control theory can be interpreted through information
theory [134]. This essential relationship is utilised in feedback thermodynamic
engines based on reversibility of feedback [63]. Reversibility helps in determining
energy cost in control as well [62].
Relationship of control theory and reversibility is twofold. On one hand,
systems with the reversibility property (specifically time reversibility) both in
classical and quantum setting are well-known and their properties are utilised
in control [129]. Another important notion of reversibility in control theory is
the reversibility of chemical reactions, so reversible control has been introduced
there as well [61].
On the other hand, application of reversible computing in control theory
takes several different forms as well. Time-reversible integration, applicable in
classical feedback control algorithms, is a notable example [55] . Finally, the
question of classical reversible computing, reversible gates and reversible programming in control theoretical applications needs to be addressed. While there
have been straightforward conversions of controllers (such as PID) from classical
building blocks to reversible logic gates, interesting attempts have been made
to provide framework for modern control tools such as fuzzy logic, starting from
theoretical foundations [116], proposed real implementations [6] to applications
in robotics [112]. Control based on reversible gates for robots has been subject
to research in light of quantum computing applications [68, 111].
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