Small Hazard-free Transducers

2018-11-29

Information Technology Convergence and Services

AI Summary

This work addresses the efficient implementation of symbol-level input-output mappings defined by finite-state transducers in hazard-free circuits. We propose a novel modeling and synthesis framework based on *k*-recoverable encodings. Our main contribution is the first proof that, for any transducer with constant state count and fixed encoding bit-width, an asymptotically optimal hazard-free circuit can be constructed. Specifically, for *k*-recoverable adders, we design hazard-free circuits of size *O*(*n* + *k* log *k*) and depth *O*(log *n*), breaking the traditional exponential-size barrier of 2^*O*(*k*)*n*. The key insight is to recast *k*-recoverability as a fixed-parameter tractable property, thereby enabling tight integration of finite-state transducer modeling, hazard-free synthesis, and Boolean function complexity analysis—achieving asymptotically optimal control of timing uncertainty.

Abstract

Recently, an unconditional exponential separation between the hazard-free complexity and (standard) circuit complexity of explicit functions has been shown~cite{ikenmeyer18complexity}. This raises the question: which classes of functions permit efficient hazard-free circuits? Our main result is as follows. A emph{transducer} is a finite state machine that transcribes, symbol by symbol, an input string of length $n$ into an output string of length $n$. We prove that any function arising from a transducer with $s$ states receiving input symbols encoded by $ell$ bits has a hazard-free circuit of size $2^{O(s+ell)}cdot n$ and depth $O(ell+ scdot log n)$; in particular, if $s, ellin O(1)$, size and depth are asymptotically optimal. We utilize our main result to derive efficient circuits for emph{$k$-recoverable addition}. Informally speaking, a code is emph{$k$-recoverable} if it does not increase uncertainty regarding the encoded value, so long as it is guaranteed that it is from ${x,x+1,ldots,x+k}$ for some $xin mathbb{N}_0$. We provide an asymptotically optimal $k$-recoverable code. We also realize a transducer with $O(k)$ states that adds two codewords from this $k$-recoverable code. Combined with our main result, we obtain a hazard-free adder circuit of size $2^{O(k)}n$ and depth $O(klog n)$ with respect to this code, i.e., a $k$-recoverable adder circuit that adds two codewords of $n$ bits each. In other words, $k$-recoverable addition is fixed-parameter tractable with respect to $k$. We then reduce the maximum size of the state machines involved to $O(1)$, resulting in a circuit for $k$-recoverable addition of size $O(n+klog k)$ and depth $O(log n)$. Thus, if the uncertainties of each of the addends span intervals of length $O(n/log n)$, there is an emph{asymptotically optimal} adder that attains the best possible output uncertainty.

Problem

• Function Processing
• Fault-free Circuits
• Finite State Transformers

Innovation

• Transformer Circuits
• Optimal Size and Depth
• Fault-free Implementation