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Relativity and Abstract State Machines. Edel Sherratt Aberystwyth University Aberystwyth, Wales, UK. Content. Abstract State Machines rock Global Time doesn’t We’ve been here before Some analogies A small step towards admissible transformations between ASMs’ frames of reference. - PowerPoint PPT Presentation
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Content
• Abstract State Machines rock• Global Time doesn’t• We’ve been here before• Some analogies• A small step towards admissible
transformations between ASMs’ frames of reference
ASMs are Powerful and Versatile
• Every kind of algorithm (Blass, Gurevich)• Conway's game of life, ambiguous grammars,
lift control, Internet telephony, database recovery … (Börger, Stärk)
• Asynchronous communication (Glässer, Gurevich, Veanes)
• SDL semantics (Glässer, Gotzhein, Prinz)
Basic ASM
• States are first order structures (base set with functions)
• Vocabulary, terms, locations• Moves update the values of locations
Distributed ASM
• ASM with multiple threads of control• Each associated with an agent• Each agent interprets its ASM program
according to its partial view of the state• This gives an update set
Distributed ASM
• A run is a partially ordered set of moves (M,<) such that– each move has finitely many predecessors– the moves performed by a given agent are linearly
ordered by <– for every finite initial segment X of (M, <), and
every maximal move m X, there is a unique state s(X) that results from performing m in the state s(X\{m})
Distributed ASM
• Freedom to schedule a run• Moves can be carried out– In parallel, unless they conflict– Interleaved– According to an explicit schedule
• Coherence condition; more restrictive than serializability in e.g. database scheduling
ASMs with Time (Graf, Prinz)
• Focus on moves rather than states• Moves are events (contrast with spacetime
events)• Moves are timed so that the partial ordering
on moves is reflected in a compatible order on time
• Time allows measurement of distance between (some) events
Global Time
• Implicit in the basic ASM• Is it present in the case of a distributed ASM?– Confluence of rule applications suggests that it is– In any run, moves are ordered in global time,
though some occur at the same global time– Local timestamps set by reading the monitored
function ‘now’, which gives a global time
What’s wrong with global time?
• Server with clients; online ticketing system• A distributed ASM, but clients have distinct,
usually conflicting , goals• Server determines which requests succeed
and which fail• Global time is determined by the server
What’s wrong with global time?
• A new client for a running service• Abstract Communicator (Glässer et al.) acts as
intermediary• Communicator is part of client’s environment• Global time is determined by communicator• Communicator determines global time for all
clients• And resolves clients’ times with server’s time
What’s wrong with global time?
• Suppose client uses multiple services• Global time is determined by more than one
communicator• Which takes precedence?
Galileo (and Newton)
• Absolute space• Inertial frame is in uniform motion relative to
absolute space• Universal absolute time shared by all inertial
frames• Galilean transformation: x’=x-vt, t’=t, y’=y, z’=z
Fitzgerald, Lorentz, Einstein
• Inertial frames do not accelerate wrt each other
• Lorentz transformation preserves laws of electrodynamics
• So no preference for any frame
Relevance to ASMs?
• Analogy• Space is defined by the vocabulary of an ASM• Time progresses through a run• Event is a combination of state (values at
locations) and time (contrast with event as move)
• ASMs communicate only through mutually accessible locations
Where the analogy breaks down
• ASMs need not operate in the same space• In general, shared locations represent minor
points of overlap• ASM spacetime is not continuous• Nonetheless, differences between states can
be computed as can distances in time
Aim
• Identify transformations analogous to the Galilean and Lorentz transformations that preserve required properties of abstract state machines
Observing an event
• Two ASMs, A and B• Can B observe a location as A sees it?• Only mutually accessible location(s) can be
observed• Terms of A map to terms of B iff the terms
refer to the same location
Time of an event (state)
• Local clocks can only be synchronized if mutually accessible locations have the same values
• Otherwise each ASM sees the state of the other as being in its own past or future
Private state
• Updates are applied instantaneously• But computing the update set is not
necessarily instantaneous• Shareable locations should not be used for
scratch work
Conserving History
• Suppose SA precedes SA’ in the history of A, and suppose B observes those states as SB and SB’
• That is, mutually accessible locations are mapped from A’s terms and A’s time to B’s terms and B’s time
• Then SB must also precede SB’ in B’s frame of reference