Uniform Proof Research Articles

Lower complexity bounds in justification logic

Justification logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. The proof method is then extended to provide a uniform proof that the corresponding full pure justification logics are Π2p-hard, reproving and generalizing an earlier result by Milnikel.

G -complete reducibility and semisimple modules

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre's notion of G-complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index (H:H°) is prime to p and that p > 2 dim V−2 for some faithful G-module V. Then the following hold: (i) V is a semisimple H-module if and only if H is G-completely reducible; (ii) H° is reductive if and only if H is G-completely reducible. We also discuss two new related results. (i) If p ⩾ dim V for some G-module V and H is a G-completely reducible subgroup of G, then V is a semisimple H-module; this generalizes Jantzen's semisimplicity theorem (which is the case H = G). (ii) If H acts semisimply on V ⊗ V* for some faithful G-module V, then H is G-completely reducible.

An inductive approach to Coxeter arrangements and Solomon’s descent algebra

In our recent paper (Douglass et al. arXiv:1101.2075 (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik---Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.

A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

AbstractWe build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

Cellularity and the Jones basic construction

We establish a framework for cellularity of algebras related to the Jones basic construction. Our framework allows a uniform proof of cellularity of Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras, partition algebras, and others. Our cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for the class of algebras that we study, we show that the cellular structures are compatible with restriction and induction of modules.

Sortable elements in infinite Coxeter groups

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Formats of Winning Strategies for Six Types of Pushdown Games

The solution of parity games over pushdown graphs (Walukiewicz '96) was the first step towards an effective theory of infinite-state games. It was shown that winning strategies for pushdown games can be implemented again as pushdown automata. We continue this study and investigate the connection between game presentations and winning strategies in altogether six cases of game arenas, among them realtime pushdown systems, visibly pushdown systems, and counter systems. In four cases we show by a uniform proof method that we obtain strategies implementable by the same type of pushdown machine as given in the game arena. We prove that for the two remaining cases this correspondence fails. In the conclusion we address the question of an abstract criterion that explains the results.

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.

Term Sequent Logic

We consider a term sequent logic for the lambda-calculus. Term sequents are a judgement form similar to the logical judgement form of entailment between sentences, but denoting equality or reducibility between terms. Using term sequents, it is possible to treat lambda-terms almost like logical sentences, and to use proof-theoretic methods to establish their properties. We prove a cut-elimination result for untyped lambda-calculus and describe how this generalises the usual confluence result. We give a notion of uniform proof for lambda-terms, and suggest how this can be viewed as a mixed logic-programming/functional programming framework with the ability to assume arbitrary reductions. Finally, we discuss related and future work.

Bellman function and bilinear embedding theorem for Schrödinger-type operators

We discuss bilinear embedding theorems for a certain class of Schrödinger operators on L p . The obtained estimates are dimension-free and linear in p. We outline a uniform proof of the theorem which relies on establishing three crucial properties of the concrete Bellman function we consider. To cite this article: O. Dragičević, A. Volberg, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On Tits' Centre Conjecture for fixed point subcomplexes

We give a short and uniform proof of a special case of Tits' Centre Conjecture using a theorem of J.-P. Serre and a result from the authors in 2005. We consider fixed point subcomplexes X H of the building X = X ( G ) of a connected reductive algebraic group G, where H is a subgroup of G. To cite this article: M. Bate et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Two modular equations for squares of the Rogers–Ramanujan functions with applications

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R2(q2). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q4) and ν:=R2(q1/2)R(q)/R(q2). New proofs are offered for modular relations connecting R(q) to R(−q), R(q2), and R(q4), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.

A simple proof of Pommerening's theorem

Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p ⩾ 0 . Assume that p is good for G. Pommerening's theorem [K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977) 525–536; K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, II, J. Algebra 65 (1980) 373–398] asserts that any distinguished nilpotent element in the Lie algebra g of G is a Richardson element for a distinguished parabolic subgroup of G. This theorem implies the Bala–Carter theorem in good characteristic. In this paper we give a short proof of Pommerening's theorem, which is a further simplification of Premet's first uniform proof [A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003) 338–366]. We also simplify Premet's proof of the existence theorem for good transverse slices to the nilpotent Ad ( G ) -orbits in g .

