Recursive Proof Research Articles

Faster Montgomery multiplication and Multi-Scalar-Multiplication for SNARKs

The bottleneck in the proving algorithm of most of elliptic-curve-based SNARK proof systems is the Multi-Scalar-Multiplication (MSM) algorithm. In this paper we give an overview of a variant of the Pippenger MSM algorithm together with a set of optimizations tailored for curves that admit a twisted Edwards form. We prove that this is the case for SNARK-friendly chains and cycles of elliptic curves, which are useful for recursive constructions. Our contribution is twofold: first, we optimize the arithmetic of finite fields by improving on the well-known Coarsely Integrated Operand Scanning (CIOS) modular multiplication. This is a contribution of independent interest that applies to many different contexts. Second, we propose a new coordinate system for twisted Edwards curves tailored for the Pippenger MSM algorithm.Accelerating the MSM over these curves is critical for deployment of recursive proof< systems applications such as proof-carrying-data, blockchain rollups and blockchain light clients. We implement our work in Go and benchmark it on two different CPU architectures (x86 and arm64). We show that our implementation achieves a 40-47% speedup over the state-of-the-art implementation (which was implemented in Rust). This MSM implementation won the first place in the ZPrize competition in the open division “Accelerating MSM on Mobile” and will be deployed in two real-world applications: Linea zkEVM by ConsenSys and probably Celo network.

The [formula omitted]-Euler theorem and the combinatorics of [formula omitted]-sequences

In 1997, Bousquet-Mélou and Eriksson stated a broad generalization of Euler’s distinct-odd partition theorem, namely the (k,l)-Euler theorem. Their identity involved the (k,l)-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers–Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the l-lecture-hall partitions (the case k=l), whose specialization in l=2 corresponds to Sylvester’s bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case k,l≥2.In the spirit of Savage and Yee’s work, we provide and prove in this paper slight variations of the suggested bijection, not only for the case k,l≥2, but also for the cases (k,1) and (1,k) with k≥4. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-Mélou and Eriksson in their recursive proof of the (k,l)-lecture hall theorem and finally provide the analogous recursive bijection for the (k,l)-Euler theorem.

Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators

AbstractWilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. In this paper, we study the structure of the associated positroids, as well as the structure of the denominator of the integrand defined by each diagram. We give an algorithm to derive the Grassmann necklace of the associated positroid directly from the Wilson loop diagram, and a recursive proof that the dimension of these cells is thrice the number of propagators in the diagram. We also show that the ideal generated by the denominator in the integrand is the radical of the ideal generated by the product of Grassmann necklace minors.

Efficient HW/SW partitioning of Halo: FPGA-accelerated recursive proof composition in blockchain

Efficient HW/SW partitioning of Halo: FPGA-accelerated recursive proof composition in blockchain

Motion Optimization for Musculoskeletal Dynamics: A Flatness-Based Polynomial Approach

A new approach for trajectory optimization of musculoskeletal dynamic models is introduced. The model combines rigid-body and muscle dynamics described with a Hill-type model driven by neural control inputs. The objective is to find input and state trajectories that are optimal with respect to a minimum-effort objective and meet constraints associated with musculoskeletal models. The measure of effort is given by the integral of pairwise average forces of the agonist-antagonist muscles. The concepts of flat parameterization of nonlinear systems and sum-of-squares optimization are combined to yield a method that eliminates the numerous set of dynamic constraints present in collocation methods. With terminal equilibrium, optimization reduces to a feasible linear program, and a recursive feasibility proof is given for more general polynomial optimization cases. The methods of the article can be used as a basis for fast, and efficient solvers for hierarchical, and receding-horizon control schemes. Two simulation examples are included to illustrate the proposed methods.

Monostationarity and Multistationarity in Tree Networks of Goldbeter-Koshland Loops.

