Current giant 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 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 turn into 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 necessary query:
Will CoT-based test-time scaling hit onerous ceilings?
Sadly, most likely sure. Numerous limitations will emerge for tougher duties: (1) chains will inevitably exceed mannequin’s context home windows, (2) important data turns into buried and almost not possible 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 traditional “write-only” CoT reasoning paradigm that dominates present LLM architectures, from each theoretical and sensible views. Moreover, we are going to discover a essentially completely different reasoning strategy that permits LLM to not solely generate ideas, but additionally erase ideas. This capability for thought erasure not solely presents important sensible advantages in efficiency and effectivity, however proves elementary for attaining optimum reasoning effectivity from a computational principle perspective.
This publish relies on the paper C. Yang et al., “PENCIL: Lengthy ideas with quick reminiscence” accepted in Worldwide Convention on Machine Studying 2025, a collaboration with Nathan Srebro, David McAllester, Zhiyuan Li. Code can also be obtainable.
Not All the things Must Be Remembered
The concept of selectively discarding data has deep roots in laptop science historical past, from the earliest computational fashions to trendy programs. The traditional Turing machine overwrites symbols on its tape relatively than preserving each state; programming languages reclaim reminiscence via stack frames which might be robotically launched when features full their execution; and trendy rubbish collectors repeatedly establish and take away objects not accessible to this system. These mechanisms weren’t merely effectivity optimizations — they have been important design selections that made advanced computation doable 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 tougher 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 kinds till reaching a last 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 might be not wanted.
Particularly, we introduce three particular tokens, referred to 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. Every time the generated sequence fully matches the sample on the left, PENCIL triggers the discount rule, erasing ideas and merging the reply again into the context. It is very important be aware that C, T and A can themselves include particular tokens, thereby supporting recursive constructions just like nested perform calls — for instance, C might include one other [CALL] token, indicating {that a} new considering subroutine has been initiated.
Easy methods to Use PENCIL?
PENCIL’s erasure mechanism flexibly helps numerous 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 harder 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 traditional NP-Full drawback Boolean Satisfiability (SAT): given a Boolean system, 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 best 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 (pink). As drawback complexity will increase, PENCIL achieves dramatic house effectivity, notably decreasing 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 perform:
For CoT, the loss for every new token relies on the entire historic context; for PENCIL, after every “write-erase” iteration, the mannequin calculates loss for brand spanking 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 may be instantly reused, with solely the cache for the shorter half A needing recalculation.
Experimental Outcomes
Our experiments deal with three inherently onerous reasoning duties: 3-SAT (NP-Full), QBF (PSPACE-Full), and Einstein’s Puzzle (pure language reasoning). For every job, we wrote a generator to generate a coaching set the place particular tokens are included. We prepare 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 remedy larger-scale reasoning issues. As proven within the determine beneath, in SAT (left) and QBF (proper) duties, when drawback measurement is small, each CoT and PENCIL completely remedy issues; however as measurement 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 pace of CoT (blue) and PENCIL (pink) below the identical FLOPs price range. PENCIL rapidly reaches 100% accuracy whereas CoT, resulting from repeatedly increasing context size, requires extra FLOPs to strategy optimality. As the issue measurement will increase, the hole between the 2 turns into extra pronounced.
to six). Circles and vertical strains point out the primary time every methodology reaches optimum efficiency.
🧩 We additional thought of a really tough logical reasoning drawback: Einstein’s Puzzle. Every drawback consists of 5 homes and 5 attribute classes of individuals dwelling in them — coloration, nationality, drink, cigarette, and pet (e.g., Purple/Inexperienced/Blue, Brit/German/Swede, Chicken/Canine/Fish, and so forth.). Given clues like “the inexperienced home is correct subsequent to the fowl proprietor’s” and “the canine proprietor lives within the pink home,” the duty is to infer “who owns the fish?” This drawback presents an excessive problem for current LLMs: even GPT-4 struggles to unravel it. The determine beneath reveals a simplified model with solely 3 homes and three attribute classes:
As proven beneath, for this drawback that even giant 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).
Idea: Common Environment friendly Computation
We additional reveal PENCIL’s elementary benefit over conventional CoT from the theoretical expressive energy perspective: PENCIL is Turing full with optimum house complexity, and thus can remedy arbitrary computable duties effectively. That is one thing essentially not possible for CoT!
Major 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 working in T time and S house, PENCIL requires solely O(T) tokens whereas sustaining a most context size of O(S) to supply an identical outcomes. Whereas earlier 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 tougher 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 not remedy these inside polynomial context size constraints, and requires not less than exponential size that exceeds sensible reminiscence limitations of any actual system. PENCIL, in distinction, can remedy 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 collection of “Simulation-Summarization” iterations to wash 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: Repeatedly 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, protecting solely the newest Turing machine state for the subsequent spherical.
This technique maintains whole token technology at O(T) whereas limiting context size to O(S).
Step 3: Transformer Implementation To show this course of may be carried out by Transformers, we developed the Full-Entry Sequence Processing (FASP) programming language and proved that any algorithm written in FASP may 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 establishing 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 means there exists a constant-sized Transformer that may carry out the identical perform
Conclusion
In conclusion, we suggest a brand new reasoning paradigm PENCIL, which alternates between technology and erasure, and permits fashions to suppose deeper to unravel extra difficult issues. Theoretically, we show that PENCIL achieves Turing completeness with optimum time and house effectivity and thus can effectively remedy any computable issues. Wanting ahead, a promising route could 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 principle of computation.
