Summary Logic for Log-Precision Transformer Models arxiv.org
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One Line
Researchers propose the logic FO(M) as a new way to express computations performed by transformer models, which is more powerful than previous logics and can provide insights into how transformer models perform their computations.
Key Points
- FO(M) logic can express the computations performed by transformer models and handle a wider range of attention patterns.
- Log-precision transformers cannot express uniform attention patterns, which are a core algorithmic primitive of transformers.
- A method for constructing log-precision transformer models using a block mapping algorithm is presented.
- Log-precision transformer models rely on addition, conditional branching, and a finite number of functions computable in time O(log n).
- Affine transformations, layer normalization, and the output classifier head can be computed by log-uniform TC0 circuit families.
Summaries
225 word summary
This paper presents the logic for log-precision transformer models, analyzing the uniformity of common neural net building blocks within the transformer. Affine transformations, layer normalization, and output classifier head can be computed by log-uniform circuit families. The article discusses the use of log-precision transformer models in neural sequence models, covering their limitations and theoretical limitations of self-attention. A method for constructing log-precision transformer models using a block mapping algorithm is presented. The paper establishes the equivalence of log-uniform circuits and FO(M) and suggests the possibility of translating real transformers to FO(M) sentences and establishing a hierarchy theorem describing the FO(M) quantifier depth needed to simulate a TC 0 family of a certain size. Researchers have developed a logic called FO(M) that can express computations performed by transformer models. FO(M) is more powerful than previous logics and handles a wider range of attention patterns. The authors propose FO(M) as a new logic that allows majority quantifiers and can express any function computed by a log-precision transformer. The paper discusses the limitations of log-precision transformer models and the importance of uniform attention. Researchers raise concerns about the safe deployment, fairness, and accountability of increasingly complex transformer models. FO(M) has the advantage of being mechanistically interpretable, providing insights into how transformer models perform their computations. The researchers believe their work could guide the design of "transformer-complete" programming languages.
699 word summary
Researchers have developed a logic, called FO(M), that can express the computations performed by transformer models. FO(M) is more powerful than previous logics used for this purpose and can handle a wider range of attention patterns. The authors demonstrate that finite-precision transformers cannot express uniform attention patterns, which are a core algorithmic primitive of transformers. The authors propose a new logic, FO(M), which allows majority quantifiers and can express any function computed by a log-precision transformer. They provide an upper bound and first logical characterization of log-precision transformers. The paper raises concerns about the safe deployment, fairness, and accountability of increasingly complex transformer models with hundreds of billions of parameters. FO(M) has the advantage of being mechanistically interpretable, meaning that it can provide insights into how transformer models perform their computations. The researchers believe that their work could guide the design of “transformer-complete” programming languages. This paper discusses the logic behind log-precision transformer models, which are transformers where the weights are fixed and the precision used to compute the forward pass is at most O(log n). The authors propose a logic for such models, including various examples of languages that can be expressed in first-order logic with the addition of certain predicates. The document discusses the limitations of log-precision transformer models and the importance of uniform attention. Theorem 2 states that any computation of a transformer on input x can be reduced to a single integer division or a finite number of Dyck-language queries. Proposition 1 analyzes a model of transformers where the precision depends on the context length. Computation graph families and threshold circuits are also discussed as useful tools for processing inputs of arbitrary length. A logic for log-precision transformer models is presented in this paper. The main theorem shows that any log-precision transformer can be expressed in FO(M) and can be simulated by a TC 0 family C that is both a power of 2 and computable in time O(log n). A method for constructing log-precision transformer models using a block mapping algorithm is also presented. The algorithm defines a circuit family simulating the computation graph family and enforces that the block size of each node matches the size of the circuit for that node. The paper establishes the equivalence of log-uniform circuits and FO(M), proving that any log-precision transformer can be translated to an FO(M) sentence that computes the same function as the transformer. The authors also prove that a transformer can be simulated by a log-uniform TC0 family C that obeys the size and depth properties of Theorem 3. The article suggests the possibility of translating real transformers to FO(M) sentences and establishing a hierarchy theorem describing the FO(M) quantifier depth needed to simulate a TC 0 family of a certain size. The article also suggests the potential for proving that any transformer can be simulated by an FO(M) sentence of quantifier depth of at most 2. This document discusses the use of log-precision transformer models in neural sequence models, specifically focusing on the practical computational power of finite precision systems and the effects of parameter norm growth during transformer training. The article covers the limitations of log-precision transformers and the theoretical limitations of self-attention in neural sequence models, citing related works on layer normalization and transformer architecture. The paper provides definitions and proofs for various concepts related to column uniformity and computation graphs, including the concept of column uniformity in computation graph families. The article presents a method for constructing log-precision transformer models, which rely on addition, conditional branching, and a finite number of functions computable in time O(log n). Finally, the size of the model's parameters is at most poly(n). This paper presents the logic for log-precision transformer models, analyzing the uniformity of common neural net building blocks within the transformer. The blocks are computable by a log-uniform, constant-depth, poly-size threshold circuit family. Affine transformations, a core part of neural networks used in the transformer, can be computed by a log-uniform, constant-size threshold circuit family of size polynomial in the weight matrix and bias vector. Layer normalization can be computed by a log-uniform TC0 circuit family. Finally, the output classifier head is computable in log-uniform TC0 if the activation function is log-uniform.
