Frege Proofs Research Articles

Short propositional refutations for dense random 3CNF formulas

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n1.5−ε) clauses (Chvátal and Szemerédi [14], Ben-Sasson and Wigderson [10]). On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with Ω(n2/log⁡n) clauses, shown by Beame et al. [6]. In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC0 formulas (i.e., TC0-Frege proofs) for random 3CNF formulas with n variables and Ω(n1.4) clauses.The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek [23]. Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.

Lifting lower bounds for tree-like proofs

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like sequent calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and "lift" it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I Δ0(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy $${G_i^*}$$ of quantified propositional proof systems does not collapse.

Polylogarithmic Cuts in Models of V^0

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of $\mathbf{V}^0$ are models of $\mathbf{VNC}^1$ by formalizing a proof of Nepomnjascij's Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie. We can then exploit our result in Proof Complexity to observe that Frege proof systems can be sub exponentially simulated by bounded depth Frege proof systems. This result has recently been obtained by Filmus, Pitassi and Santhanam in a direct proof. As an interesting observation we also obtain an average case separation of Resolution from AC0-Frege by applying a recent result with Tzameret.

Propositional proofs and reductions between [formula omitted] search problems

Total NP search problems (TFNP problems) typically have their totality guaranteed by some combinatorial property. This paper proves that, if there is a polynomial time Turing reduction between two TFNP problems, then there are quasipolynomial size, polylogarithmic height, constant depth, free-cut free propositional (Frege) proofs of the combinatorial property associated with the first TFNP problem from the property associated with the second problem. In addition, they have Nullstellensatz derivations of polylogarithmic degree. These results extend the previous work of Buresh-Oppenheim and Morioka first by applying to Turing reductions in place of many-one reductions and second by giving tight bounds on the height of the Frege proofs. As a corollary, PLS-complete problems are not polynomial time Turing reducible to PPA problems relative to a generic oracle. We establish a converse construction as well, by showing that a polynomial time Turing reduction can be obtained from a family of quasipolynomial size, polylogarithmic depth, propositional proofs which are based on “decision tree” substitutions. This establishes the optimality of our constructions, and partially resolves an open question posed by Buresh-Oppenheim and Morioka.We observe that the classes PPA, PPAD, PPADS, and PLS are closed under Turing reductions, and give an example of a TFNP class that is not closed under Turing reductions.

Algebraic proofs over noncommutative formulas

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook–Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003). We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas ( PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitinʼs formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991). The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.

On theories of bounded arithmetic for [formula omitted

We develop an arithmetical theory VNC ∗ 1 and its variant VNC ¯ ∗ 1 , corresponding to “slightly nonuniform” NC 1 . Our theories sit between VNC 1 and VL , and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of Σ 0 B ( L VNC ¯ ∗ 1 ) -formulas provable in VNC ¯ ∗ 1 admit L -uniform polynomial-size Frege proofs.

A note on propositional proof complexity of some Ramsey-type statements

A Ramsey statement denoted \({n \longrightarrow (k)^2_2}\) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n k) and with terms of size \({\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}\) . Let r k be the minimal n for which the statement holds. We prove that RAM(r k , k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4k, k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.

Lower bounds for bounded depth Frege proofs via Pudlák-Buss games

We present a simple proof of the bounded-depth Frege proof lower bounds of Pitassi et al. [1993] and Krajíček et al. [1995] for the pigeonhole principle. Our method uses the interpretation of proofs as two player games given by Pudlák and Buss. Our lower bound is conceptually simpler than previous ones, and relies on tools and intuition that are well known in the context of computational complexity. This makes the lower bound of Pitassi et al. [1993] and Krajíček et al. [1995] accessible to the general computational complexity audience. We hope this new view will open new directions for research in proof complexity.

Syntax in Basic Laws §§29–32

In order to accommodate his view that quantifiers are predicates of predicates within a type theory, Frege introduces a rule which allows a function name to be formed by removing a saturated name from another saturated name which contains it. This rule requires that each name has a rather rich syntactic structure, since one must be able to recognize the occurrences of a name in a larger name. However, I argue that Frege is unable to account for this syntactic structure. I argue that this problem undermines the inductive portion of Frege's proof that all of the names of his system denote in §§29–32 of The Basic Laws.

