---
title: 'Pacheck 2.0: Algebraic Proof Recycling'
url: https://www.emergentmind.com/topics/pacheck-2-0
type: topic
---

# Pacheck 2.0: Algebraic Proof Recycling

Searching arXiv for the specified paper and closely related prior work on PAC/LPAC and Pacheck.
Pacheck 2.0 is an algebraic proof checker for the Practical Algebraic Calculus with linear combinations (LPAC), extended with two “recycling” rules, `PatternNew` and `PatternApply`, that permit capture and reuse of recurring proof fragments. It verifies algebraic proof certificates showing that a target polynomial \(f\) belongs to the ideal generated by a set of axioms \(G \subseteq K[X]\), with particular emphasis on Boolean variables modeled by the axioms \(B(X)=\{x^2-x \mid x \in X\}\) [2507.20267]. The system is situated in the proof-logging and proof-checking tradition familiar from SAT and QBF, but specialized to algebraic derivations, especially those arising in arithmetic circuit verification. Its central contribution is the condensation of certificates by factoring repeated derivations into patterns that are checked once and later instantiated under suitable substitutions, thereby reducing proof size and verification time [2507.20267].

## 1. Formal setting and verification objective

Pacheck takes as input a field \(K\), a set of variables \(X\), a set of polynomials \(G \subseteq K[X]\), a target polynomial \(f \in K[X]\), and an LPAC proof certificate intended to establish
\[
f \in \langle G \rangle.
\]
In the Boolean setting, for each \(x \in X\), the polynomial \(x^2-x\) is added to the axioms, so Booleanity is represented by the ideal generated by \(B(X)=\{x^2-x \mid x \in X\}\) [2507.20267].

An LPAC proof is a sequence of states. In the original formulation, the state is \((X,P)\), where \(X\) is the current variable set and \(P\) is a sequence indexed by integers, with each \(P(i)\) either a polynomial in \(K[X]\) or “undefined” \((\bot)\). Initially, \((X,P)=(\emptyset,[])\). In the extended system implemented by Pacheck 2.0, the state becomes \((X,P,C)\), where \(C\) is a sequence of reusable patterns. Initially, \(C=[]\), and the legacy rules do not modify it [2507.20267].

The checker parses each proof step, checks syntactic and semantic side conditions, recomputes algebraic operations, and verifies the claimed conclusion polynomial. For `LinComb`, it explicitly computes the linear combination and reduces modulo \(\langle B(X)\rangle\), then compares the result with the declared polynomial. Acceptance occurs when the target polynomial \(f\) is derived and all prior steps are valid; otherwise the checker reports the first failed check [2507.20267].

This verification model places Pacheck 2.0 between two familiar algebraic proof paradigms. Polynomial Calculus supports derivations via ideal-theoretic algebraic rules, whereas Nullstellensatz-style certificates use explicit linear combinations. LPAC combines PC-style derivability with an NSS-like linear-combination layer while retaining local checkability and error localization, and Pacheck is the corresponding checker for that calculus [2507.20267]. This suggests that the design objective is not only proof existence but operational certification under machine-checkable local conditions.

## 2. LPAC rules and proof-state semantics

The core LPAC rules implemented by earlier Pacheck versions and retained in Pacheck 2.0 are `Axiom`, `Deletion`, `LinComb`, and `Ext` [2507.20267].

The `Axiom` rule inserts a polynomial \(p \in K[X]\) at an undefined index \(i\) and enlarges the current variable set with the variables occurring in \(p\):
\[
\text{Axiom}(i,p)\quad
\frac{(X,P)\quad P(i)=\bot \quad p\in K[X]}
{(X\cup Var(p), P(i\mapsto p))}.
\]

The `Deletion` rule sets \(P(i)\) to \(\bot\):
\[
\text{Deletion}(i)\quad
\frac{(X,P)}
{(X,P(i\mapsto \bot))}.
\]
Its role is operational rather than deductive, since it reduces the memory footprint during checking [2507.20267].

