The Parallelism Tradeoff: Limitations of Log-Precision Transformers

This work aims to characterize the computational model implicit in transformer neural networks (Vaswani et al., 2017), which form the basis of recent breakthroughs in large language models such as BERT (Devlin et al., 2019), T5 (Raffel et al., 2020), and GPT-3 (Brown et al., 2020). What computational primitives can the transformer’s components implement, and what problems can the full system solve in aggregate? These questions are important for interpreting transformers in a principled way, understanding potential limitations of their reasoning capabilities, and building trust in deployed transformer-based systems.

Early theoretical work on transformers established their Turing completeness, albeit with assumptions like infinite precision and arbitrarily powerful feedforward subnets (Pérez et al., 2019; Dehghani et al., 2019). On the other hand, a strand of more recent work uses techniques from circuit complexity theory to derive strong limitations on the types of problems transformers can solve given restrictions on the form of attention allowed in the transformer. Specifically, Hahn (2020) and Hao et al. (2022) showed that transformers restricted to hard attention are very limited: they can only solve problems in a weak complexity class (non-uniform AC⁰) that doesn’t even contain basic problems like majority of n bits. Merrill et al. (2022) extended this to a more general class of “saturated attention” transformers with a floating point datatype, and showed a larger class of problems (non-uniform TC⁰) as an upper bound. This motivates analyzing a setting that strikes a middle ground: Can we characterize transformers whose precision and feedforward nets’ computational power are realistically bounded, but where attention is also realistically expressive?

An important practical limitation of these prior results is the “non-uniform” nature of the considered circuit classes, which makes these classes non-realizable and the findings difficult to interpret. This is because non-uniform AC⁰ and TC⁰, while highly limited in computation, also contain some problems that are not even decidable, i.e., for which there doesn’t exist any exact algorithm. Thus, non-uniform classes cannot be directly compared with standard algorithmic complexity classes such as P, NP, etc. This motivates our second key question: Can we derive uniform upper bounds on transformers?

We show that one can achieve both of these goals by making the modest assumption that all values in the transformer have $O(logn)$ precision (where n is the number of input tokens), and, similarly, that transformer’s subnetworks are computable in $O(logn)$ space. Log precision is enough to represent the positional encodings at the input layer of the transformer, and to encode pointers to all other positions in the sequence at later transformer layers. Assuming log precision across all layers captures the idea that the hidden representations contain a constant number of hidden states whose precision (16 or 32 bits) is small relative to the length of the input (2048 in GPT-3). On long sequences, the precision will not be enough to losslessly encode the full input sequence into a single vector. Instead, the processing of the sequence must somehow be distributed in each layer and performed in parallel.

Our main contribution is proving that log-precision transformers can be simulated by uniform constant-depth threshold circuits. Thus, such transformers can only solve problems in uniform TC⁰. This characterization is strikingly weak compared to the Turing-completeness of infinite-precision transformers. Since we believe log precision is more realistic for practical transformers than infinite precision, these results point to the conclusion that transformers are not Turing-complete in practice.

In contrast to past results, our upper bound on transformers is a uniform circuit class, enabling direct comparison of log-precision transformers to many natural complexity classes. These connections reveal specific problems that define the upper limits of log-precision transformers’ capabilities, as discussed further in §2.

Intuitively, our upper bound says that log-precision transformers are computationally shallow, and that this shallowness can be understood to emerge from their parallelizability. Transformers’ inherent parallelism is useful for training them efficiently at massive scale, but may limit the complexity of the computations they can express. We introduce the term parallelism tradeoff to capture this idea, which represents a potential fundamental weakness of the current paradigm of scaling language models. Formally characterizing reasoning capabilities relevant to language models and understanding whether they likely fall outside upper bounds implied by the tradeoff would clarify the practical implications of this limitation of scaling.

It could also be that the limitations of parallelism are not a curse but a blessing, if they constrain the hypothesis space in a way useful for learning. We have no evidence that this is true, but mention it as an alternate interpretation of the results that could be clarified in future work.

We also consider an instruction following setting (Brown et al., 2020) where the transformer is provided the description of a task along with an input on which to execute the instruction. We construct a practically parameterizable transformer that can execute instructions perfectly if they are provided in the form of TC⁰ circuits. This complements recent work that studies transformers’ ability to follow other forms of instructions such as regular expressions (Finlayson et al., 2022).

