Formal Methods for Schedulings of Latency-Insensitive Designs

: ( Latency-Insensitive Design ) theory was invented to deal with SoC timing closure issues, by allowing arbitrary (cid:12)xed integer latencies on long global wires. Latencies are coped with using a resynchronization protocol that performs dynamic scheduling of data transportation. Functional behaviour is preserved. This dynamic scheduling is implemented using speci(cid:12)c synchronous hardware elements: Relay-Stations ( RS ) and Shell-Wrappers ( SW ). Our (cid:12)rst goal is to provide a formal modeling of RS and SW , that can then be formally veri(cid:12)ed. As turns out, resulting behaviour is k -periodic, thus amenable to static scheduling. Our second goal is to provide formal hardware modeling here also. It initially performs Throughput Equalization , adding integer latencies wherever possible; residual cases require introduction of Fractional Registers ( FRs ) at speci(cid:12)c locations. Benchmark results are presented, run on our KPassa tool implementation.


Introduction
Long wire interconnect latencies induce time-closure difficulties in modern SoC designs, with propagation of signals across the die in a single clock cycle is problematic.The theory of Latency-Insensitive Design (LID), proposed originally by Luca Carloni, Kenneth McMillan and Alberto Sangiovanni-Vincentelli [7,8], offers solutions for this issue.This theory can roughly be described as such: an initial fully synchronous reference specification is first desynchronized as an asynchronous network of synchronous block components (a GALS system); it is then re-synchronized, but this time with proper interconnect mechanisms allowing specified (integer-time) latencies.
Interconnects consist of fixed-sized lines of so-called Relay-Stations.These Relay-Stations, together with Shell-Wrapper around the synchronous Pearl IP blocks, are in charge of managing the signal value flows.With their help proper regulation of the signal traffic is performed.Computation blocks may be temporarily paused at times, either because of input signal unavailability, or because of the inability of the rest of the network to store their outputs if they were produced.This latter issue stems from the limitation of fixed-size buffering capacity of the interconnects (Relay-Station lines).
Since their invention Relay-Stations have been a subject of attention for a number of research groups.Extensive modeling, characterization and analysis were provided in [9,14,13].
We mentioned before that the process of introducing latencies into synchronous networks introduced, at least conceptually, an intermediate asynchronous representation.This corresponds to Marked Graphs [16], a well-studied model of computation in the literature.The main property of Marked Graph is the absence of choice which matches with the absence of control in LID.
Marked Graphs with latencies were also considered under the name of Weighted Marked Graphs (WMG) [19].We shall reduce WMGs to ordinary Marked Graphs by introducing new intermediate Transportation Nodes (TN), akin to the previous Computation Nodes (CN) but with a single input and output link.In fact LID systems can be thought of as WMGs with buffers of capacity 2 (exactly) on link between Computation and/or Transportation Nodes.The Relay-Stations and Shell-Wrappers are an operational means to implement the corresponding flow-control and congestion avoidance mechanisms with explicit synchronous mechanisms.
The general theory of WMG provides many useful insights.In particular it teaches us that there exists static repetitive scheduling for such computational behaviors [6,2].Such static kperiodic schedulings have been applied to software pipelining problems [18,5], and later SoC LID design problems in [12].But these solutions pay in general little attention to the form of buffering elements that are holding values in the scheduled system, and their adequacy for hardware circuit representation.We shall try to provide a solution that "perfectly" equalizes latencies over reconvergent paths, so that token always arrive simultaneously at the Computation Node.Sadly, this cannot always be done by inserting an integer number of latency under the form of additional transportation sections.One sometimes need to hold back token for one step discriminatingly ans sometimes does not.We provide our solution here under the form of Fractional Registers (FR), that may hold back values according to an (input) regular pattern that fits the need for flowcontrol.Again we contribute explicit synchronous descriptions of such elements, with correctness properties.We also rely deeply on a syntax for schedule representation, borrowed from the theory of N-synchronous processes [15].
Explicit static scheduling that uses predictable synchronous elements is desirable for a number of issues.It allows a posteriori precise re-dimensioning of glue buffering mechanisms between local synchronous elements to allow the system to work, and this without affecting the components themselves.Finally, the extra virtual latencies introduced by equalization could be absorbed by the local computation times of CN, to resynthesize them under relaxed timing constraints.
We built a prototype tool for equalization of latencies and Fractional Registers insertion.It uses a number of elaborated graph-theoretical and linear-programming algorithms.We shall briefly describe this implementation.
Contributions: Our first contribution is to provide a formal description of Relay-Stations and Shell-Wrappers as synchronous elements [4], something that was never done before in our knowl-edge (the closest effort being [10]).We introduce local correctness properties that can easily be model-checked; these generic local properties, when combined, ensure the global property of the network.
We introduce the Equalization process to statically schedule a LID Specification: slowing down "too fast" cycles while maintaining the original throughput of the LID Specification.The goal is to simplify the LID protocol.
But rational difference of rates may still occur after Equalization process, we solve it by adding Fractional Registers (FR), that may hold back values according to a regular pattern that fits the need for flow-control.
We introduce a new class of smooth schedules that optimally-minimizes the number of FRs used on a statically scheduled LID design.
Article Outline: In the next Section we provide some definitional and notational background on various models of computations involved in our modeling framework, together with an explicit representation of periodic schedules and firing instants; with this we can state historical results on k-periodic scheduling of WMGs.In Section 3 we provide the synchronous reactive representation of Relay-Stations and Shell-Wrappers, show their use in dynamic scheduling of Latency-Insensitive Design, and describe several formal local correctness properties that help with the global correctness property of the full network.Statically scheduled LID systems are tackled in Section 4; we describe an algorithm to build a statically scheduled LID, possibly adding extra virtual integer latencies and even Fractional Registers.We provide a running example to highlight potential difficulties.We also present benchmarks result of a prototype tool which implements the previous algorithms and their variations.We conclude with considerations on potential further topics.