The `LinComb` rule is the central algebraic inference mechanism. It derives a polynomial \(p\) as a linear combination of existing polynomials modulo the Boolean ideal:
\[
\text{LinComb}(i,(j_1,\ldots,j_n),(q_1,\ldots,q_n),p)
\]
\[
\frac{
\begin{array}{l}
(X,P)\quad P(j_1)\neq \bot,\ldots,P(j_n)\neq \bot,\quad P(i)=\bot \\
p,q_1,\ldots,q_n\in K[X] \\
p=q_1\cdot P(j_1)+\cdots+q_n\cdot P(j_n)\;\text{mod }\langle B(X)\rangle
\end{array}}
{(X,P(i\mapsto p))}.
\]
The side condition expresses equality modulo \(\langle B(X)\rangle\), which is the algebraic representation of Boolean semantics in the system [2507.20267].

The `Ext` rule introduces a new Boolean variable \(v\) as an alias for a Boolean expression \(q\):
\[
\text{Ext}(i,v,q)\quad
\frac{
\begin{array}{l}
(X,P)\quad P(i)=\bot \quad v\notin X \\
q\in K[X]\quad q^2-q\in \langle B(X)\rangle
\end{array}}
{(X\cup\{v\},P(i\mapsto -v+q))}.
\]
The proof object stores the equation as \(-v+q\), thereby enforcing \(v=q\), and the condition \(q^2-q\in \langle B(X)\rangle\) ensures that the introduced alias is Boolean modulo the Boolean axioms [2507.20267].

These rules define LPAC as a practical system for machine-verifiable ideal-membership derivations. Pacheck 2.0 preserves this operational core and adds a mechanism for explicitly factoring repeated subderivations.

## 3. Pattern-based proof recycling

The distinctive extension in Pacheck 2.0 is the addition of `PatternNew` and `PatternApply`, which implement proof recycling by storing and reusing verified proof fragments [2507.20267].

A pattern stored at index \(i\) consists of a sequence of input polynomials \(I\), a sequence of internal LPAC proof steps \(S\), a set of selected output polynomials \(O\), and the set \(V_{\ext}\) of extension variables occurring in \(O\). The rule
\[
\text{PatternNew}(i,I,S,O)
\]
checks that \((I,S)\) forms a correct standalone LPAC proof and then stores only the summary \((i,I,O,V_{\ext})\) in the pattern table \(C\), discarding the internal derivation \(S\) afterward [2507.20267]. Formally,
\[
\text{PatternNew}(i,I,S,O)\quad
\frac{
(X,P,C)\quad C(i)=\bot \quad \text{Correct-LPAC-Proof}(I,S)\quad O\subseteq \text{Conclusions}(S)\quad V_{\ext}=\text{ExtensionVar}(O)}
{(X,P,C\cup (i,I,O,V_{\ext}))}.
\]

The companion rule `PatternApply` instantiates a stored pattern using a substitution \(\varphi\), a set \(X_{\ext}\) of fresh extension variables for the current application, indices of existing input polynomials, and designated output locations:
\[
\text{PatternApply}\left(i,X_{\ext},\varphi,(j_1,\ldots,j_m),\{(k_1,p_1),\ldots,(k_n,p_n)\}\right).
\]
Its side conditions require that the pattern exists, the extension variables are fresh, extension variables from the stored pattern are mapped injectively to variables in \(X_{\ext}\), every variable image under \(\varphi\) is Boolean modulo \(\langle B(X\cup X_{\ext})\rangle\), the instantiated pattern inputs match the currently stored polynomials \(P(j_1),\ldots,P(j_m)\), and the instantiated outputs match the declared \(p_1,\ldots,p_n\) [2507.20267]. The resulting state is
\[
(X\cup X_{\ext}, P(k_1\mapsto p_1,\ldots,k_n\mapsto p_n), C).
\]

The semantic intuition is that `PatternNew` validates a local LPAC derivation once, while `PatternApply` reuses the corresponding input-output relation without replaying the internal steps. The paper characterizes this as “recycling” algebraic proof fragments, especially when the same proof steps recur for different variable sets [2507.20267]. A plausible implication is that the pattern mechanism introduces a proof-engineering layer over LPAC, separating local derivational correctness from repeated deployment across isomorphic contexts.

