Proof Length Research Articles

Looking From The Inside And From The Outside

Many times in mathematics there is a natural dichotomy betweendescribing some object from the inside and from the outside. Imaginealgebraic varieties for instance; they can be described from theoutside as solution sets of polynomial equations, but one can also tryto understand how it is for actual points to move around inside them,perhaps to parameterize them in some way. The concept of formalproofs has the interesting feature that it provides opportunities forboth perspectives. The inner perspective has been largely overlooked,but in fact lengths of proofs lead to new ways to measure informationcontent of mathematical objects. The disparity between minimallengths of proofs with and without ``lemmas'' provides an indicationof internal symmetry of mathematical objects and their descriptions.A principal observation of this paper is that mathematicalstructures can be embedded into spaces of logical formulae and inheritadditional structure from proofs. We shall look at finitely-generatedgroups, rational numbers and SL(2,Z), and examples fromtopology and analysis.

Bounded arithmetic, proof complexity and two papers of Parikh

This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs.

Automating the Search for Elegant Proofs

The research reported in this article was spawned by a colleague’s request to find an elegant proof (of a theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request was met by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far shorter proof is presented in detail through a sequence of experiments. Although clearly not an algorithm, the methodology is sufficiently general to enable its use for seeking elegant proofs regardless of the domain of study. In addition to (usually) being more elegant, shorter proofs often provide the needed path to constructing a more efficient circuit, a more effective algorithm, and the like. The methodology relies heavily on the assistance of McCune’s automated reasoning program OTTER. Of the aspects of proof elegance, the main focus here is on proof length, with brief attention paid to the type of term present, the number of variables required, and the complexity of deduced steps. The methodology is iterative, relying heavily on the use of three strategies: the resonance strategy, the hot list strategy, and McCune’s ratio strategy. These strategies, as well as other features on which the methodology relies, do exhibit tendencies that can be exploited in the search for shorter proofs and for other objectives. To provide some insight regarding the value of the methodology, I discuss its successful application to other problems from Boolean algebra and to problems from lattice theory. Research suggestions and challenges are also offered.

Length and Structure of Proofs

Etude du probleme de la mesure de la longueur des preuves mathematiques chez Godel, fonde sur le critere du nombre de symboles utilises dans la demonstration. Appliquant les techniques de calcul developpees par R. Parikh dans le domaine de l'arithmetique de Peano au contexte des paradoxes sorites, l'A. compare le nombre des symboles a celui des lignes de la preuve, d'une part, et en mesure les consequences sur les differents moyens de formalisation (systeme schematique, graphique du flux logique), d'autre part. L'A. evoque les travaux recents consacres au lien entre la preuve et les problemes de complexite dans le calcul des propositions selon S. Cook et dans les systemes fregeens

Using induction and rewriting to verify and complete parameterized specifications

In software engineering there is a growing demand for formal methods for the specification and validation of software systems. The formal development of a system might give rise to many proof obligations. We must prove the completeness of the specification and the validity of some inductive properties. In this framework, many provers have been developed. However they require much user interaction even for simple proof tasks. In this paper, we present new procedures to test sufficient completeness and to prove or disprove inductive properties automatically in para-meterized conditional specifications. The method has been implemented in the prover SPIKE. Computer experiments illustrate the improvements in length and structure of proofs, due to parameterization. Moreover, SPIKE offers facilities to check and complete specifications.

Formalizing forcing arguments in subsystems of second-order arithmetic

We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL +0 is conservative over RCA 0 with no significant increase in the lengths of proofs.

Using induction and rewriting to verify and complete parameterized specifications

In software engineering there is a growing demand for formal methods for the specification and validation of software systems. The formal development of a system might give rise to many proof obligations. We must prove the completeness of the specification and the validity of some inductive properties. In this framework, many provers have been developed. However they require much user interaction even for simple proof tasks. In this paper, we present new procedures to test sufficient completeness and to prove or disprove inductive properties automatically in para-meterized conditional specifications. The method has been implemented in the prover SPIKE. Computer experiments illustrate the improvements in length and structure of proofs, due to parameterization. Moreover, SPIKE offers facilities to check and complete specifications.

Proof Lengths for Equational Completion

We first show that ground term-rewriting systems can be completed in a polynomial number of rewriting steps, if the appropriate data structure for terms is used. We then apply this result to study the lengths of critical pair proofs in non-ground systems, and obtain bounds on the lengths of critical pair proofs in the non-ground case. We show how these bounds depend on the types of inference steps that are allowed in the proofs.