Substitution Frege and extended Frege proof systems in non-classical logics

We investigate the substitution Frege ( SF ) proof system and its relationship to extended Frege ( EF ) in the context of modal and superintuitionistic (si) propositional logics. We show that EF is p-equivalent to tree-like SF , and we develop a “normal form” for SF -proofs. We establish connections between SF for a logic L , and EF for certain bimodal expansions of L . We then turn attention to specific families of modal and si logics. We prove p-equivalence of EF and SF for all extensions of KB , all tabular logics, all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual EF system for classical logic with respect to a naturally defined translation. On the other hand, we establish exponential speed-up of SF over EF for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result.

Quantum deduction rules

We define propositional quantum Frege proof systems and compare them with classical Frege proof systems.

Frege on Definitions

AbstractThis article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo‐Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre‐definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension.

Polynomal-size Frege proofs of Bollobás’ theorem on the trace of sets

In this note we show that Bollobas' theorem on the trace of sets has polynomial-size Frege proofs.

HOW DID FREGE FALL INTO THE CONTRADICTION?

AbstractQuine made it conventional to portray the contradiction that destroyed Frege's logicism as some kind of act of God, a thunderbolt that descended from a clear blue sky. This portrayal suited the moral Quine was antecedently inclined to draw, that intuition is bankrupt, and that reliance on it must therefore be replaced by a pragmatic methodology. But the portrayal is grossly misleading, and Quine's moral simply false. In the person of others – Cantor, Dedekind, and Zermelo – intuition was working pretty well. It was in Frege that it suffered a local and temporary blindness. The question to ask, then, is not how Frege was overtaken by the contradiction, but how it is that he didn't see it coming. The paper offers one kind of answer to that question. Starting from the very close similarity between Frege's proof of infinity and the reasoning that leads to the contradiction, it asks: given his understanding of the first, why did Frege did not notice the second? The reason is traced, first, to a faulty generalization Frege made from the case of directions and parallel lines; and, through that, to Frege's having retained, and attempted incoherently to combine with his own, aspects of a pre‐Fregean understanding of the generality of logical principles.

The proof theoretic strength of the Steinitz exchange theorem

We show that the logical theory QLA proves the Cayley–Hamilton theorem from the Steinitz exchange theorem together with a strengthening of the linear independence principle. Since QLA is a fairly weak theory (in the sense that its quantifier-free fragment, LA , translates into tautologies with TC 0 -Frege proofs—when restricted to the field Q of the rationals), it follows that the proof complexity of matrix algebra can be distilled to the Steinitz exchange theorem.

Polynomial-size Frege and resolution proofs of st-connectivity and Hex tautologies

A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st-connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere. We prove that the propositional tautologies which encode the st-connectivity principle have polynomial-size Frege proofs and polynomial-size TC 0 -Frege proofs. For bounded-width grid graphs, the st-connectivity tautologies have polynomial-size resolution proofs. A key part of the proof is to show that the group with two generators, both of order two, has word problem in alternating logtime (Alogtime) and even in TC 0 . Conversely, we show that constant depth Frege proofs of the st-connectivity tautologies require near-exponential size. The proof uses a reduction from the pigeonhole principle, via tautologies that express a “directed single source” principle SINK, which is related to Papadimitriou's search classes PPAD and PPADS (or, PSK). The st-connectivity principle is related to Urquhart's propositional Hex tautologies, and we establish the same upper and lower bounds on proof complexity for the Hex tautologies. In addition, the Hex tautology is shown to be equivalent to the SINK tautologies and to the one-to-one onto pigeonhole principle.

Upper bounds on complexity of Frege proofs with limited use of certain schemata

The paper considers a commonly used axiomatization of the classical propositional logic and studies how different axiom schemata in this system contribute to proof complexity of the logic. The existence of a polynomial bound on proof complexity of every statement provable in this logic is a well-known open question.

Bounded-Depth Frege Lower Bounds for Weaker Pigeonhole Principles

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle $PHP^{m}_n$ where $m= (1+1/{\polylog n})n$. This lower bound qualitatively matches the known quasi-polynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for bounded-depth Frege proofs, is novel in that the tautology to which this switching lemma is applied remains random throughout the argument.

Non-Automatizability of Bounded-Depth Frege Proofs

In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that bounded-depth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the Diffie-Hellman function is sufficiently hard. It follows as a corollary that bounded-depth Frege is not automatizable; in other words, there is no deterministic polynomial-time algorithm that will output a short proof if one exists. A notable feature of our argument is its simplicity.

Weak theories of linear algebra

We investigate the theories Open image in new window of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment Open image in new window in which we can interpret a weak theory V1 of bounded arithmetic and carry out polynomial time reasoning about matrices - for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, Open image in new window proves the commutativity of inverses.