Computation nets
We start from a very general definition, describing what is common of all our models.Definition 1 (Computation Network Scheme).We call Computation Network Scheme (CNS) a graph whose vertices are called Computation Nodes, and whose arcs are called links.We also allow arcs without a source vertex, called input links, or without target vertex, called output links.
An instance of a CNS is depicted on Figure 1 (a).The intention is that Computation Nodes perform computations by consuming a data on each of its incoming links, and producing as a result a new data on each of its outgoing links.
The occurrence of a computation thus only depends on data presence and not their actual values, so that data can be safely abstracted as tokens.A CNS is choice free.
In the sequel we shall often consider the special case where the CNS forms a strongly connected graph, unless specified explicitly.
This simple model leaves out the most important features, that are mandatory to define its operational semantics under the form of behavioral firing rules.Such features are: the initialization setting (where do tokens reside initially), the nature of links (combinatorial wires, simple registers, bounded or unbounded place, ...), and the nature of time (synchronous, with computations firing simultaneously as soon as they can, or asynchronous, with distinct computations firing independently).
Setting up choices in these features provides distinct Models of Computation.The S/R model conforms to synchronous digital circuits or (single-clock) synchronous reactive formalisms [3].The network operates "at full speed": there is always a value present in each register, so that CNs operates at each instant.As a result, they consume all values (from registers and through wires), and replace them again with new values produced in each register.The system is causal iff there is at least one register along each cycle in the graph.Causal S/R nets are well-behaved in the sense that their semantics is well-founded.Definition 3. A Marked Graph is a CNS where time is asynchronous: computations are performed independently, provided they find enough tokens in their incoming links; links have a place holding a number of tokens; in other words, Marked Graphs form a subclass of Petri Nets.The initial marking of the graph is the number of tokens held in each place.In addition a Marked Graph is said to be of capacity k if each place can hold no more than k tokens.

Boucaron & Millo & de Simone
There is a simple way to encode Marked Graphs with capacity as Marked Graphs with unbounded capacity: this requires to add a reverse link for each existing one, which contains initially a number of tokens equal to the difference between the capacity and the initial marking of the original link.
It was proved that a strongly connected Marked Graph is live (each computation can always be fired in the future) iff there is at least one token in every cycle in the graph [16].Also, the total number of tokens in a cycle is an invariant, so strongly connected Marked Graphs are k-safe for a given capacity k.
Under proper initial conditions S/R nets and Marked Graphs behave essentially the same, with S/R systems performing all computations simultaneously "at full rate", while similar computations are now performed independently in time in Marked Graph.Definition 4. A Synchronous Marked Graph (SMG) is a Marked Graph with an ASAP (As Soon As Possible) semantics: each Computation Node (transition) that may fire due to the availability of it input tokens immediately does so (for the current instant).
SMGs and the ASAP firing rule are underlying the works of [6,2], even though they are not explicitly given name there.
Figure 1 (c) shows a Synchronous Marked Graph.Note that SMGs depart from S/R models: here all tokens are not always available.

Adding latencies and time durations
We now add latency information to indicate transportation or computation durations.These latencies shall be all along constant integers (provided from "outside").Definition 5. A Weighted Marked Graph (WMG) is a CNS with (constant integer) latency labels on links.This number indicates the time spent while performing the corresponding token transportation along the link.
We avoid computation latencies on CNs, which can be encoded as transportation latencies on links by splitting the actual CN into a begin/end CN.Since latencies are global time durations, the relevant semantics which take same into account is necessarily ASAP.The system dynamics also imposes that one should record at any instant "how far" each token is currently in its travel.This can be modeled by an age stamp on token, or by expanding the WMG links with new Transportation Nodes (TN) to divide them into as many sections of unit latency.TNs are akin to CNs, with the particularity that they have unique source and target links.This expansion amounts to reducing WMGs to (much larger) plain SMGs.Depending on the concern, the compact or the expanded form may be preferred.For correctness matters there should still be at least one token along each cycle in the graph, and less token on a link than its prescribed latency.This corresponds to the correctness required on the expanded SMG form.Definition 6.A Latency-Insensitive Design (LID) is a WMG where the expanded SMG obtained as above uses places of capacity 2 in between CNs and TNs.This definition reads much differently than the original one in [8].This comes partly from an important concern of the authors then, which is to provide a description built with basic components (named Relay-Stations and Shell-Wrappers) that can easily be implemented in hardware.Next Section 3 provides a formal representation of Relay-Stations and Shell-Wrappers, together with their properties.
Summary CNS lead themselves quite naturally to both synchronous and asynchronous interpretations.Under some easily expected initial conditions, these variants can be shown to provide the same input/output behaviours.With explicit latencies to be considered in computation and data transportation this remains true, even if congestion mechanisms may be needed in case of bounded resources.The equivalence in the ordering of event between a synchronous circuit and a LID circuit is shown in [7], and equivalence between a MG and a S/R design is shown in [20].