1940 word summary
The paper discusses the logic for log-precision transformer models. It analyzes the uniformity of common neural net building blocks used within the transformer and shows that they are computable by a log-uniform, constant-depth, poly-size threshold circuit family. The paper also discusses affine transformations, which are a core part of neural networks used in various parts of the transformer. It shows that for p = O(log n), any p-precision affine transformation where the weight matrix and bias vector are log-uniform is computable by a log-uniform, constant-size threshold circuit family of size polynomial in the weight matrix and bias vector. The paper also discusses layer normalization and shows that it can be computed by a log-uniform TC0 circuit family. Finally, the paper discusses output classifier head and shows that if the activation function is log-uniform, then the output classifier head is computable in log-uniform TC0. The document presents a method for constructing log-precision transformer models. The activation block of the model consists of two parts: the first simulates the pooling part of the self-attention sublayer, and the second applies layer-norm and simulates the feedforward subnetwork. The self-attention component is computable in log-uniform TC0, and the value function is defined as v(hi) = Whhi?i+bh. The similarity function computes queries and keys and then takes the scaled dot-product between them. The model can compute self-attention, binary multiplication, binary division, and iterated addition in log-uniform TC0. Finally, the size of the model's parameters is at most poly(n). This document discusses the logic for log-precision transformer models. The transformer embedding function represents token positions with a vector and can be expressed as a constant-size computation graph. The parameter vector for each function is log-uniform. The transformer column components rely on addition, conditional branching, and a finite number of functions computable in time O(log n). The size of F? will be of size bsize(n). The article discusses the use of log-uniform TC 0 families in transformer models. Lemma 2 shows that transformer components are computable in log-uniform TC 0, and Lemma 5 states that each component in F is computable in log-uniform TC 0 if T is a log-uniform transformer with log-uniform ? T. The embedding component, self-attention mechanism, activation block, and output classifier head are all computable in log-uniform TC 0. Lemma 3 establishes that there exists a function bsize(n) that is a power of 2 and computable in O(log n) time. Lemma 4 shows that circuit families can be padded to log-time size upper bounds. The article concludes by providing a formula for edge G T (n, i, j) and discussing causally masked attention. The paper discusses the concept of column uniformity in computation graph families, specifically in transformer models. It is shown that any transformer model is a log-column-uniform computation graph family, meaning that it can be computed in log-uniform time using a lookup table. The paper provides definitions and proofs for various concepts related to column uniformity and computation graphs. Common notation for computation graph and circuit families is summarized in Table 1. The paper cites related works on layer normalization and transformer architecture. The document discusses logic for log-precision transformer models, including the practical computational power of finite precision systems and the effects of parameter norm growth during transformer training. It also covers the limitations of log-precision transformers and the theoretical limitations of self-attention in neural sequence models. The article includes references to various papers and conferences on language recognition, neural information processing, and automata. The article discusses the development of logic for log-precision transformer models. It references various works that reconcile deep learning with symbolic artificial intelligence and provide theoretical foundations for guiding mechanistic interpretability work. The article suggests the possibility of translating real transformers to FO(M) sentences and establishing a hierarchy theorem describing the FO(M) quantifier depth needed to simulate a TC 0 family of a certain size. The article also suggests the potential for proving that any transformer can be simulated by an FO(M) sentence of quantifier depth of at most 2. The paper presents a method to translate log-precision transformers to a simple logic, providing the tightest known upper bound on such models. The authors conjecture that it is possible to simulate transformers in an even simpler logic. The results challenge the idea of a rigid division between symbolic and neural models. The paper establishes the equivalence of log-uniform circuits and FO(M), proving that any log-precision transformer can be translated to an FO(M) sentence that computes the same function as the transformer. The authors also prove that a transformer can be simulated by a log-uniform TC0 family C that obeys the size and depth properties of Theorem 3. The paper includes corollaries that establish the applicability of the method to various computation graph families. The results have implications for understanding the inner workings of transformers and for developing frameworks that unify different types of neural models. The document presents a method for constructing log-precision transformer models using a block mapping algorithm. The algorithm creates contiguous blocks of circuit gates simulating each node in the computation graph and routes inputs and outputs between blocks appropriately. The algorithm depends on a block mapping that defines the block node, block start, and block size for each node in the computation graph. The algorithm defines a circuit family simulating the computation graph family and enforces that the block size of each node matches the size of the circuit for that node. The algorithm satisfies three premises that ensure the circuit family and computation graph family compute the same function. The algorithm uses induction over circuit gates with topological ordering to show that the premises hold up to a certain point. The algorithm defines a uniform threshold circuit family for each primitive function using log-uniform threshold circuit families. This document presents a logic for log-precision transformer