A major challenge in systems biology is to elicit general properties in the face of molecular complexity. Here, we introduce a class of enzyme-catalysed biochemical networks and examine how the existence of a single positive steady state (monostationarity) depends on the network structure, enzyme mechanisms, kinetic rate laws and parameter values. We consider Goldbeter-Koshland (GK) covalent modification loops arranged in a tree network, so that a substrate form in one loop can be an enzyme in another loop. GK loops are a canonical motif in cell signalling and trees offer a generalisation of linear cascades which accommodate network complexity while remaining mathematically tractable. In particular, they permit a modular, recursive proof strategy which may be more widely applicable. We show that if each enzyme follows its own complex reaction mechanism under mass action kinetics, then any network is monostationary for all appropriate parameter values. If the kinetics is non-mass action with a plausible monotonicity requirement, and each enzyme follows the Michaelis-Menten mechanism, then monostationarity is preserved. Surprisingly, a single GK loop with a complex enzyme mechanism under non-mass action monotone kinetics can have more than one positive steady state (multistationarity). The broader interplay between network structure, enzyme mechanism and kinetics remains an intriguing open problem.

Renormalization of a diagonal formulation of first order Yang-Mills theory

We study the BRST renormalization of an alternative formulation of the Yang-Mills theory, where the matrix-propagator of the gluon and the complementary fields is diagonal. This procedure involves scalings as well as non-linear mixings of the fields and sources. We show, in the Landau gauge, that the BRST identities implement a recursive proof of renormalizability to all orders.

All-loop Mondrian diagrammatics and 4-particle amplituhedron

Based on 1712.09990 which handles the 4-particle amplituhedron at 3-loop, we have found an extremely simple pattern, yet far more non-trivial than one might naturally expect: the all-loop Mondrian diagrammatics. By further simplifying and rephrasing the key relation of positivity in the amplituhedron setting, remarkably, we find a completeness relation unifying all diagrams of the Mondrian types for the 4-particle integrand of planar mathcal{N} = 4 SYM to all loop orders, each of which can be mapped to a simple product following a few plain rules designed for this relation. The explicit examples we investigate span from 3-loop to 7-loop order, and based on them, we classify the basic patterns of Mondrian diagrams into four types: the ladder, cross, brick-wall and spiral patterns. Interestingly, for some special combinations of ordered subspaces (a concept defined in the previous work), we find failed exceptions of the completeness relation which are called “anomalies”, nevertheless, they substantially give hints on the all-loop recursive proof of this relation. These investigations are closely related to the combinatoric knowledge of separable permutations and Schröder numbers, and go even further from a diagrammatic perspective. For physical relevance, we need to further consider dual conformal invariance for two basic diagrammatic patterns to correct the numerator for a local integrand involving one or both of such patterns, while the denominator encoding its pole structure and also the sign factor, are already fixed by rules of the completeness relation. With this extra treatment to ensure the integrals are dual conformally invariant, each Mondrian diagram can be exactly translated to its corresponding physical loop integrand after being summed over all ordered subspaces that admit it.

BRST renormalization of the first order Yang–Mills theory

We examine the renormalization of the first order formulation of the Yang–Mills theory, by using the BRST identities. These preserve the gauge invariance of the theory and enable a recursive proof of renormalizability to higher orders in perturbation theory. The renormalization involves non-linear mixings as well as re-scalings of the fields and sources, which lead to a renormalized action at all orders.

Scalable Zero Knowledge Via Cycles of Elliptic Curves

Non-interactive zero-knowledge proofs of knowledge for general NP statements are a powerful cryptographic primitive, both in theory and in practical applications. Recently, much research has focused on achieving an additional property, succinctness, requiring the proof to be very short and easy to verify. Such proof systems are known as zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs), and are desired when communication is expensive, or the verifier is computationally weak. Existing zk-SNARK implementations have severe scalability limitations, in terms of space complexity as a function of the size of the computation being proved (e.g., running time of the NP statement’s decision program). First, the size of the proving key is quasilinear in the upper bound on the computation size. Second, producing a proof requires “writing down” all intermediate values of the entire computation, and then conducting global operations such as FFTs. The bootstrapping technique of Bitansky et al. (STOC ’13), following Valiant (TCC ’08), offers an approach to scalability, by recursively composing proofs: proving statements about acceptance of the proof system’s own verifier (and correctness of the program’s latest step). Alas, recursive composition of known zk-SNARKs has never been realized in practice, due to enormous computational cost. Using new elliptic-curve cryptographic techniques, and methods for exploiting the proof systems’ field structure and nondeterminism, we achieve the first zk-SNARK implementation that practically achieves recursive proof composition. Our zk-SNARK implementation runs random-access machine programs and produces proofs of their correct execution, on today’s hardware, for any program running time. It takes constant time to generate the keys that support all computation sizes. Subsequently, the proving process only incurs a constant multiplicative overhead compared to the original computation’s time, and an essentially-constant additive overhead in memory. Thus, our zk-SNARK implementation is the first to have a well-defined, albeit low, clock rate of “verified instructions per second”.