Periodic behaviors, throughput and explicit schedules
We now provide the definitions and classical results needed to justify the existence of static scheduling.This will be used mostly in Section 4, when we develop our formal modeling for such scheduling using again synchronous hardware elements.
Definition 7 (Rate, throughput and critical cycles).Let G be a WMG graph, and C a cycle in this graph.The rate R of the cycle C is equal to T L , where T is the number of tokens in the cycle, and L is the sum of latencies of the arcs of this given cycle.The throughput of the graph is defined as the minimum rate among all cycles of the graph.A cycle is called critical if its rate is equal to the throughput of the graph.
A classical result states that, provided simple structural correctness conditions, a stronglyconnected WMG runs under a ultimately k-periodic schedule, with the throughput of the graph [6,2].We borrow notation from the theory of N -synchronous processes [15] to represent these notions formally, as explicit analysis and design objects.Definition 8 (Schedules, periodic words, k-periodic schedules).A pre-schedule for a CNS is a function Sched : N → w N assigning an infinite binary word w N ∈ {0, 1} ω to every Computation Node and Transportation Node N of the graph.Node N is activated (or triggered, or fired, or run) at global instant i iff w N (i) = 1, where w(i) is the i th letter of word w.
A pre-schedule is a schedule if the allocated activity instants are in accordance with the token distribution (the lengthy but straightforward definition is left to the reader).Furthermore, the schedule is called ASAP if it activates a node N whenever all its input tokens have arrived (according to the global timing).
An infinite binary word w ∈ {0, 1} ω is called ultimately periodic: if it is of the form u.(v) ω where u and v ∈ {0, 1} , u represents the initialization phase, and v the periodic one.The length of v is noted |v| and called its period.The number of occurrences of 1s in v is denoted |v| 1 and called its periodicity.The rate R of an ultimately periodic word w is defined as |v|1 |v| .A schedule is called k-periodic whenever for all N , w N is a periodic word.
Thus a schedule is constructed by simulating the CNS according to its (deterministic) ASAP firing rule.
Furthermore, it has been shown in [2] that the length of the stationary periodic phase (called period) can be computed based on the structure of the graph and the (static) latencies of cycles: for a CSCC (Critical Strongly Connected Component) the length of the stationary periodic phase is the GCD (Greatest Common Divisor) over latencies of its critical cycles.For instance assume a CSCC with 3 critical cycles having the following rates: 2/4, 4/8, 6/12, the GCD of latencies over its critical cycles is: 4. For the graph, the length of its stationary periodic phase is the LCM (Least Common Multiple) over the ones computed for each CSCCs.For instance assume the previous CSCC and another one having only one critical cycle of rate 1/2 then the length of the stationary periodic phase of the whole graph is 2.
Figure 1(d) shows the schedules obtained on our example.If latencies were "well-balanced" in the graph, tokens would arrive simultaneously at their consuming node; then, the schedule of any Node should exactly be the one of its predecessor(s) shifted right by one position.However it is not the case in general when some input tokens have to stall awaiting others.The "difference" (target schedule minus 1-shifted source schedule) has to be coped with by introducing specific buffering elements.This should be limited to the locations where it is truly needed.Computing the static scheduling this allows to avoid adding the second register that was formerly needed everywhere in RSs, together with some of the backpressure scheme.
The issue arises in our running example only at the top-most Computation Node.We indicate it by prefixing some of the inactive steps (0) in its schedule by symbols: lack of input from the right input link ('), or from the left one (').

Synchronous to LID: Dynamic Schedule
In this Section we shall briefly recall the theory of Latency-Insensitive Design, and then focus on formal modeling with synchronous components of its main features [4].
LID theory was introduced in [7].It relies on the fact that links with latency, seen as physical long wires in synchronous circuits, can be segmented into sections.Specific elements are then introduced in between sections.Such elements are called Relay-Stations (RS).They are instantiated at the oval places in Figure 1(c).Instantaneous communication is possible inside a given section, but the values have to be buffered inside the RS before it can be propagated to the next section.The problem of computing realistic latencies from physical wire lengths was tackled in [11], where a physical synthesis floor-planner provides these figures.
Relay-Stations are complemented with so-called Shell-Wrappers (SW), which compute the firing condition for their local synchronous component (called Pearl in LID theory).They do so from the knowledge of availability of input token and output storage slots.

Relay-Stations
The signaling interface of a Relay-Station is depicted in Figure 2. The val signals are used to propagate tokens, the stop signal are used for congestion control.For symmetry here stop out is an input and stop in an output.Intuitively the Relay-Station behaves as follows: when traffic is clear (no stop), each token is propagated down at the next instant from the one it was received.When a stop out signal is received because of downward congestion, the RS keeps its token.But then, the previous section and the previous RS cannot be warned instantly of this congestion, and so the current RS can perfectly well receive another token at the same time it has to keep the former one.So there is a need for the RS to provide a second auxiliary register slot to store this second token.Fortunately there is no need for a third one: in the next instant the RS can propagate back a stop in control information to preserve itself from receiving yet another value.Meanwhile the first token can be sent as soon as stop out signals are withdrawn, and the RS remains with only one value, so that in the next step it can already allow a new one and not send its congestion control signal.Note that in this scheme there is no undue gap between the token sent.
This informal description is made formal with the description of a synchronous circuit with two registers describing the RS in Figure 3, and its corresponding syncchart [1] (in Mealy FSM style) in Figure 4.
The syncchart contains 4 states: empty when no token are currently buffered in the RS; in this state the RS simply waits for a valid input token coming, and store it in its main register then it goes to state half.stop out signals are ignored, and not propagated upstream, as this RS can absorb traffic.half when it holds one token; Then the RS only transmits its current, previously received token if ever it does not receive an halting stop out signal.If halting is requested, (stop out), then it retains its token, but must also accept a potential new one coming from upstream (as it has not sent any back-pressure holding signal yet).In the second case it becomes full, with the second value occupying its "emergency" auxiliary register.If the RS can transmit (stop out = f alse), it either goes back to empty or retrieve a new valid signal (val in), remaining then in the same state.On the other hand it still makes no provision to propagate back-pressure (in the next clock cycle), as it is still unnecessary due to its own buffering capacity.
full when it contains two tokens; then it raises in any case the stop in signal, propagating to the upstream section the hold-out stop out signal received in the previous clock cycle.If it does not itself receive a new stop out, then the line downstream was cleared enough so that it can transmit its token; otherwise it keeps it and remains halted.
error is a state which should never be reached (in an assume/guarantee fashion).The idea is that there should be a general precondition stating that the environment will never send

Boucaron & Millo & de Simone
the val in signal whenever the RS emits the stop in signal.This should be extended to any combination of RS, and build up a "sequential care-set" condition on system inputs.The property is preserved as a postcondition as each RS will guarantee correspondingly that val out is not sent when stop out arrives.