Based on the fundamental property that transformers can correctly evaluate any given TC⁰ circuit on a given input, we introduce the notion of advice transformers akin to advice-taking Turing machines. We show that transformers can recognize any (non-uniform) TC⁰ language if provided appropriate poly-size advice.

In summary, our findings provide new insights on both the abilities and the limitations of transformers, and bring out bounded precision, threshold computations, and parallelism as key notions for understanding the implicit computational model of transformers in practice.

Before diving into technical details, we discuss in §2 the implications of our results on both fundamental as well as practical abilities of transformers. §3 provides a brief primer on circuits as a model of computation. It then discusses a way of serializing a circuit into a string; we later show how to generate such serializations using a resource-bounded algorithm, which is the key to proving containment of transformers in uniform circuit classes. §4 defines our formal model of bounded-precision transformers. §5 derives our first formal bound on log-precision transformers. This bound involves non-uniform circuit families, similar in spirit to prior results in this area. §6 proves our more technical main result: the first uniform circuit complexity upper bound for transformers (specifically, uniform TC⁰). Finally, §7 provides a lower bound on transformers, introduces the notion of an Advice Transformer, and connects these to the machine learning problems of Instruction Learning and Following.

Before diving into technical details, we discuss the general implications of our findings on the abilities and limitations of transformers. We will focus here on our main result (Theorem 2), which shows that log-precision transformers are in the complexity class logspace-uniform TC⁰.

One interpretation of complexity classes such as NC⁰, AC⁰, and TC⁰ is sets of poly-time solvable problems that are parallelizable to a very high degree—they can be solved in parallel in constant time with enough parallel processors. This gives some intuitive explanation of our result: log-precision transformers end up in TC⁰ because they were designed to be highly parallelizable. Since parallelism is an important property of today’s dominant paradigm of training models at massive scale, this points to the conclusion that any massively scaled up model—transformer or otherwise—will likely obey restrictions similar to the ones derived here for log-precision transformers. There is thus an important tradeoff between the massive parallelizability of today’s networks and their representation power.

Our result places log-precision transformers in the complexity class logspace-uniform TC⁰. This has immediate implications on the kinds of problems such transformers can and cannot accurately solve.

Consider any problem X that is complete for a complexity class C that contains logspace-uniform TC⁰. By definition of completeness, every problem log-precision transformers can solve perfectly is efficiently reducible to X and is thus no harder than X. This implies that—despite their massive size—the computation performed by such transformers is, for instance, no harder than solving basic L-complete problems like graph connectivity: the problem of checking whether there is a path between two nodes in an undirected graph (Lewis and Papadimitriou, 1982; Reingold, 2008).

By the same token, if C is strictly larger than logspace-uniform TC⁰, then such transformers cannot perfectly solve X. Thus, log-precision transformers cannot perfectly solve the following reasoning problems:

• Linear equalities: find x s.t. Ax = b¹

• Universal context-free recognition¹,²

• Propositional satisfiability (SAT)³

• Horn-clause satisfiability (HORN-SAT)¹

• AI planning (Bylander, 1991)

• Permanent computation⁴

This highlights the limits of practical transformers with limited-precision arithmetic, indicating that they are far from being universal or all-powerful as suggested by some prior studies.

One important caveat about these negative results is that they are asymptotic in nature—they apply for “large enough” input size n. It’s possible for log-precision transformers to solve such problems easily when n is small. Further, these negative results are about exact solutions, but they often also extend beyond this when formal hardness-of-approximation results are known.

Prior formal characterizations of transformers either make unrealistically strong assumptions (Pérez et al., 2019; Dehghani et al., 2019) or place unrealistic restrictions (Hahn, 2020; Hao et al., 2022; Merrill et al., 2022). In contrast, we make only one assumption—namely, all intermediate values in the transformer are limited to $O(logn)$ bits, where n is the number of input tokens. We next discuss some implications of this assumption and what our findings mean for practical transformers.

As mentioned above, our bounds are asymptotic in nature and thus apply when n is sufficiently large. In practice, transformers use fixed precision at each computation node, which is more restrictive than precision growing with the input sequence length n, as $O(logn)$ bits. However, this constant could be large and thus, for relatively small n, our results do not rule out practical transformers solving difficult problems. Our results, however, do show that as n grows sufficiently large, log-precision transformers are fundamentally limited to problems within TC⁰ and cannot accurately solve various commonly studied problems mentioned earlier under “What Transformers Cannot Compute”. Extending our analysis to small n will help close the gap to practice.