models. The main theorem shows that any log-precision transformer can be expressed in FO(M), and can be simulated by a TC 0 family C that is both a power of 2 and computable in time O(log n). Lemma 4 shows that if F is a log-uniform TC 0 family, then size F (n) ? bsize(n) for all F ? F. Lemma 3 shows that each component in F is computable in log-uniform TC 0. Lemma 1 shows that a transformer T is a log-uniform computation graph family where F contains embedding, self-attention, feedforward, and output components. The proof of Theorem 2 views transformers as computation graphs and focuses on simulating computation graph families with threshold circuit families. Uniform computation graph families are families where node G and edge G can be computed efficiently, i.e., under some constraints on space or time. Computation graph families are a useful tool for processing inputs of arbitrary length. They generalize computation graphs to define functions over unbounded-length strings as inputs. Threshold circuits are a special case of computation graphs where D=(0,1) and the nodes are labeled by a function f?F computed by this node. The edges represent a value D flowing as output from one node into another node. A computation graph G of arity k parameterizes a function Dk?D in the standard way. The output node is considered the output of the function. The size, depth, and arity of computation graph families become functions that can grow in size and precision with n. Theorem 2 can also be extended to apply to log-precision transformers with log-uniform weights. Threshold circuits can simulate neural network components that output multiple bits. This paper discusses the logic behind log-precision transformer models. Theorem 2 states that any computation of a transformer on input x can be reduced to a single integer division or a finite number of Dyck-language queries. Theorem 2 is the tightest known upper bound for any transformer with up to O(log n) precision. Proposition 1 analyzes a model of transformers where the precision depends on the context length. It shows that fixed-precision transformers are artificially limited because they can only attend over bounded-length windows, making them similar to hard-attention transformers. The document discusses the limitations of log-precision transformer models and proposes a logic for such models. The logic includes various examples of languages that can be expressed in first-order logic with the addition of certain predicates, such as majority quantifiers and parentheses matching. The authors argue that finite precision cannot represent uniform attention over long sequences and provide examples of tasks that require higher precision, such as iterated addition and skip-bigram matching. They also discuss the importance of uniform attention and its potential applications in transformer models. This document discusses Logic for Log-Precision Transformer Models. It introduces FO(M) formulas, which can express conditional majority quantifiers, counting and threshold quantifiers, and define a formal language. FO(M) formulas are constructed using indices, variables ranging over positions 1 to n, and predicates. The document also defines log-precision transformers as transformers where the weights defining T are fixed and the precision used to compute the forward pass is at most O(log n). The core functions in T are embeddings, self attention, activation, and the classifier head. The network prediction on x ? ? n is ?(h dn). The Logic for Log-Precision Transformer Models document discusses the computation and output of transformer models as a function of input x. The transformer is defined as a function of context length n and can be expressed as a sentence in FO(M). Fixed-precision transformers can only attend to a fixed number of tokens, while log-precision transformers are capable of attending over contexts of length n. This sheds new light on how to interpret transformer models and their computation. However, an exact logical characterization of transformers remains an open problem. The empirical results of Bhattamishra et al. (2020) are for a harder variant of transformers, and hard attention is weaker than general attention. Researchers have developed a logic, called FO(M), that can be used to express the computations performed by transformer models. FO(M) is more powerful than previous logics used for this purpose, and can handle a wider range of attention patterns. The researchers prove that any transformer model with finite precision can be expressed using FO(M). They also identify several problems that log-precision transformers cannot solve, such as computing boolean matrix permanents. FO(M) has the advantage of being mechanistically interpretable, meaning that it can provide insights into how transformer models perform their computations. The researchers believe that their work could guide the design of "transformer-complete" programming languages. The paper examines the limitations of finite-precision transformer models and proposes a new logic for log-precision transformer models. The authors demonstrate that finite-precision transformers are fundamentally weak and cannot express uniform attention patterns, which are a core algorithmic primitive of transformers. They also show that any function that cannot be defined in first-order counting logic with modular indexing cannot be expressed by the transformer. The authors propose a new logic, FO(M), which allows majority quantifiers and can express any function computed by a log-precision transformer. They provide an upper bound and first logical characterization of log-precision transformers. The paper raises concerns about the safe deployment, fairness, and accountability of increasingly complex transformer models with hundreds of billions of parameters. Precision transformers are a weak variant of transformers that can be equivalently expressed in a generalization of first-order logic. Recently, Chiang et al. showed that finite-precision transformers can be used to describe the types of logical rules they can resolve over some input text. One way to interpret the reasoning power of transformer-based language models is to describe the types of logical rules they can resolve. The paper "A Logic for Expressing Log-Precision Transformers" explores this concept. The authors are Ashish Sabharwal from Allen Institute for AI and William Merrill from New York University.