NB:
The notation val out(main) or val out(aux) means emit the signal val out taking its value in the buffer, respectively, main or aux.
Correctness properties Global correctness depends upon an assumption on the environment (see description of error state above).We now list a number of properties that should hold for Relay-Stations, and further links made of a connected line L n (k) of n successive RS elements and currently containing k values (remember that a line of n RS can store 2n values).
On a single RS: (¬stop out ⇒ ¬X n (stop in) ) (free slots propagate backwards); ♦ val in ∧ (♦(¬stop out) ⇒ ♦val out (if traffic is not completely blocked from below from a point on, then tokens get through) The first property is true of any line of length n, the second of any line containing initially at least k tokens, the third of any line.
We have implemented RSs and lines of RSs in the Esterel synchronous language, and modelchecked combinations of these properties using EsterelStudio1 .

Shell-Wrappers
The purpose of Shell-Wrappers is to trigger the local Computation Node exactly when tokens are available from each input link, and there is storage available for result in output links.It corresponds to a notion of clock gating in circuits: the SW provides the logical clock that activates the IP component represented by the CN.Of course this requires that the component is physically able to run on such an irregular clock (a property called patience in LID vocabulary), but this technological aspect is transparent to our abstract modeling level.Also, it should be remembered that the CN is supposed to produce data on all its outputs while consuming on all its inputs in each computation step.This does not imply a combinatorial behavior, since the CN itself can contain internal registers of course.A more fancy framework allowing computation latencies in addition to our communication latencies would have to be encoded in our formalism.This can be done by "splitting" the node into a begin CN and a end CN nodes, and installing internal transportation links with desired latencies between them; if the outputs are produced with different latencies one should even split further the node description.We shall not go into further details here, and keep the same abstraction level as in LID and WMG theories.
The signal interface of SWs consists of val in and stop in signals indexed by the number of input links to the SW, and of val out and stop out signals indexed by the number of its output links.There is an output clock signal in addition, to fire the local component.This last signal stop_in [1] stop_in so, internal Pearl's clock is set to false whenever a backward stop out j occurs as true, or a forward val in i is false.In such case the registers already busy hold their true value, while others may receive a valid token "just now"; stop in i signals are raised towards all channels whose corresponding register was already loaded (a token was received before, and still not consumed), to warn them not to propagate any value in this clock cycle.Of course such signal cannot be sent in case the token is currently received, as it would raise a causality paradox (and a combinatorial cycle).
flip-flop registers are reset when the Pearl's clock is raised, as it consumes the input token.Following the previous remark, the signal stop in i holding back the traffic in channel i is raised for these channels where the token have arrived before the current instant, even in this case.
Correctness properties Again we conducted a number of model-checking experiments on SWs using Esterel Studio: (∃j, stop out j )∨ ⇒ ¬clock where j is an input index; (∃j, stop out j ) ⇒ (∀i, ¬val out i ) where j/i is an input/output index respectively; (∀j, ¬stop out j ∧ ¬X(stop out j )) ⇒ (X(clock) ⇒ ∃i, X(val in i )) where j, i are input index (if the SW was not suspended at some instant by output congestion, and it triggers its pearl the next instant, then it has to be because it received a new value toke on some input at this next instant) On the other hand, most useful properties here would require syntactic sugar extensions to the logics to be easily formulated (like "a token has had to arrive on each input before or when the SW triggers its local Pearl", but they can arrive in any order).
As in the case of RSs, correctness also depends on the environmental assumption that ∀i, stop in i ⇒ ¬val in i , meaning that upward components must not send a value while this part of the system is jammed.

Tool implementation
We built a prototype tool named Kpassa2 to simulate and analyze a LID system made of a combination of previous components.
Simulation is eased by the following fact: given that the ASAP synchronous semantics of LID ensures determinism, for closed systems each state has exactly one successor.So we store states that were already encountered to stop the simulation as soon as a state already visited is reached.
While we will come back to the main functions of the tool in the next Section, it can be used in this context of dynamic scheduling to detect where the back-pressure control mechanisms are really been used, and which Relay-Stations actually needed their secondary register slot to preserve from traffic congestion.

Synchronous to LID: Static Scheduling
We now turn to the issue of providing static periodic schedules for LID systems.According to the previous philosophy governing the design of Relay-Stations, we want to provide solutions where tokens are not allowed to accumulate into places in large numbers.In fact we will attempt to equalize the flows so that tokens arrive as much as possible simultaneously at their joint Computation Nodes.
We try to achieve our goal by adding new virtual latencies on some paths that are faster than others.If such an ideal scheme could lead to perfect equalization then the second buffering slot mechanism of Relay-Stations and the back-pressure control mechanisms could be done without altogether.However it will appear that this is not always feasible.Nevertheless integer latency equalization provides a close approximation, and one can hope that the additional correction can be implemented with smaller and simpler Fractional Registers.
Extra virtual latencies can often be included as computational latencies, thereby allowing the re-design of local Computation Nodes under less stringent timing budget.
As all connected graphs, general (connected) CNS consist of Directed Acyclic Graphs of strongly connected components.If there is at least one cycle in the net it can be shown that all cycles have to run at the rate of the slowest to avoid unbounded token accumulation.This is also true of input token consumption, and output token production rates.Before we deal with the (harder) case of strongly connected graphs that is our goal, we spend some time on the (simpler) case of acyclic graphs (with a single input link).

