A tour of free monads and their motivating use cases
Dynamic task delegation for hierarchical agents
This is the fourth installment in a series of papers offering models of hierarchical structure for dynamical systems, using the language of polynomial functors. The operad underlying the symmetric monoidal category $(\mathbf{Poly}, \otimes, \mathcal{y})$ can be viewed as defining the behavior of hierarchical delegation. In particular, a morphism $\mathbf{Poly}(p_1 \otimes \cdots \otimes p_m, q)$ turns the outputs of subordinates with interfaces $p_i$ into the output of an agent with interface $q$ and turns a task given to the agent into a task for each of the subordinates. In this article, we extend the framework so that subordinates may be invoked asynchronously depending on the outcomes of other subordinates. We prove that the free (co)monad (co)monad extends to a (co)monad on $\mathbf{Org}$. From the perspective of programs/pattern, this extension implies the existence of a $\mathbf{Cat}$-enriched operad $\mathbf{Org}_\mathfrak{m}$, and from the perspective of behavior/matter, it implies the existence of a $\mathbf{Cat}$-enriched operad $\mathbf{Org}^\mathfrak{c}$. Second, we crispen the relationship between the programmatic and behavioral perspectives via a functor $[-, t] \colon \mathbf{Org}_{\mathfrak{m}}^\textrm{op} \to \mathbf{Org}^\mathfrak{c}$ for any polynomial monad $t$.
A Modern Architecture for FP
Death to IO monads: FP has better tools that point the way towards the future.
What I did with Monad the last 2 weeks: Oak, a web stack, and a path toward univalence - Monad dev blog
October 31, 2026 The last two weeks were the busiest stretch this repository has seen. Between October 16 and October 30 the tree took 370 commits — roughly 185...
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Subtyping is Overrated - Caleb Meredith aka @calebmer at @ReactEurope 2018
Why is Scala's type inference not as powerful as Haskell's?
The type inference engine of Haskell is much more powerful than Scala's. In Haskell I rarely have to explicitly write the types whereas in Scala the types can only be inferred in expressions but no...
Abstracting Definitional Interpreters
In this functional pearl, we examine the use of definitional interpreters as a basis for abstract interpretation of higher-order programming languages. As it turns out, definitional interpreters, especially those written in monadic style, can provide a nice basis for a wide variety of collecting semantics, abstract interpretations, symbolic executions, and their intermixings. But the real insight of this story is a replaying of an insight from Reynold's landmark paper, Definitional Interpreters for Higher-Order Programming Languages, in which he observes definitional interpreters enable the defined-language to inherit properties of the defining-language. We show the same holds true for definitional abstract interpreters. Remarkably, we observe that abstract definitional interpreters can inherit the so-called "pushdown control flow" property, wherein function calls and returns are precisely matched in the abstract semantics, simply by virtue of the function call mechanism of the defining-language. The first approaches to achieve this property for higher-order languages appeared within the last ten years, and have since been the subject of many papers. These approaches start from a state-machine semantics and uniformly involve significant technical engineering to recover the precision of pushdown control flow. In contrast, starting from a definitional interpreter, the pushdown control flow property is inherent in the meta-language and requires no further technical mechanism to achieve.
Haskell: A Great Procedural Language
These sound like dismissals or absurdities from the outside, but once you learn what they really mean, they take on a new light. In this article, I want to explain the third. (See the appendix if you are curious about the first two.)
Performative Thinking? The Brittle Correlation Between CoT Length and Problem Complexity
Intermediate token generation (ITG), where a model produces output before the solution, has been proposed as a method to improve the performance of language models on reasoning tasks. While these reasoning traces or Chain of Thoughts (CoTs) are correlated with performance gains, the mechanisms underlying them remain unclear. A prevailing assumption in the community has been to anthropomorphize these tokens as "thinking", treating longer traces as evidence of higher problem-adaptive computation. In this work, we critically examine whether intermediate token sequence length reflects or correlates with problem difficulty. To do so, we train transformer models from scratch on derivational traces of the A* search algorithm, where the number of operations required to solve a maze problem provides a precise and verifiable measure of problem complexity. We first evaluate the models on trivial free-space problems, finding that even for the simplest tasks, they often produce excessively long reasoning traces and sometimes fail to generate a solution. We then systematically evaluate the model on out-of-distribution problems and find that the intermediate token length and ground truth A* trace length only loosely correlate. We notice that the few cases where correlation appears are those where the problems are closer to the training distribution, suggesting that the effect arises from approximate recall rather than genuine problem-adaptive computation. This suggests that the inherent computational complexity of the problem instance is not a significant factor, but rather its distributional distance from the training data. These results challenge the assumption that intermediate trace generation is adaptive to problem difficulty and caution against interpreting longer sequences in systems like R1 as automatically indicative of "thinking effort".
Performative Thinking? The Brittle Correlation Between CoT Length and Problem Complexity
Intermediate token generation (ITG), where a model produces output before the solution, has been proposed as a method to improve the performance of language models on reasoning tasks. While these reasoning traces or Chain of Thoughts (CoTs) are correlated with performance gains, the mechanisms underlying them remain unclear. A prevailing assumption in the community has been to anthropomorphize these tokens as "thinking", treating longer traces as evidence of higher problem-adaptive computation. In this work, we critically examine whether intermediate token sequence length reflects or correlates with problem difficulty. To do so, we train transformer models from scratch on derivational traces of the A* search algorithm, where the number of operations required to solve a maze problem provides a precise and verifiable measure of problem complexity. We first evaluate the models on trivial free-space problems, finding that even for the simplest tasks, they often produce excessively long reasoning traces and sometimes fail to generate a solution. We then systematically evaluate the model on out-of-distribution problems and find that the intermediate token length and ground truth A* trace length only loosely correlate. We notice that the few cases where correlation appears are those where the problems are closer to the training distribution, suggesting that the effect arises from approximate recall rather than genuine problem-adaptive computation. This suggests that the inherent computational complexity of the problem instance is not a significant factor, but rather its distributional distance from the training data. These results challenge the assumption that intermediate trace generation is adaptive to problem difficulty and caution against interpreting longer sequences in systems like R1 as automatically indicative of "thinking effort".
Abstract syntax and variable binding
We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with compatible algebra and substitution structures. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.
Integrating functional and imperative programming | Proceedings of the 1986 ACM conference on LISP and functional programming
We present a new model, based on monads, for performing input/output in a non-strict, purely functional language. It is composable, extensible, efficient, requires no extensions to the type system, and extends smoothly to incorporate mixed-language ...
Generalised algebraic theories and contextual categories
Kan Extensions in Haskell
Previously: Tabulation Tribulations. If you think of functor composition as a form of multiplication, Kan extensions are an attempt to construct inverses of this multiplication. But unlike multipli…