## 4. Illustrative derivations and proof structure

The paper gives two representative examples. The first reuses a linear-combination pattern. With
\[
G=\{x+2y-2,\,-y-z+1,\,a+2b-2,\,-b-z+1,\,a-x+1\},
\]
the goal is to prove \(1\in \langle G\rangle\). A pattern is defined with inputs \(p_1=v_1-2v_2\) and \(p_2=v_2-v_3\), one internal `LinComb` step deriving \(p_3=v_1-2v_3\), and output \(\{p_3\}\). It is then instantiated twice, first with
\[
v_1\mapsto x,\quad v_2\mapsto 1-y,\quad v_3\mapsto z,
\]
yielding \(x-2z\), and again with
\[
v_1\mapsto a,\quad v_2\mapsto 1-b,\quad v_3\mapsto z,
\]
yielding \(a-2z\). A final `LinComb` step combines these with \(a-x+1\) to derive \(1\) [2507.20267]. The example isolates the precise redundancy targeted by the system: a linear combination computed once in a pattern and then reused at multiple instantiations.

The second example shows that patterns may include extension-variable reasoning. From the polynomials encoding \(\neg x\lor \neg y\) and \(y\lor z\),
\[
xy=0,\qquad yz-y-z+1=0,
\]
the proof introduces a new variable \(\bar z\) for \(1-z\) and derives \(x\bar z\). The pattern contains both an `Ext` step introducing \(w_3\) as \(1-v_3\) and a `LinComb` step deriving \(v_1w_3\). Instantiation with
\[
v_1\mapsto x,\quad v_2\mapsto y,\quad v_3\mapsto z,\quad w_3\mapsto \bar z
\]
produces the relation \(-\bar z+1-z\) and the resolvent polynomial \(x\bar z\) [2507.20267]. This demonstrates that proof recycling is not limited to repeated coefficient patterns in linear combinations; it also covers recurring extension-variable constructions.

These examples are significant because they make the granularity of the abstraction explicit. The reusable object is neither a macro over syntax alone nor a global lemma in an external meta-language; it is a checked LPAC fragment with declared inputs and outputs, instantiated under substitutions that satisfy the same Boolean side conditions as ordinary LPAC steps.

## 5. Checker architecture and operational consequences

Pacheck 2.0 reads proof commands from the set `Axiom`, `Deletion`, `LinComb`, `Ext`, `PatternNew`, and `PatternApply`, and maintains the state \((X,P,C)\), where \(X\) contains current variables including Boolean extension variables, \(P\) is the polynomial array, and \(C\) is the pattern table whose entries store \((I,O,V_{\ext})\) [2507.20267].

Processing `PatternNew(i,I,S,O)` requires three checks: that \(C(i)=\bot\), that `Correct-LPAC-Proof(I,S)` holds by running the standard LPAC verifier on the local proof fragment \((I,S)\), and that every output in \(O\) occurs among the conclusions of \(S\). The checker then computes \(V_{\ext}=\text{ExtensionVar}(O)\), stores \((I,O,V_{\ext})\) at \(C(i)\), and discards \(S\) [2507.20267]. The discarding of internal steps is operationally important because it allows the persistent certificate state to retain only the summary needed for later applications.

Processing `PatternApply` requires verifying pattern existence, freshness of the supplied extension variables, injective mapping of stored extension variables into fresh variables, Boolean consistency of every variable image under \(\varphi\), matching of instantiated inputs against already-defined polynomials, and matching of instantiated outputs against the newly declared result polynomials [2507.20267]. If these checks succeed, the checker extends the state by the fresh variables and stores the instantiated outputs at the designated indices.

The paper contrasts the operational cost of repeated derivations with and without patterns. Without patterns, repeated subderivations must be replayed as multiple `LinComb` or `Ext` steps, each of which triggers algebraic verification. With patterns, the internal derivation is checked once in `PatternNew`, and each later application performs structural and substitution checks instead of full replay [2507.20267]. The paper does not derive formal complexity bounds, but explicitly states that experimentally the pattern mechanism reduces both the number of steps to check and the overall runtime [2507.20267].

This suggests a modularization of proof checking analogous to memoization at the certificate level. The stored object is not a cached arithmetic result but a certified derivational schema.

## 6. Benchmark evaluation

The implementation was evaluated on multiplier-verification benchmarks, where the task is to prove that a circuit specification polynomial belongs to the ideal generated by circuit-encoding polynomials. This benchmark family is particularly suitable because multipliers contain repeated substructures such as adders, so proof recycling can exploit isomorphic reasoning patterns [2507.20267].