Our formal model is based on a binary classification view of transformers. However, our results apply directly to multi-class classification as well and can be extended to generation problems by viewing, for instance, next word prediction in NLP as a multi-class classification problem. However, if the transformer decoder is allowed to condition on its previous output in a generation problem, then this would violate our formal setup.

Elhage et al. (2021) propose extracting circuits⁵ that capture the computational structure of transformers. Our results suggest threshold circuit families are a good formalism for expressing mechanisms extracted from transformers. Constructively converting transformers to threshold circuits is beyond the scope of the current paper, although we hope to explore this in more detail in future work.

Theorem 2 also motivates a paradigm for quickly testing complexity theory conjectures. If a problem is believed to separate TC⁰ and NC¹, a transformer can be trained on problem instances. If the transformer generalizes perfectly to harder instances than it was trained on, this gives an empirical hint that the problem is in TC⁰, providing evidence against the conjecture.

Let {0,1}^* be the set of finite binary strings. For x ∈{0,1}^*, let |x| be its length. We refer to a function from {0,1}^* to {0,1}^* as a Boolean function. Boolean functions can implement arithmetic operations if we define a semantics for binary strings as numbers. We will treat the intermediate values in a transformer as binary strings, and the internal operations as Boolean functions.

Circuits are a model of computation for computing Boolean functions of fixed-length binary strings.⁶ Formally, a circuit is a directed acyclic computation graph. The leaf nodes represent binary variables and their negations. The internal nodes represent functions in some set $G$⁠, and the directed edges represent the flow of function outputs into inputs of other functions. One or more nodes in the circuit are marked such that their value is the output of the circuit.

The size of a circuit is the total number of gates in it, including negation. The depth of a circuit is the length of the longest path from any input node to any output node.

A circuit family generalizes a circuit to take variable-length binary strings as input. Formally, a circuit family is a sequence of circuits C_n : {0,1}ⁿ →{0,1} for n ∈ℕ. A circuit family implicitly recognizes a formal language defined as follows:

We now define classes of languages by constraining the complexity of the circuit families needed to recognize them:

For k ∈ ℕ, a threshold gateθ_≤k takes m input bits and returns whether $∑i=1mxi≤k$⁠. We define θ_≥k analogously. For example, θ_≤3(110011) = 0.

The gates ¬, ∧, and ∨ are all just special cases of thresholds, so we can imagine TC⁰ circuits to have access to these as well. Thus, TC⁰ circuits can implement AC⁰ circuits.

We identify a circuit with its serialization in a formal language that identifies each node’s label and adjacency list. We will adopt a specific grammar for concreteness, but our construction can be adapted to other string representations of circuits.

We define a circuit serialization as a traversal of a circuit ordered by some topological sort. In this serialization, leaf nodes (variables) are represented by the string X. An internal node (gate) is represented in Polish notation by the function it computes (AND, OR, or NOT) followed by a list of pointers to its arguments. Each argument &1^j of gate i encodes (in a unary) a zero-indexed pointer to the j-th gate in the circuit, where j < i. The final node is interpreted as the circuit output.

The circuit families we have defined above are non-uniform, meaning that we do not enforce that the circuits processing different input sizes must be related in any way. In degenerate cases, non-uniform circuit families can solve undecidable problems⁹ because they have infinite description length, making them a physically unrealizable model of computation. Complexity theorists have thus introduced uniform circuit families. Uniform circuit families are a realizable model of computation with relations to classes in computational complexity and formal language theory.

Intuitively, in a uniform circuit family, the circuits for different input sizes must be “somewhat similar” to each other. We formalize this (cf. Arora and Barak, 2009) by saying that there exists a resource-constrained Turing machine that maps the input 1ⁿ to a serialization of circuit C_n.

This notion of uniformity is more general than the standard notion in that the input size I(n) is a function of the problem complexity n. The reason for this is that we will apply uniformity to subcomputations with different input sizes I(n) within a larger computation of input size n. The standard notion of uniformity corresponds to I(n) = n.

Furthermore, we will refer to a circuit family as uniform if it is uniformly computable with $S(n)=O(logn)$ (cf. Arora and Barak, 2009). We can define uniform versions of AC⁰ and TC⁰ by adopting the previous definitions exactly, but also enforcing uniformity. For the rest of the paper we will clarify whether we mean the uniform or non-uniform variant of TC⁰ when unclear from context, since both classes will come up.

A transformer (Vaswani et al., 2017) is a neural network architecture made up of a constant number of transformer layers. A transformer layer is a module that computes self-attention over a sequence followed by an elementwise transformation of the output vectors.