Graph Decomposition and Parity

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph Gn,p. We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo q. We determine the asymptotic distribution of all fixed two-component graphs in Gn,p for all q, and we give infinite families of many-component graphs with a uniform asymptotic distribution for all q. We also prove a negative result that no recursive proof of the simplest form exists for a uniform asymptotic distribution for arbitrary graphs.

Then We Characterize Primes and Composite Numbers Via Divisibility

In this paper, we show a Theorem which helps us to characterize prime numbers and composite numbers via divisibility; and we use the characterizations of primes and composite numbers to characterize twin primes, Mersenne primes, even perfect numbers, Sophie Germain primes, Fermat primes, Fermat composite numbers and Mersenne composite numbers (we recall that a logic (non recursive) proof of problems posed by twin primes, Mersenne primes, perfect numbers, Sophie Germain primes, Fermat primes, Fermat composite numbers and Mersenne composite numbers, is given in [12]. [[ Prime numbers are well kwown ( see [15] or [19]) and we recall that a composite number is a non prime number. We recall (see [1] or [2] or [3] or [6] or [9] or [10] or [12] or [13] or [14] or [17]) that a Fermat prime is a prime of the form Fn = 2 2 n + 1, where n is an integer 0; and a Fermat composite is a non prime number of the form Fn = 2 2 n + 1, where n is an integer 1; it is known that for every j 2 f0; 1; 2; 3; 4g, Fj is a Fermat prime, and it is also known that F5 and F6 are Fermat composite. We recall (see [11]) that a prime h is called a Sophie Germain prime, if both h and 2h + 1 are prime; the

Recursive proofs for inductive tree data-structures

We develop logical mechanisms and procedures to facilitate the verification of full functional properties of inductive tree data-structures using recursion that are sound, incomplete, but terminating. Our contribution rests in a new extension of first-order logic with recursive definitions called Dryad, a syntactical restriction on pre- and post-conditions of recursive imperative programs using Dryad, and a systematic methodology for accurately unfolding the footprint on the heap uncovered by the program that leads to finding simple recursive proofs using formula abstraction and calls to SMT solvers. We evaluate our methodology empirically and show that several complex tree data-structure algorithms can be checked against full functional specifications automatically, given pre- and post-conditions. This results in the first automatic terminating methodology for proving a wide variety of annotated algorithms on tree data-structures correct, including max-heaps, treaps, red-black trees, AVL trees, binomial heaps, and B-trees.

An abstract interpretation framework for termination

Proof, verification and analysis methods for termination all rely on two induction principles: (1) a variant function or induction on data ensuring progress towards the end and (2) some form of induction on the program structure. The abstract interpretation design principle is first illustrated for the design of new forward and backward proof, verification and analysis methods for safety. The safety collecting semantics defining the strongest safety property of programs is first expressed in a constructive fixpoint form. Safety proof and checking/verification methods then immediately follow by fixpoint induction. Static analysis of abstract safety properties such as invariance are constructively designed by fixpoint abstraction (or approximation) to (automatically) infer safety properties. So far, no such clear design principle did exist for termination so that the existing approaches are scattered and largely not comparable with each other. For (1), we show that this design principle applies equally well to potential and definite termination. The trace-based termination collecting semantics is given a fixpoint definition. Its abstraction yields a fixpoint definition of the best variant function. By further abstraction of this best variant function, we derive the Floyd/Turing termination proof method as well as new static analysis methods to effectively compute approximations of this best variant function. For (2), we introduce a generalization of the syntactic notion of struc- tural induction (as found in Hoare logic) into a semantic structural induction based on the new semantic concept of inductive trace cover covering execution traces by segments, a new basis for formulating program properties. Its abstractions allow for generalized recursive proof, verification and static analysis methods by induction on both program structure, control, and data. Examples of particular instances include Floyd's handling of loop cutpoints as well as nested loops, Burstall's intermittent assertion total correctness proof method, and Podelski-Rybalchenko transition invariants.