The reported benchmark set includes `abc`, `abc-rsn`, and `abc-cmp` with 64 axioms, and `sp-ar-rc`, `sp-bd-rc`, `sp-dt-rc`, `sp-os-rc`, and `sp-ar-cl` with 96–108 axioms [2507.20267]. The comparison distinguishes LPAC without patterns from LPAC with patterns. For the baseline, the metrics are number of `LinComb` steps (“Steps”, \(\times 10^3\)), proof file size (“File” in MB), memory usage (“Mem” in MB), and verification time (“Time” in seconds). For the extended system, the metrics additionally include the number of `PatternNew` commands (“#”), the number of `PatternApply` commands (“Apply”), and the maximum number of internal steps per pattern \((\max |S|)\) [2507.20267].

| Benchmark | Without patterns | With patterns |
|---|---|---|
| `abc` | 196k steps, 454 MB file, 982 MB memory, 10.17 s | 48k steps, 9 `PatternNew`, 4032 `PatternApply`, max pattern length 46, 439 MB file, 913 MB memory, 9.61 s |
| `sp-bd-rc` | 284k steps, 1130 MB file, 2098 MB memory, 32.33 s | 111k steps, 42 `PatternNew`, 6520 `PatternApply`, max pattern length 87, 1117 MB file, 2038 MB memory, 29.45 s |
| `sp-ar-cl` | 2043k steps, 959 MB file, 1867 MB memory, 25.98 s | 1820k steps, 79 `PatternNew`, 3999 `PatternApply`, max pattern length 421, 924 MB file, 1749 MB memory, 24.37 s |

Across all benchmarks, the pattern extension *significantly* reduces the number of proof steps, reliably reduces proof file size, and usually reduces memory usage and verification time; no benchmark worsened under the pattern-based system in these metrics [2507.20267].

The empirical profile is notable because the largest improvement is in step count, while file-size improvements are smaller in absolute terms for some instances. This suggests that the dominant effect is elimination of repeated derivational replay, with secondary gains in certificate size and checking resources.

## 7. Soundness, scope, and relation to algebraic proof certification

The soundness discussion proceeds by inheritance from LPAC. The original rules `Axiom`, `Deletion`, `LinComb`, and `Ext` preserve the invariant that all polynomials in \(P\) lie in the ideal generated by the axioms together with \(B(X)\). `PatternNew` is sound because it requires `Correct-LPAC-Proof(I,S)` and merely stores a summary without changing the current polynomial state. `PatternApply` is sound because it instantiates a sound LPAC derivation under a mapping \(\varphi\) that preserves Boolean consistency, matches actual inputs, and maps extension variables to fresh Booleans [2507.20267]. Detailed formal proofs are not given in the excerpt, but the rule design is explicitly intended to inherit soundness from the underlying LPAC calculus [2507.20267].

The mechanism is instantiated for LPAC with Boolean variables. This matters because `LinComb` is defined modulo \(\langle B(X)\rangle\), `Ext` requires Booleanity of the alias expression, and `PatternApply` requires
\[
\varphi(v)^2-\varphi(v)\in \langle B(X\cup X_{\ext})\rangle
\]
for every variable \(v\) appearing in the pattern [2507.20267]. In non-Boolean settings, the mapping constraints would have to be adjusted accordingly. The paper states that extending the idea of patterns to other polynomial proof systems is conceptually straightforward but requires dedicated instantiation of the mapping constraints [2507.20267].

Within the broader algebraic-certification landscape, Pacheck 2.0 addresses the practical problem that proof certificates often contain the same proof steps repeated for different variable sets. The pattern store \(C\), together with `PatternNew` and `PatternApply`, is the formal mechanism by which these repetitions are factored out [2507.20267]. This is especially relevant in arithmetic circuit verification, where repeated reasoning over isomorphic subcircuits such as full or half adders can be replaced by a single stored pattern and many applications [2507.20267].

The paper also identifies open directions, including combining pattern-based recycling with new methods for finding short ideal-membership proofs, cited there as [HV24], and pursuing further efficiency improvements in proof checking [2507.20267]. A plausible implication is that the long-term significance of Pacheck 2.0 lies not only in certificate compression, but in the emergence of reusable proof schemata as first-class objects within algebraic proof logging.

Source: https://www.emergentmind.com/topics/pacheck-2-0