We will assume that each transformer is resource bounded in terms of the precision of each value it computes and, for some of our results, the space it uses for the computation of key operations such as embedding, attention, and activation. Specifically, we will assume precision p, i.e., the values at all layers, as well as the outputs of all key intermediate operations in it (attention, activation, arithmetic operators, etc.), are represented using p bits. This is a realistic assumption as, in practice, today’s transformers are typically limited to the 64-bit precision of the underlying hardware. Formally, we define p-precision as follows:

This says the size of the function input and output are bounded below p. Similarly, the intermediate space used by the computation must also be bounded below p. Thus, higher precision computations cannot somehow be hidden inside f.

Definition 6 naturally applies to functions with bounded arity k. We will also need to define p precision for the summation operator in the transformer, which adds n different floats of size p.¹⁰ Adding n floats can blow up the precision needed to represent their sum. For example, imagine adding the floating points 1 · 2⁰ + 1 · 2^c. We obtain (2^c + 1) · 2⁰, whose mantissa takes c + 1 bits to represent. In practice, computers do not preserve full precision in such situations: instead, small terms like 1 · 2⁰ are discarded. Thus, we define the transformer’s addition operation ⊕ to be similarly approximate (and thus preserve precision); see §A.

The core building block of a transformer is an attention head. We define this at a high level of abstraction as follows:

Let h₁,…,h_n ∈{0,1}^p be the input sequence to a p-precision attention head, and let ⊕ be approximate floating-point addition (§A).

Standard transformer attention heads (Vaswani et al., 2017) are a special case of this definition where s is scaled dot-product similarity between keys and queries. Standard transformers also have a linear or affine value function applied to each $hjℓ$ in the sum over j. By its affineness, the value function can, without loss of generality, be removed from the attention head and considered to be part of the transformer layer (i.e., applied to the output of the attention head).

A p-precision transformer layer is then a tuple of heads and a function f used to combine them.

A p-precision transformer layer can be understood to define a sequence of vectors $h1ℓ+1,…,hnℓ+1$ in terms of an input sequence of vectors $h1ℓ,…,hnℓ$ (coming from the previous layer in the transformer) by first computing k attention heads in parallel and then combining their output using f. The first k inputs to f will correspond to the attention head outputs, and the additional input is the original input from the previous layer. Recall that $aihℓ+1$ is the output of head $Hihℓ+1$ on input h^ℓ at position i. The function computed by a transformer layer can be described formally as follows.

f can be understood to encapsulate layernorm, residual connections, and the feedforward sublayer of a standard transformer (Vaswani et al., 2017). $hiℓ$ is given to f to allow residual connections. As mentioned in §4.3, f can also encapsulate the value function for each head.

Finally, we define a transformer of depth d as a cascade of d transformer layers:

For a position embedding function ϕ and w ∈ Σⁿ, let ϕ(w) be the position-wise broadcasted embedding of w: for 1 ≤ i ≤ n, $ϕi(w)≜ϕ(wi,i)$⁠.

We will use n to denote the length of w, and take the transformer’s depth d to be fixed w.r.t. n.

The input to the transformer can thus be represented with $N=nlog∣Σ∣$ bits using a binary encoding for the vocabulary. The circuits we construct in subsequent sections to simulate transformers will also have input size N. We will assume transformers have log-precision relative to the size of the input, specifically, $O(logN)$-precision. Since |Σ| is fixed (typically 30000 in practice), we will think in terms of $O(logn)$-precision. Thus, by Definition 6, all of the intermediate functions of such transformers are computable in $O(logn)$ space and output (at most) these many bits. Note that this is enough precision to represent positional encodings and for each position to point to a constant number of other values, but not enough precision for non-lossy pooling of the entire input into a single value.

Our log-precision transformers do not enforce that s (Definition 7) and f (Definition 9) follow the transformer structure. However, a feedforward net whose primitive operations (e.g., scalar multiplication) are defined over $O(logn)$-size numbers can be computed in $O(logn)$ space. Thus, bounded-precision practical transformers are a special case of our log-precision transformers. This makes our setup appropriate for proving upper bounds on transformers, which is our main contribution.

We first show that log-precision transformers can be simulated by non-uniform threshold circuits, before presenting the more technical uniform version of the results in §6. The initial non-uniform result extends the findings of Merrill et al. (2022), who showed that saturated attention transformers¹² can be simulated in TC⁰. Here, we remove the simplifying saturated attention assumption and other restrictions on the underlying datatype. Instead, we show that our log-precision assumption is enough to prove that a transformer can be simulated in TC⁰ with any attention function.