DAG Case
We consider the problem of equalizing latencies in the case of Directed Acyclic Graphs (DAGs) with a single source Computation Node (one can reduce DAGs to this sub-case if all inputs are arriving at the same instant), and no initial token is present in the DAG.Definition 9 (DAG Equalization).In this case the problem is to equalize the DAG such that all paths arriving to a Computation Node are having the same latency from inputs.
We provide a sketch of the abstract algorithm and its correction proof.
Definition 10 (Critical Arc).We define an arc as critical if it belongs to a path of maximal latency M ax l (N ) from the global source Computation Node to the target Computation Node N of this arc.
Definition 11 (Equalized Computation Node).We define a Computation Node N which is having only incoming critical arcs to be an equalized Computation Node, i.e from any path from the source to this Computation Node we have the same latency M ax l (N ).If a Computation Node has only one incoming arc then this arc will be critical and this Computation Node will be equalized by definition.
The core idea of the algorithm is first to find for each Computation Node N of the graph what is its maximal latency M ax l (N ) and to mark incoming critical arcs; Then the second idea is to saturate all non-critical arcs of each Computation Node of the DAG in order to obtain an equalized DAG.
The first part of the algorithm is done through a modified longest-path algorithm, marking incoming critical arcs for each Computation Node of the DAG and putting for each Computation Node N its maximal latency M ax l (N ) (as shown in algorithm 1).
The second part of the algorithm is done as follows (see algorithm 2): Since it may exist incoming arcs of a Computation Node N that are not critical: it exists an integer number that we can add such that the non-critical arc becomes critical.We can compute this integer number easily through this formula: M ax l (N ) = M ax l (N ) + non critical arc l + , where N is the source Computation Node passing through the non-critical arc and reaching the Computation Node N .Now, the non-critical arc through the add of is critical.We apply this for all non-critical arcs of the Computation Node N , then the Computation Node is equalized.Finally, we apply this for all Computation Nodes of the DAG, then the DAG is equalized.
An instance of the unequalized, critical arcs annotated and equalized DAG is shown in Figure 6: Starting from the unequalized graph in Figure 6 (a): The first pass of the algorithm is determining for each Computation Node, its maximal latency M ax l (in circles) and incoming critical arcs denoted using large links as in Figure 6 (b).
The second part of the algorithm is adding "virtual" latencies (the ) on non-critical incoming arcs, since we known what are the critical arcs coming through each Computation Node (large links), then we just have to add the needed amount ( ) in order that the non-critical arc is now critical: the sub between the value of the target Computation Node, minus the sum between the arriving critical arc and its source Computation Node maximal latency.For instance, consider the Computation Node holding a 9, the left branch is not critical, hence we are just solving 9 = 6+1+ and = 2 thus the arc will now have a latency of 3 = 1 + and is so critical by definition.Finally the whole graph will be fully-critical and thus equalized by definition as in Figure 6 (c).Definition 12.A critical path is composed only of critical arcs.

Theorem 1. DAG equalization algorithm is correct
Proof.For all Computation Nodes, there is at least one critical arc incoming by definition; then if there is more than one incoming arc, we add the result of the sub between the maximum latency of the path passing through the so-called critical arc and the add between the non-critical arc latency and the maximum latency of the path arriving to the Computation Node where the non-critical arc starts.Now any arc on this given Computation Node are all critical and thus this Computation Nodes is equalized by definition.And this is done for any Computation Node, thus the graph is equalized.Since in any case we do not modify any critical arc, we still have the same maximum latency on critical paths.

Strongly Connected Case
In this case, the successive algorithmic steps involved in the process of equalization consist in: 1. Evaluate the graph throughput; 2. Insert as many additional integer latencies as possible (without changing the global throughput); 3. Compute the static schedule and its initial and periodic phases; 4. Place Fractional Registers where needed;

Optimize the initialization phase (optional)
These steps can be illustrated on our example in Figure 1.Inspecting closely those schedules one can notice that in all cases the schedule of a CN is the one of its predecessors shifted right by one position, except for the schedule of the topmost Computation Node.One can deduce from the differences in scheduling exactly when the additional buffering capacity was required, and insert dedicated Fractional Registers which delay selectively some tokens accordingly.This only happens for the initial phase for tokens arriving from the right, and periodically also for tokens arriving from the left; 4. It could be noticed that, by advancing only the single token at the bottom of the up going rightmost link for one step, one reaches immediately the periodic phase, thus saving the need for a FR element on the right cycle used only in the initial phase.Then only one FR has to be added past the regular latch register colored in grey.
We describe now the equalization algorithm steps in more details: Graph throughput evaluation: For this we enumerate all elementary cycles and compute their rates.While this is worst-case exponential, it is often not the case in the kind of applications encountered.An alternative would be to use well-known "minimum mean cycle problem" algorithms (see [17] for a practical evaluation of those algorithms).But the point here is that we need all those elementary cycle for setting up Linear Programming (LP) constraints that will allow to use efficient LP solving techniques in the next step.We are currently investigating alternative implementations in Kpassa.
Integer latency insertion: This is solved by LP techniques.Linear equation systems are built to express that all elementary cycles, with possible extra variable latencies on arcs, should now be of rate R, the previously computed global throughput.The equations are also formed while enumerating the cycles in the previous phase.An additional requirement entered to the solver can be that the sum of added latencies be minimal (so they are inserted in a best factored fashion).Rather than computing a rational solution and then extracting an integer approximate value for latencies, the particular shape of the equation system lends itself well to a direct greedy algorithm, stuffing incremental additional integer latencies into the existing systems until completion.This was confirmed by our prototype implementations.
The following example of Figure 7 shows that our integer completion does not guarantee that all elementary cycles achieve a rate very close to the extremal.But this is here because a cycle "touches" the slowest one in several distinct locations.While the global throughput is of 3  16 , given by the inner cycle, no integer latency can be added to the outside cycle to bring its rate to 1  5 from 1 4 .Instead four fractional latencies should be added (in each arc of weight 1).

