Combinationality of cyclic definitions

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Combinationality of cyclic definitions

• EECS 290A Spring 2005
• UC Berkeley

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Outline

• Beyond ternary-simulation formulation?
• Cyclic definitions
• From circuits to functions
• From hardware to software
• When cyclic definitions behave combinationally
• Validity of functional-level analysis
• From combinational to sequential sequential
  determinism

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Circuit-level analysis

• Advantage
• Simple (compared to whats coming)
• Weakness
• Analysis at low abstraction level (after circuit
  netlists are derived)
• Not combinational! Now what?
• Combinationality depends on circuit structures
• The schizophrenia problem in Esterel compilation

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Escape the trap!

• Is ternary-simulation formulation the only
  solution?
• What if cyclicities are to be broken
• What if synthesis targets are software

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Functional-level analysis

• Why ?
• Analysis at circuit level
• Too conservative
• when cyclic definitions are to be broken
• when synthesis targets are software
• At very low level
• needs to translate high-level descriptions into
  gate-level representations
• yields inconsistent conclusions about
  combinationality
• Analysis at functional level
• admits more general combinationality formulation
• avoids the translation from functional to gate
  level representation
• yields consistent conclusions about
  combinationality
• How ?

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Different levels of abstraction

• Distinctions between functional and circuit
  levels of abstraction
• Functional level
• Valuations take no time (timing information is
  abstracted away)
• Circuit level
• Gates and wires are associated with delay (in
  particular, under the UIN delay model)

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Cyclic definitions

• Example
• D a x a ? c, b x (a ? b) ? c, c x
  b, y x (a ? a b) ? x (a c ? a c)
• Input I x, output O y, internal a, b,
  c
• Break cyclic definitions of D w.r.t. a minimal
  cutset C a, b
• Excitation functions
• a x a ? x b, b x (a ? b) ? x b
• Observation function
• y x (a ? a b) ? x (a b ? a b)

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State evolution graphs

• Any SEG is defined for some input assignment
• Example (contd)
• Excitation functions
• a x a ? x b,
• b x (a ? b) ? x b
• Observation function
• y x (a ? a b) ? x (a b ? a b)

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Facts about SEGs

• An SEG is defined for some fixed input assignment
  
• Every state in an SEG has exactly one outgoing
  edge
• Any state is either in a loop or on a path
  leading to a loop
• Any two loops of an SEG are disjoint

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Combinationality at functional level

• Definition. Combinationality A set D of cyclic
  definitions w.r.t. some cutset C is combinational
  at the functional level if, for any input
  assignment, all states in loops of any SEG have
  the same output observation.
• Example (contd)
• I x, O y, C a, b
• a x a ? x b
• b x (a ? b) ? x b
• y x (a ? a b) ? x (a b ? a b)

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Combinationality test

• Algorithm combinationality test
• Compute states in loops of SEGs by a greatest
  fixed-point computation (iterative image
  computation w/o the quantification of input
  variables)
• Assert the states in loop of any SEG have one
  observation label (apply cprojection operation
  Lin Newton 91)

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Compute states in loops

• Let L denote the set of states in loops and T be
  the transition relation for SEGs
• The greatest fixed-point computation corresponds
  to
• Let L0(x,cs) 1 (i.e. all states for any
  input)
• Repeat
• Li1(x,cs) ?cs. Li(x,cs) ? T(x,cs,ns)
  ns?cs
• Until Li1 Li
• Return L

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Assert consistent observations

• Let O(x,cs,o) be the observation relation and
  ?(x,o) ? ?cs. L(x,cs) ? O(x,cs,o) .
• The set of cyclic definitions is combinational
  iff ?(x,o) cprojection(?(x,o),a), where a is an
  arbitrary minterm in the o-space

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Cycle breaking

• Algorithm cycle breaking
• From combinationality test, rewrite definitions
  of PO variables as functions of PI variables.
• Or, rewrite definitions of cutset variables as
  functions of PI variables.

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Combinationality at functional level

• Theorem. The combinationality analysis is
  independent of the choice of minimal cutsets.

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Combinationality at functional level

• Theorem. Generality There exists a feasible
  combinational implementation of a set D of cyclic
  definitions if, and only if, D satisfies the
  combinationality test.
• The combinationality formulation is the most
  general

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Combinationality at functional level

• Theorem. Complexity Combinationality analysis
  at the functional level is PSPACE-complete in the
  cutset size.
• Thus the applicability of the analysis strongly
  depends on the cutset size
• Notice the freedom of choosing minimal cutsets

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Validaty of combinationality analysis

• Conditions of legitimacy
• Cyclicities are to be broken, or
• Synthesis targets are software
• Cyclic definitions can be maintained
• Assume that execution follows the evolutions in
  SEGs

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Ternary simulation at functional level

• If all valuations must stabilize
• For any input assignment, there exists a set of
  signals valuating to either 0 or 1 such that all
  cyclic definitions are broken
• Correspondingly, every SEG G has a single
  self-loop state to which all other states of G
  evolve
• If only output signals must stabilize
• Partition input assignments into two sets one
  fully and the other partially determines output
  valuations w/o valuating internal signals
• The first set imposes no restriction on SEGs
• The second set imposes the same restriction as
  the previous case (all valuations must stabilize)
• Therefore, ternary simulation at functional level
  is very restricted compared to the introduced
  analysis

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Some extensions

• Stable cyclic definitions
• Loops of length greater than one are unstable
• Undesirable because of dynamic power dissipation
  (though valid for software synthesis)
• Rewrite definitions to consist of only stable
  loops
• Input-output determinism of state transition
  systems
• Extend combinationality analysis to systems with
  state-holding elements (c.f. GMW analysis used in
  Shiple 96)

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Sequential determinism

• For an FSM to behave deterministically in IO, it
  is allowed to have non-deterministic transition
  functions
• The cyclic definitions can be more flexible (in
  observations induced by the transition functions)

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Summaries

• What we learned
• Combinationality formulation at functional level
• Valid if cyclicities are to be broken, or
  synthesis targets are software
• Combinationality analysis and the rewrite
  procedure to make cyclic definitions acyclic
• Sequential determinism

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References

• Lin Newton 91 B. Lin and A. Newton. Implicit
  manipulation of equivalence classes using binary
  decision diagrams. In Proc. ICCD, pp.81-85, 1991.
• JMB 04 On breakable cyclic definitions. In
  Proc. ICCAD, 2004.