Hao et al. observed that any Boolean function of $O(logn)$ bits can be computed by a poly(n) size circuit. We extend this to m-bit outputs, which is both more convenient and more efficient than constructing m separate Boolean circuits:

We now use Lemma 1 to prove the following non-uniform result. We note that the proof goes through even if the notion of p-precision (Definition 6) is relaxed to not require computability in space p. This requirement will, however, become important for our subsequent result in §6.

We will now extend the argument from the last section to show that $O(logn)$-precision transformers can be simulated by uniform constant-depth threshold circuits by capitalizing on the assumption that ϕ, s, and f are log-precision, and thus can be computed in $O(logn)$ space. The overall proof idea is similar, but due to the uniformity condition, the proof becomes substantially more technical. We must not just show the existence of a threshold circuit family computing a transformer, but also show that this circuit family can be generated by a log-space Turing machine.

We first extend Lemma 1 to respect uniformity:

We can leverage this lemma to derive the uniform analog of Theorem 1, as follows.

Because the depth derived in Theorem 2 is constant with respect to n, it follows that:

So far, we have shown that uniform TC⁰ is an upper bound for log-precision transformers. Is this upper bound tight, i.e., also a lower bound? While we do not answer this question here, we address a related question as a first step: we construct a transformer that can evaluate TC⁰ circuits on binary inputs, showing that transformers can compute any TC⁰ function when their input is augmented with the right “instructions”.

More formally, we consider the Circuit Value Problem (CVP) (Ladner, 1975), also referred to as the Circuit Evaluation Problem, where the input is a Boolean circuit C and a string x ∈{0,1}ⁿ, and the task is to return the value of C(x) ∈{0,1}. This problem is known to be complete for the class P under AC⁰ reductions (Ladner, 1975). We will assume C is serialized as described in §3 and prove that log-precision transformers can evaluate any TC⁰ circuit. Note that this is an extension of the typical CVP since the circuit has threshold gates, not just standard AND/OR gates.

It is known that LSTMs cannot evaluate Boolean formulae (Merrill, 2020), a special case of the CVP. In contrast, we show that transformers can.

To demonstrate the practicality of our lower bound construction, we will not just prove the existence of transformers that can evaluate TC⁰ circuits but also specify concrete choices for the positional embedding scheme and the class of attention functions that are sufficient to do so.

After normalization, saturated attention creates a distribution that is uniform over a subset of positions. Thus, it is capable of parameterizing hard attention, uniform attention over the full sequence, and various attention patterns in between.

For simplicity, we assume pooling functions f are thresholded linear functions of their inputs. Thus, they could be implemented by a feedforward neural net. Without loss of generality, we let attention heads have a value function, which can be folded into the pooling function from the last layer (see §4).

Now, we are ready to present the main result. Our construction below is specific to the circuit serialization scheme discussed in §3, but can be extended to other serializations as well.

CVP is closely related to instruction learning (Brown et al., 2020) and instruction following tasks (Finlayson et al., 2022). The latter task setup provides a transformer two inputs: a regular expression r as an “instruction”, and blackblackz ∈{0,1}^*. The goal of the task is to return whether z belongs to the regular language represented by r. Viewed from this lens, the circuit evaluation setup asks: Can transformers follow instructions provided in the form of a circuit? As discussed below, our result says the answer is yes for all constant depth threshold circuits. This, to the best of our knowledge, provides the first non-trivial lower bound for transformers in the instruction learning setting.

Formally, an instruction I is any description¹⁷ of a function f_I of {0,1}^*. We say a transformer correctly follows an instruction I if, for all x ∈{0,1}^*, it correctly computes f_I(x) on input $〈I,x〉$⁠. A non-uniform instruction description is a family of length-specific descriptions ${In}n=1∞$⁠. We say a transformer correctly follows a non-uniform instruction family {I_n} if, for all n and all x ∈{0,1}ⁿ, it correctly computes f_I(x) on input $〈In,x〉$⁠. The non-uniform description {I_n} may take any form. When it forms a TC⁰ circuit family, we refer to it as a TC⁰ instruction description. Since Theorem 3 constructs a transformer that can evaluate any TC⁰ circuit, it follows that:

Thus, transformers with simple position embeddings, attention, and pooling functions can simulate any instruction provided in the form of a TC⁰ circuit. We note that while it is unknown whether the class of regular languages, considered by Finlayson et al. (2022), is contained in TC⁰, the other side is known: There are problems computable by TC⁰ circuits that are not computable by a regular language. These include problems involving counting and arithmetic, which are beyond regular languages. Our results thus expand the known kinds of instructions transformers are able to follow, at least with hand-constructed weights.