Initial and Periodic phase Schedule computation:
In order to compute the explicit schedules of the initial and stationary phases we currently need to simulate the system's behavior.We also need to store visited state, as a termination criterion for the simulation whenever an already visited state is reached.The purpose is to build (simultaneously or in a second phase) the schedule patterns of Computation Nodes, including the quote marks (') and ('), so as to determine where residual fractional latency elements have to be inserted.In a synchronous run each state will have only one successor, and this process stops as soon as a state already encountered is reached back.The main issue here consists in the state space representation (and its complexity).Further simplification of the state space in symbolic BDD model-checking fashion is also possible but it is out of the scope of this paper.
We are currently investigating (as "future work") analytic techniques so as to estimate these phases without relying on this state space construction.
Fractional Register insertion: In an ideally equalized system, the schedules of distinct Computation/Transportation Nodes should be precisely related: the schedule of the "next" CN should be that of the "previous" CN shifted one slot right.If not, then extra Fractional Registers need to be inserted just after the regular register already set between "previous" and "next" nodes.This FR should delay discriminatingly some tokens (but not all).
We shall introduce a formal model of our FR in the next SubSection.The block diagram of its interfaces are displayed in Figure 8.
We conjecture that, after integer latency equalization, such elements are only required just before Computation Nodes to where cycles with different original rates re-converge.We prove in subsection 4.4 that this is true under general hypothesis on smooth distribution of tokens along critical cycles.In our prototypal approach we have decided to allow them wherever the previous step indicated their need.The intention is that a regular register coupled with a FR one should almost amount to a RS, with the only difference that the backpressure control stop {in/out} signal mechanisms could be simplified due to static scheduling information computed previously.

Computation Node
Computation Node Register FR next previous Optimized initialization.So far we have only considered the case where all components did fire as soon as they could.Sometimes delaying some computations or transportations in the initial phase could lead faster to the stationary phase, or even to a distinct stationary phase that may behave more smoothly as to its scheduling.Consider in the example of Figure 1 (c) the possibility of firing the lower-right Transportation Node alone (the one on the backward up arc) in a first step.This modification allows the graph to reach immediately the stationary phase (in its last stage of iteration).
Initialization phases may require a lot of buffering resources temporarily, that will not be used anymore in the stationary phase.Providing short and buffer-efficient initialization sequences becomes a challenge.One needs to solve two questions: first, how to generate efficiently states reachable in an asynchronous fashion (instead of the deterministic asap single successor state); second, how to discover very early that a state may be part of a periodic regime?These issues are

Boucaron & Millo & de Simone
still open.We are currently experimenting with Kpassa on efficient representation of asynchronous firings and resulting state spaces.
Remark When applying these successive transformation and analysis steps, which may look quite complex, it is predictable that simple sub-cases often arise, due to the well-chosen numbers provided by the designer.Exact integer equalization is such a case.The case when fractional adjustments only occur at reconvergence to a critical paths are also noticeable.We built a prototype implementation of the approach, which indicates that these specific cases are indeed often met in practice.

