Latest massive language fashions (LLMs) — comparable to OpenAI’s o1/o3, DeepSeek’s R1 and Anthropic’s Claude 3.7 — reveal that permitting the mannequin to suppose deeper and longer at check time can considerably improve mannequin’s reasoning functionality. The core strategy underlying their deep considering functionality is known as chain-of-thought (CoT), the place the mannequin iteratively generates intermediate reasoning steps and appends them to the present context till producing the ultimate reply.
Nevertheless, as duties grow to be more and more advanced, the steps wanted to unravel them develop dramatically. As an example, contemplate fixing NP-hard issues utilizing CoT — the reasoning hint would inevitably span exponential steps, assuming a fixed-size Transformer as the bottom mannequin and P ≠ NP. This raises an essential query:
Will CoT-based test-time scaling hit laborious ceilings?
Sadly, most likely sure. Numerous limitations will emerge for more durable duties: (1) chains will inevitably exceed mannequin’s context home windows, (2) crucial data turns into buried and almost unattainable to retrieve from quite a few previous tokens, and (3) the self-attention complexity makes producing every new token prohibitively costly.
On this article, we problem the standard “write-only” CoT reasoning paradigm that dominates present LLM architectures, from each theoretical and sensible views. Moreover, we’ll discover a basically completely different reasoning strategy that enables LLM to not solely generate ideas, but in addition erase ideas. This capability for thought erasure not solely presents important sensible advantages in efficiency and effectivity, however proves basic for attaining optimum reasoning effectivity from a computational idea perspective.
This put up relies on the paper C. Yang et al., “PENCIL: Long thoughts with short memory” accepted in Worldwide Convention on Machine Learning 2025, a collaboration with Nathan Srebro, David McAllester, Zhiyuan Li. Code can be obtainable.
Not The whole lot Must Be Remembered
The thought of selectively discarding data has deep roots in pc science historical past, from the earliest computational fashions to fashionable methods. The basic Turing machine overwrites symbols on its tape slightly than preserving each state; programming languages reclaim reminiscence via stack frames which are robotically launched when features full their execution; and fashionable rubbish collectors repeatedly determine and take away objects not accessible to this system. These mechanisms weren’t merely effectivity optimizations — they had been important design selections that made advanced computation attainable inside finite sources.
This concept additionally applies to human reasoning. In theorem proving, as soon as a lemma is established, we discard its detailed derivation whereas preserving the consequence; when exploring problem-solving approaches, we merely mark unproductive paths as “failed” with out retaining their full traces. All through advanced reasoning, we naturally compress data, retaining conclusions whereas discarding the scaffolding used to achieve them.
✏️ PENCIL: A New Reasoning Paradigm
Subsequently, we suggest ✏️ PENCIL, a brand new reasoning paradigm for LLMs. In contrast to ✒️ CoT that solely generates ideas, PENCIL recursively generates and erases ideas till reaching the ultimate reply. It maintains solely the minimal context required for producing future ideas, so the mannequin can suppose longer and deeper to unravel more durable duties utilizing shorter working reminiscence. The next determine illustrates how PENCIL works
How Do Fashions Erase Ideas?
PENCIL’s erasure mechanism attracts on two classical concepts. First, from rewriting guidelines in logic and classical automated theorem proving, which repeatedly apply predefined guidelines to simplify advanced logical or arithmetic expressions into canonical varieties till reaching a remaining reply. Second, from practical programming languages, which creates stack frames to retailer native variables when calling features and releases corresponding reminiscence when features return, robotically discarding intermediate states which are not wanted.
Particularly, we introduce three particular tokens, known as [CALL], [SEP], and [RETURN], and use the next discount rule to implement erasure:
the place C stands for context, T stands for intermediate ideas, and A stands for reply. At any time when the generated sequence utterly matches the sample on the left, PENCIL triggers the discount rule, erasing ideas and merging the reply again into the context. You will need to word that C, T and A can themselves comprise particular tokens, thereby supporting recursive buildings just like nested operate calls — for instance, C might comprise one other [CALL] token, indicating {that a} new considering subroutine has been initiated.