We can also view circuit evaluation abilities of transformers (Lemma 3) from the lens of advice-taking Turing machines which, in addition to their usual input, are also provided an input length dependent (but input independent) advice string. For instance, P/poly is the class of problems decidable in polynomial time when the Turing machine is given an advice string of size polynomial in the input length (cf. Arora and Barak, 2009).

In the same vein, let T/poly be the class of log-precision, constant-depth transformers with polynomial advice strings. In other words, on an input of size n, we allow the transformer to receive an additional poly(n) bits of input that cannot depend on the standard input. Now let ${Cn}n=1∞$ be a circuit family demonstrating that a problem is in non-uniform TC⁰. Then, by passing the description of C_n as advice for input length n, it immediately follows from Lemma 3 that advice transformers can simulate non-uniform TC⁰:

Since non-uniform TC⁰ even contains some undecidable languages (Arora and Barak, 2009, Claim 6.8), T/poly is clearly a very powerful class and a strict superset of T, the class of decision problems recognized by transformers (which are all decidable). Thus, a problem in T/poly cannot always be solved by a transformer on its own. However, if given a description of how to do so (“advice”) in the form of a TC⁰ circuit, our result shows that a transformer could solve that problem.

Answering two open questions from Merrill et al. (2022), we prove log-precision transformers with any (including soft) attention can be simulated by uniform constant-depth threshold circuits. This establishes thresholded addition as a fundamental operation for understanding the computational model of transformers: Any log-precision transformer can be re-expressed as a polynomial number of threshold gates stacked to a constant depth. This result also establishes potential limits on the computational power of log-precision transformers; e.g., if L ⊂P, transformers cannot compute all poly-time functions. They are certainly very far from being universal. The intuition at the heart of this result is that forcing a model to be highly parallelizable likely sacrifices its expressiveness. Since parallelism seems essential to pretraining any massive model at scale, any large language model—transformer or otherwise—may suffer from a similar tradeoff.

The authors are grateful for the valuable feedback from the anonymous reviewers and the TACL action editor. They also thank Paul Beame and colleagues at AI2 including Kyle Richardson, Michal Guerquin, Peter Clark, Tushar Khot, and especially Matthew Finlayson, whose empirical findings about instruction learning inspired §7. Feedback from Sam Bowman, Arya McCarthy, Roma Patel, and Lena Strobl, and discussions with the FLaNN, ML for Code (MILA), and Foundations of Language Processing (Umeå) research groups helped improve earlier drafts. The authors also appreciate Rahul Santhanam’s feedback. This work was funded in part by NSF award 1922658. William Merrill was supported by an NSF graduate research fellowship and by AI2.

We interpret a p-bit string x as a p-precision float by taking the first p/2 bits¹⁸ of x as a signed integer m encoding the mantissa and the remaining p/2 bits of x as another signed integer e encoding the exponent. A float with mantissa m and exponent e, denoted $〈m,e〉$⁠, encodes m · 2^e.

Computing the sum of n n-bit integers (known as iterated addition or simply summation) is well-known to be in uniform TC⁰ (Hesse, 2001; Chiu et al., 2001). We leverage this fact to show that the same holds for the sum of n$O(logn)$-precision floats. A subtlety of adding p-precision floats is that their sum can require more than p bits to represent precisely as a float. For instance, while each of 2^r and 1 is representable with a (signed) mantissa of only 2 bits, their exact sum, 2^r + 1, requires a mantissa of r + 1 bits. Hence, p-precision transformers must sacrifice some precision when performing summation.

We define float addition by mapping the floats to integers, adding the integers exactly, and then mapping the sum back to a float (with possible loss of precision). Let $Iqmax=2q−1$ be the greatest q-bit signed integer, and $Iqmin=−Iqmax$⁠. Let $Fpmax$ be the greatest value representable by a p-precision float. Since the exponent of a float ϕ can be negative and represent a fraction, we rescale ϕ by $2−Ip/2min$ when mapping it to an integer g_p(ϕ):

We first verify that Definition 14 closely approximates exact addition.

Finally, we show that, with log precision, computing ⊕ (Definition 14) is in uniform TC⁰.