Formal polytypic programs and proofs

AbstractThe aim of our work is to be able to do fully formal, machine-verified proofs over Generic Haskell-style polytypic programs. In order to achieve this goal, we embed polytypic programming in the proof assistantCoqand provide an infrastructure for polytypic proofs. Polytypic functions are reified within Coq as a datatype and they can then be specialized by applying a dependently typed term specialization function. Polytypic functions are thus first-class citizens and can be passed as arguments or returned as results. Likewise, we reify polytypic proofs as a datatype and provide a lemma that a polytypic proof can be specialized to any datatype in the universe. The correspondence between polytypic functions and their polytypic proofs is very clear: programmers need to give proofs for, and only for, the same cases that they need to give instances for when they define the polytypic function itself. Finally, we discuss how to write (co)recursive functions and do (co)recursive proofs in a similar way that recursion is handled in Generic Haskell.

Recursive proof of the Bell–Kochen–Specker theorem in any dimension [formula omitted

We present a method to obtain sets of vectors proving the Bell–Kochen–Specker theorem in dimension n from a similar set in dimension d ( 3 ⩽ d < n ⩽ 2 d ). As an application of the method we find the smallest proofs known in dimension five (29 vectors), six (31) and seven (34), and different sets matching the current record (36) in dimension eight.

Ramsey degrees of bipartite graphs: A primitive recursive proof

In this paper, we give a primitive recursive proof for a theorem that provides a measure of the extent to which a finite bipartite graph is a Ramsey object in the class of finite bipartite graphs.

On the convexity of the multiplicative potential and penalty functions and related topics

It is well known that a function f of the real variable x is convex if and only if (x,y)→yf(y-1x),y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from old ones. In particular, it allows to extend the aforementioned property to functions of the form (x,y)→g(y)f(g(y)-1x) and provides a new tool for the study of the multiplicative potential and penalty functions.

A proof of existence of whiskered tori with quasi flat homoclinic intersections in a class of almost integrable hamiltonian systems

Rotators interacting with a pendulum via small, velocity independent, potentials are considered: the invari- ant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds (“whiskers”), whose intersections define a set of homoclinic points. The homoclinic splitting can be introduced as a mea- sure of the splitting of the stable and unstable manifolds near to any homoclinic point. In a previous paper, [G1], cancellation mechanisms in the perturbative series of the homoclinic splitting have been investigated. This led to the result that, under suitable conditions, if the frequencies of the quasi periodic motion on the tori are large, the homoclinic splitting is smaller than any power in the frequency of the forcing (“quasi flat homoclinic intersections”). In the case l = 2 the result was uniform in the twist size: for l > 2 the discussion relied on a recursive proof, of KAM type, of the whiskers existence, (so loosing the uniformity in the twist size). Here we extend the non recursive proof of existence of whiskered tori to the more than two dimensional cases, by developing some ideas illustrated in the quoted reference.

New architectures for optical TDM switching

A systematic theoretical study of architectures for optical TDM switching, using lithium niobate optical switches and optical fiber delay lines for storage, is undertaken. The architectures allow the bit rate and wavelength transparency of these devices to be exploited. A technique involving recursive definition and proof is used to define the networks, which are mathematically related to Benes and Waksman networks. This produces architectures that are very different from existing optical TDM networks. They exhibit economical use of components, which reaches the theoretical minimum in some cases. The use of feed-forward rather than feed-back delays give these networks superior crosstalk performance and more uniform attenuation than existing designs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>