How you can Use PENCIL?
PENCIL’s erasure mechanism flexibly helps varied reasoning patterns, comparable to:
1️⃣ Job Decomposition: Utilizing [CALL] to provoke subproblems, generate intermediate outcomes, after which use [SEP] and [RETURN] to merge outputs and erase subproblem reasoning particulars;
2️⃣ Department and Backtrack: Utilizing a [CALL], [SEP], [RETURN] triplet to handle an exploration department in a search tree, erasing invalid paths upon conflicts or failures.
3️⃣ Summarization / Tail Recursion: Condensing a prolonged reasoning hint into concise abstract, just like tail recursion optimization in programming:
the place T represents the unique advanced reasoning course of (or a tougher drawback), and T’ represents the summarized or simplified content material (or an equal, extra tractable drawback).
Instance on a NP-Full Job
For instance, contemplate a basic NP-Full drawback Boolean Satisfiability (SAT): given a Boolean formulation, decide whether or not there exists a variable project that makes it true. This drawback is (extensively believed to) require exponential time however solely polynomial house to unravel, with the only strategy being traversing a binary search tree of depth n.
Conventional CoT would accumulate intermediate calculations, inflicting the context size to develop proportionally with the variety of nodes within the search tree, which is exponential time complexity of O(2^n). As compared, PENCIL can recursively department to strive True/False for a variable, backtracking upon battle and erasing all ideas inside that department. This thus retains the context size proportional to the search depth, which is house complexity of solely O(n).
The next determine compares the utmost context size of the vanilla CoT with out discount (blue) and PENCIL with discount (crimson). As drawback complexity will increase, PENCIL achieves dramatic house effectivity, notably lowering context size from 151,192 to only 3,335 tokens for Einstein’s Puzzle.
Coaching and Experiments
The core distinction between CoT and PENCIL throughout coaching is the calculation of the loss operate:
For CoT, the loss for every new token relies on the whole historic context; for PENCIL, after every “write-erase” iteration, the mannequin calculates loss for brand new tokens solely on the lowered sequence. Though each generate the identical variety of tokens, PENCIL considerably shortens the context size corresponding to every token and thus is extra environment friendly.
It’s additionally worthwhile to notice that after every discount, the KV cache for the shared prefix C will be straight reused, with solely the cache for the shorter half A needing recalculation.
Experimental Outcomes
Our experiments deal with three inherently laborious reasoning duties: 3-SAT (NP-Full), QBF (PSPACE-Full), and Einstein’s Puzzle (pure language reasoning). For every activity, we wrote a generator to generate a coaching set the place particular tokens are included. We practice a small transformer (SAT/QBF with 10.6M parameters; Einstein’s Puzzle with 25.2M parameters) beginning with random initialization for these duties.
📊 In comparison with CoT, we discovered PENCIL can resolve larger-scale reasoning issues. As proven within the determine under, in SAT (left) and QBF (proper) duties, when drawback dimension is small, each CoT and PENCIL completely resolve issues; however as dimension will increase, conventional CoT accuracy drops considerably (e.g., solely about 50% for SAT at n=10), whereas PENCIL maintains excessive accuracy ≥ 99%. That is primarily as a result of CoT’s context sequence size explodes exponentially, whereas PENCIL avoids explosion by dynamic discount.
⚡️ Moreover, PENCIL considerably saves computational sources. As proven within the determine, for QBF (n=3–6) duties, we in contrast the convergence velocity of CoT (blue) and PENCIL (crimson) beneath the identical FLOPs finances. PENCIL rapidly reaches 100% accuracy whereas CoT, because of repeatedly increasing context size, requires extra FLOPs to strategy optimality. As the issue dimension will increase, the hole between the 2 turns into extra pronounced.
to six). Circles and vertical strains point out the primary time every technique reaches optimum efficiency.