Integration of a security type system into a program logic

Type systems and program logics are often thought to be at opposing ends of the spectrum of formal software analyses. In this paper we show that a flow-sensitive type system ensuring non-interference in a simple while-language can be expressed through specialised rules of a program logic. In our framework, the structure of non-interference proofs resembles the corresponding derivations in a state-of-the-art security type system, meaning that the algorithmic version of the type system can be used as a proof procedure for the logic. We argue that this is important for obtaining uniform proof certificates in a proof-carrying code framework. We discuss in which cases the interleaving of approximative and precise reasoning allows us to deal with delimited information release. Finally, we present ideas on how our results can be extended to encompass features of realistic programming languages such as Java.

Focusing the Inverse Method for LF: A Preliminary Report

In this paper, we describe a proof-theoretic foundation for bottom-up logic programming based on uniform proofs in the setting of the logical framework LF. We present a forward uniform proofs calculus which is a suitable foundation for the inverse method for LF and prove its correctness. We also present some preliminary results of an implementation for the Horn Fragment as part of the logical framework Twelf, and compare its performance with the tabled logic programming engine.

Theorem proving for conditional logics: CondLean and GOALDUCK

In this paper we focus on theorem proving for conditional logics. First, we give a detailed description of CondLean, a theorem prover for some standard conditional logics. CondLean is a SICStus Prolog implementation of some labeled sequent calculi for conditional logics recently introduced. It is inspired to the so called “lean” methodology, even if it does not fit this style in a rigorous manner. CondLean also comprises a graphical interface written in Java. Furthermore, we introduce a goal-directed proof search mechanism, derived from the above mentioned sequent calculi based on the notion of uniform proofs. Finally, we describe GOALDUCK, a simple SICStus Prolog implementation of the goal-directed calculus mentioned here above. Both the programs CondLean and GOALDUCK, together with their source code, are available for free download at.

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $\sigma$ such that $\sigma |_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite and infinite dimensional cases, and for both discrete and continuous spectra.

A sequent calculus and a theorem prover for standard conditional logics

In this paper we present a cut-free sequent calculus, called SeqS, for some standard conditional logics. The calculus uses labels and transition formulas and can be used to prove decidability and space complexity bounds for the respective logics. We also show that these calculi can be the base for uniform proof systems. Moreover, we present CondLean, a theorem prover in Prolog for these calculi.

The Vaught Conjecture: Do Uncountable Models Count?

We give a model theoretic proof, replacing admissible set theory by the LopezEscobar theorem, of Makkai’s theorem: Every counterexample to Vaught’s conjecture has an uncountable model which realizes only countably many Lω1,ω-types. The following result is new. Theorem. If a first order theory is a counterexample to the Vaught conjecture then it has 2א1 models of cardinality א1. In this paper we prove several properties of putative counterexamples to the Vaught conjecture. Specifically, these results concern the number of models the counterexample has in power א1. One of these results was proved 30 years ago using admissible model theory; we give a more straightforward argument. The following question guides our discussion. Is the Vaught Conjecture model theory? Here are some possible ways in which this question would have a clear answer. Shelah, Buechler, Newelski have shown using rather difficult techniques from stability theory that the conjecture holds for first order theories that are ‘simple’ from the stability theoretic standpoint: ω-stable or superstable with finite U -rank. If a counterexample were found for a first order theory of slightly greater complexity (e.g. a stable but not superstable first order theory), this would indicate the issue was a model theoretic one. If on the other hand, a uniform proof for sentences of Lω1,ω were given using methods of descriptive set theory, then it would not be a model theoretic problem. The results below give partial answers to the following methodological questions. What specific model theoretic as opposed to descriptive set theoretic techniques can attack the problem? Can one use more direct model theoretic arguments to obtain some result of admissible model theory? I would argue the problem is model theoretic if its solution is different for Lω,ω and Lω1,ω. So we will investigate the differences between properties known about counterexamples to

Weakly metrizable pseudocompact groups

We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weaklyextremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex-tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.