Fractional Register element (FR)
We now formally describe the specific FR, both as a synchronous circuit in Figure 9(b) and as a corresponding syncchart (in Mealy FSM style) in Figure 9(a).
The FR interface consists of two input wires val in and hold, and one output wire val out.Its internal state consists of a register catch reg.The register will be used to "kidnap" the valid data (and its value in a real setting) for one clock cycle whenever hold holds.We note pre(catch reg) the (boolean) value of the register computed at the previous clock cycle.It indicates whether the slot is currently occupied or free.It is possible that the same data is held several instants in a row.But meanwhile there should be no new data arriving, as the FR can store only one value; otherwise this would cause a conflict.It is also possible that a full sequence of consecutive datas are held back one instant each in a burst fashion.But then each data/value should leave the element in the very next instant to be consumed by the subsequent Computation Node; otherwise this would also cause a conflict.Stated formally, when hold ∧ pre(catch reg) holds then either val in holds, in which case the new data enters and the current one leaves (by scheduling consistency the Computation Nodes that consumes it should then be active), or val in does not hold, in which case the current data remains (and, again by scheduling consistency, then the Computation Node should be inactive).Furthermore the two extra conditions are requested: [hold ⇒ (val in ∨ pre(catch reg)):] if nothing can be held, the scheduling does not attempt to; [(val in∧pre(catch reg)) ⇒ hold:] otherwise the two datas could cross the element and be output simultaneously.
The FR behavior amounts to the two equations: [catch reg = hold:] the register slot is used only when the scheduling demands; [val out = val out 1 ∨ val out 2 :] val out 1 = val in ⊕ pre(catch reg) ∧ ¬hold.
either a new value directly falls across, or an old one is chased by a new one being held in its place.
Our main design problem is now to generate hold signals exactly when needed.Its schedule should be the difference between the schedule of its source (Computation or Transportation) Node shifted by one instant, and the schedule of its target node; indeed, a token must be held when the target node does not fire while the source CN did fire to produce a token last instant, or if the token was already held at last instant.Provided the hold signals Consider again Figure 8, we shall name w the schedule of the previous source CN, and w the schedule of the next target CN.After the regular register delay the datas are produce to the FR entry on schedule 0.w (shifted one slot/instant right).The Fractional Register should hold the data exactly when the k th active step at this entry is not the k th activity step at its target CN that must consume it.In other words the FR resynchronize its input and output, which cannot be away be more than one activity step.This last property is true as the schedules were computed using the LID approach with Relay-Stations, which do not allow more than one extra token in addition to the regular one on each arc between Computation or Transportation Nodes.
Stated formally, this property becomes: It says that at a given instant n we should kidnap a value if the number of occurrences of 1 up to instant n on the previous CN is different than the number of occurrences of 1 on the next Computation Node.More precisely, the −|w 0 | 1 term takes care of a possible initial activity at the target CN, not caused by the propagation of tokens from the source CN, that would have to be removed.Correctness properties It can be formally proved that, under proper assumptions, a full RS is sequentially equivalent to a system made of a regular register followed by a fractional one, with the respective stop out and hold signal equated (as in figure 11).The exact assumption is that a stop out/hold signal is never received when the systems considered are already full (both registers occupied in each case).Providing this assumption to a model-checker is cumbersome, as it deals with internal states.It can thus be replaced by the fact that never in history there are more than one val in signal received in excess of the val out signals sent.This can easily be encoded by a synchronous observer.
In essence the previous property states that the two systems are equivalent safe for the emission of stop in on a full RS.This emission can also be shown to be simulated by inserting the previous HOLD component with proper inputs.Of course this does not mean that the implementation will use such a dynamic HOLD pattern, but that simulating its effect (because the static scheduling instructs us of when to generate the signal) would make things equal to the former RS case.

Issues of optimal FR allocation
As already mentioned in the case of a SCC we still do not have a proof that in the stationary phase it is enough to include such elements at the entry points of Computation Nodes only, so they can be installed in place of more Relay-Stations also.Furthermore it is easy to find initialization phases where tokens in excess will accumulate at any locations, before the rate of (the) slowest cycle(s) distribute them in a smoother, evenly distributed pattern.Still we have several hints that partially deal with the issue.It should be remembered here that, even without the result, we can equalize latencies (it just needs adding more FRs).
Definition 13 (Smoothness).A schedule is called smooth if the sequences of successive 0 (inactive) instants in the schedule in between two consecutive 1 cannot differ by more than 1.The schedule (1001) is not smooth since they are two consecutive 0 between the first and second occurrences of 1, while there is none between the second and the third.
Conjecture 1.If all Computation Node schedules are smooth, rates can be equalized using FR only at Computation Node entry points.
Counter example 1.We originally thought that the conjecture 1 should be sufficient, but the counter example of the figure 12  The reason of this failure is that the definition of smoothness is not restrictive enough.In the schedule of the counter-example 12, the pattern 10 is repeated 3 times at the beginning and we have 3 occurrences of 1 (which are not followed by any 0) at the end.0 and 1 are not spread regularly enough in the schedule.However, if the schedule of the CN become (01011011011), we now need only one FR on the non critical cycle.
We propose a new definition: Definition 14 (Extended Smoothness).A schedule w is said extended smooth if any subword, with a length l, contains either n bits at 1 or n + 1 bits at 1, where n is equal to l * |w|1 |w| , |w| 1 is the number of occurrences of 1 in w and |w| is the length of w.