🧩 We additional thought-about a really tough logical reasoning drawback: Einstein’s Puzzle. Every drawback consists of 5 homes and 5 attribute classes of individuals residing in them — shade, nationality, drink, cigarette, and pet (e.g., Purple/Inexperienced/Blue, Brit/German/Swede, Chook/Canine/Fish, and many others.). Given clues like “the inexperienced home is true subsequent to the chicken proprietor’s” and “the canine proprietor lives within the crimson home,” the duty is to infer “who owns the fish?” This drawback presents an excessive problem for current LLMs: even GPT-4 struggles to solve it. The determine under exhibits a simplified model with solely 3 homes and three attribute classes:
As proven under, for this drawback that even massive fashions battle with, PENCIL achieves 97% accuracy utilizing solely a small 25.2M parameter mannequin, whereas conventional CoT achieves solely 25% accuracy (near random guessing).
Principle: Common Environment friendly Computation
We additional reveal PENCIL’s basic benefit over conventional CoT from the theoretical expressive energy perspective: PENCIL is Turing full with optimum house complexity, and thus can resolve arbitrary computable duties effectively. That is one thing basically unattainable for CoT!
Important Outcomes
Particularly, we show: Utilizing a hard and fast, finite-sized Transformer, PENCIL can simulate any Turing machine with optimum time and house complexity, thereby effectively fixing all computable issues.
In different phrases, for any Turing machine operating in T time and S house, PENCIL requires solely O(T) tokens whereas sustaining a most context size of O(S) to provide similar outcomes. Whereas previous work established that conventional CoT could make Transformers Turing full, it calls for O(T) context size with every token representing an intermediate computation step. This distinction between most context size turns into essential as a result of for many algorithms, house complexity S is considerably smaller than time complexity T, particularly for more durable issues.
Contemplate NP-Full issues like Touring Salesman or Hamiltonian Circuit, that are extensively believed to require exponential time however solvable in polynomial house. Conventional CoT can’t resolve these inside polynomial context size constraints, and requires at the very least exponential size that exceeds sensible reminiscence limitations of any actual system. PENCIL, in distinction, can resolve them utilizing solely polynomial most context size, making beforehand intractable reasoning duties possible.
Proof Sketch
We now briefly introduce our proof thought, the place the important thing perception is to have PENCIL use a sequence of “Simulation-Summarization” iterations to scrub the reminiscence.
Step 1: Utilizing CoT to Encode Turing Machine Transitions As illustrated within the left a part of the determine above, we encode every Turing machine state transition as a token encoding “new state”, “written image”, and “head motion route” triplet within the embedding. The mannequin can use self-attention to calculate the present head place and decide the image at this place. With out discount, this course of generates T tokens with context size O(T).
Step 2: Alternating “Simulation-Summarization” PENCIL achieves house/time optimality via alternating:
- Simulation: Constantly generate Turing machine state transition tokens, simulating a number of computation steps;
- Summarization: When new tokens exceed twice the house wanted, summarize the computation utilizing S tokens. The discount rule then discards earlier ideas, retaining solely the most recent Turing machine state for the following spherical.
This technique maintains whole token era at O(T) whereas limiting context size to O(S).
Step 3: Transformer Implementation To show this course of will be carried out by Transformers, we developed the Full-Entry Sequence Processing (FASP) programming language and proved that any algorithm written in FASP will be carried out by a fixed-sized Transformer. In a FASP program, every variable corresponds to a Transformer sub-module, and every line of code transforms current variables to a brand new variable via predefined features, which is equal to developing a extra advanced Transformer primarily based on sub-modules. The variable returned by this system is the specified Transformer that encodes the algorithm. We wrote a FASP program that implements the “Simulation-Summarization” operation, which suggests there exists a constant-sized Transformer that may carry out the identical operate
Conclusion
In conclusion, we suggest a brand new reasoning paradigm PENCIL, which alternates between era and erasure, and permits fashions to suppose deeper to unravel extra sophisticated issues. Theoretically, we show that PENCIL achieves Turing completeness with optimum time and house effectivity and thus can effectively resolve any computable issues. Wanting ahead, a promising route can be to fine-tune LLMs to include PENCIL’s memory-efficient reasoning capabilities. We hope these findings will encourage reexamining present reasoning fashions from the attitude of idea of computation.