Some remarks on lengths of propositional proofs

Some remarks on lengths of propositional proofs

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(md+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed.

On interprocess communication and the implementation of multi-writer atomic registers

Two protocols for implementing n-writer m-reader atomic registers with 1-writer m-reader atomic registers are described. In order to give complete proofs, a theory of interprocess communication is presented first. The correctness of a protocol that implements an atomic register is proved here in two stages: 1. (1) a formulation of higher-level specifications and a proof that the protocol satisfies these specifications. 2. (2) a proof of atomicity assuming that the specifications hold. This division enables a better understanding of the protocols, and the fact that both protocols share the same higher-level specifications reduces the length of the correctness proof. The difference between the two protocols is that in the first the readers do not write at all, while in the second they do. The first protocol is space efficient, while the second is time efficient.

An exponential lower bound to the size of bounded depth frege proofs of the pigeonhole principle

AbstractWe prove lower bounds of the form exp(nε d), εd > 0, on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d, for any constant d. This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.

On Gödel's theorems on lengths of proofs I: Number of lines and speedup for arithmetics

AbstractThis paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact, unbounded) proof speedup of (i + l)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and • as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as axioms and allows all generalizations of axioms as axioms.Our first proof of Gödel's claim is based on self-referential sentences: we give a second proof that avoids the use of self-reference based loosely on a method of Statman.

A Note on the Length of Proofs

A Note on the Length of Proofs

The deduction rule and linear and near-linear proof simulations

AbstractWe introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

An optimality result for clause form translation

The exponential complexity in size of the standard clause form translation is often considered as a serious drawback of the resolution method. Fortunately, a polynomial translation is possible by first introducing definitions, one for each subformula of the conjecture. This exhaustiveness can however be proved inefficient when the length of proofs is considered. In order to improve this interesting technique, we first generalize it to renamings, which consist in introducing definitions only for a subset of subformulas, resulting in a wide set of possible clause forms from a single conjecture. We show how a simple and efficient algorithm yields a renaming which, on equivalence-free conjectures, minimizes the number of clauses among these clause forms. This translation has been tested on the famous challenge problem by P. Andrews, yielding a spectacular reduction in search space and time, and therefore is one of the more simple and general technique to efficiently produce a resolution proof for this problem.

Complexity of resolution proofs and function introduction

The length of resolution proofs is investigated, relative to the model-theoretic measure of Herband complexity. A concept of resolution deduction (R-deduction) is introduced which is somewhat more general than the classical concepts. It is shown that proof complexity is exponential in terms of Herband complexity and that this bound is tight. The concept of R-deduction is extended to FR-deduction, where, besides resolution, a function introduction rule (based on a re-Skolemization technique) is allowed. As an example, consider the clause P(x, y) ∨ Q(x, y): conclude P(x,f(x)) ∨ Q(a, y), where a, f(x) are new function symbols extending Herbrand's universe. (We use the derivability of (∀x)(∃y) P(x, y) ∨ (∃x)(∀) Q(x, y) from (∀x)(∀y)( P(x, y) ∨ Q(x, y)) and Skolemization). By modification of a construction of Statman it is shown that FR-deductions may be nonelementarily shorter than R-deductions (thus giving a nonelementary speedup with respect to Herbrand complexity and search space). Finally it is proved that FR-deduction has weaker effects only (maximally double exponential speedup) if clausal logic is restricted to Horn logic.

Remark on Kreisel's conjecture

At the end of the 60's, Kreisel conjectured that formal arithmetics admit infinite induction rules if the lengths of proofs of their premises are uniformly bounded by the same number. By the length of a proof we mean the number of applications of axioms and rules of inference. In this article we construct a theoryR*with a finite number of specific axioms. The language ofR* contains a constant 0, a unary function symbol, equality, and ternary predicates for addition and multiplication. It is proved that for any consistent axiomatizable extension R* it is possible to find a formula A(a) that satisfies the following conditions: a) ∀xA(x) cannot be derived in ; b) for any n the length of the proof of the formula A(0(n)) is no greater than c1[log2 (n+1)]+c2, where the constants c1 and c2 are independent of n. Here the expression 0(n) indicates 0 with n primes.

On the Length of Proofs after Eliminating Atomic Cut Inferences

On the Length of Proofs after Eliminating Atomic Cut Inferences

Propositional proof systems, the consistency of first order theories and the complexity of computations

AbstractWe consider the problem about the length of proofs of the sentences saying that there is no proof of contradiction in S whose length is < n. We show the relation of this problem to some problems about propositional proof systems.