Tool implementation
Our Kpassa tool implements the various algorithmic stages described above.Given that we could not yet prove that FR were only required at specific locations, the tool is ready to insert some anywhere.Kpassa computes and displays the system throughput, shows critical cycles and the locations of choice for extra integer latency insertions in non-critical cycles.It then computes an explicit schedule for each Computation and Transportation Node (in the future it could be helpful to display only the important ones), and provides locations for Fractional Registers insertion.It also provides log information on the numbers of elements added, and whether perfect integer equalization was achieved in the early steps.
In the future, we plan to experiment with algorithms for finding efficient asynchronous transitory initial phases that may reach the stationary periodic regime faster than with the current ASAP synchronous firing rule.
Figure 13 displays a screen copy of Kpassa on a case study drawn from [9].Using the original latency specifications our tool found a static schedule using less resources than the former implementation based on Relay-Stations and dynamic back-pressure mechanisms.And now the activation periods of components are fully predictable.The first examples were built from existing and publicly available models of structural IP block diagrams, but the latency figures were assumed and given plausible values by us (in [11] the authors use a public-domain floorplanner to synthesize approximate latency figures, based on wire lengths induced by the placement of IPs).The last two examples are based on graph shapes and latency distribution that are a priori adverse to the approach (without being formerly worst-cases).
Table 1 provides features of size that are relevant to the algorithmic complexity.Table 2 reports the results obtained, about: whether Perfect Equalization holds; on the number of Fractional Registers required in the initial and periodic phases (note that some FR elements may still be needed for the initial part even in perfectly equalized cases); on the number of integer latencies added; on time and space performances.
The current implementation of the tool is not yet optimized for complexity in time and space, until now this is not yet important.The graph state encoding is naive, and algorithms are not optimal.
Kpassa is a formal tool that is able to compute effectively the length of initialization and periodic pattern, to compute an upper-bound of the number of resources used for the implementation.The tool provides huge preliminary implementations for the static-scheduled LID, but it let us experiment new ideas to optimize those implementations.
In addition to the results shown in the Tables 1 and 2, Kpassa also provides synthetic information on the criticality of Nodes: cycles can be ordered by their rates, and then Nodes by the slowest rate of a cycle it belongs to.Then the nodes are painted from red ("Hotspot") to blue ("Coldspot") accordingly.This visual information is particularly useful before Equalization.Relaxing the firing rule: So far the theory developed here only consider the case where local synchronous components all consume and produce token on all input and output channels in each computation step, and where they all run on the same clock.In this favorable case functional determinacy and confluence are guaranteed, with latencies only impacting the relative ordering of behaviors.So it can be proved that the relaxed-synchronous version produces the same output streams from the same input streams as the fully synchronous specification (indeed the rank of a token in a stream corresponds to its time in the synchronous model, thereby reconstructing the structure of successive instants).Several papers considered extensions in the context of GALS systems, but then ignored the issue of functional correspondence with an initial well-clocked specification, which is our important correctness criterion.This relaxation may help minimize some metrics : -We certainly would like to establish that FR are needed only at Computation Nodes, minimizing their number rather intuitively; -Discovering short and efficient (minimizing number of FR) initial phases is also an important issue here.
-The distribution of integer latencies over the arcs could attempt to minimize (on average) the number of Computation Nodes that are active altogether.In other words transportation latencies should be balanced so that computations alternate in time whenever possible.The goal is here to avoid "hot spots" that is to say flatten the power peaks.It could be achieved by some sort of retiming/recycling techniques and schedules exploration still using a relaxed firing rule; Marked Graphs do not allow control-flow (and control modes).The reason is, in general case such as full Petri Nets, it can no longer be asserted that token are consumed and produced at the same rate.But explicit "branch schedules" could maybe help regulate the branching control parts similarly to the way they control the flow rate; Finally, the goal would be to define a general GALS modeling framework, where GALS components cold be put in GALS networks (to this day the framework is not compositional in the sense that local components need to be synchronous).A system would consist again of computation and interconnect communication blocks, this time each with appropriate triggering clocks, and of a scheduler providing the subclocks computation mechanism, based on their outer main clock and several signals carrying information on control flow.
Summary In this article we first introduced full formal models of Relay Stations and Shell Wrappers, the basic components for the theory of Latency-Insensitive Design.Altogether they allow to build a dynamic scheduling scheme which stalls traveling values in case of congestion ahead.We established a number of correctness properties holding between (lines of) RSs and SWs.
Then, using former results from scheduling theory we recognized the existence of a static periodic schedules for networks with fixed constant latencies.We tried to use these results to compute and optimize the allocation of buffering resources to the system.By equalization we obtain location where a full extra latency is always mandatory (these virtual latencies can later be absorbed in the redesign of more relaxed IP components).Fractional latencies still need to be inserted to provide perfect equalization of throughputs.By simulation we compute the exact schedules of Computation Nodes, and deduce the locations of Fractional Register assignments to support that.We conjectured that under simple "smoothness" assumptions on the token values distribution along graph cycles the FR elements could be inserted in an optimized fashion.We also proved properties on FR implementation, and its relation to RSs.
Finally we described a prototype implementation of the techniques used to compute schedules and allocate integer and fractional latencies to a system, together with preliminary benchmarks on several case studies.

Figure 1 :
Figure 1: (a) An example of CNS (with rectangular Computation Nodes), (b) a corresponding WMG with latency features and token information, (c) a SMG/LID with explicit (rectangular) Transportation Nodes and (oval) places/Relay-Stations, dividing arcs according to latencies, (d) a LID with explicit schedules

Figure 1 (
b) displays a Weighted Marked Graph obtained by adding latencies to figure (a), which can be expanded into the SMG of figure (c).

1 .
The left cycle in Figure (b) has rate 2/2 = 1, while the (slowest) rightmost one has rate 3/5.Throughput is thus 3/5; 2. A single extra integer latency can be added to the link going upward in the left cycle, bringing this cycle's rate to 2/3.Adding a second one would bring the rate to 2/4 = 1/2, slower than the global throughput.This leads to the expanded form in Figure 1 (c); 3. The WMG is still not equalized.The actual schedules of all CN can be computed (using Kpassa, as displayed in Figure (d).

Figure 7 :
Figure 7: An example of WMG where no integer latency insertion can bring all the cycle rates the closest to the global throughput.

Figure 8 :
Figure 8: Fractional Register insertion in the Network.

Figure 9 :
Figure 9: (a) The syncchart, (b) the interface block-diagram of the FR, and (c) the datapath

Figure 10
Figure10shows a possible implementation computing hold from signals that would explicit provide the target and source schedules as inputs.

Figure 11 :
Figure 11: Equivalence of RS and FR roles

Figure 13 :
Figure 13: An example simulation result (MPEG2 Encoder) with Kpassa.In (A), the graph; in (B), the displayed schedules for two vertices was found: Assume a simple graph formed with two cycles sharing one CN.The first critical cycle has 7 tokens and 11 latencies, the second one has 5 tokens and 7 latencies.It exists a stationary phase were the schedule of all CN is smooth (it's[10101010111]or any rotation of this word) but we need two successive FRs on the non critical cycle because only one FR should overflow.
FR Figure 12: Counter example of the conjecture 1. the FR overflow at instant 7.

Table 1 :
Example sizes before equalization

Table 2 :
Equalization performances and results (Run on P4 3.4Ghz, 1GB RAM , Linux 2.6 and JDK 1.5)6 Further TopicsConcerning the static scheduling, a number of important topics are left open for further theoretical developments: