Cyclic Implicit Complexity

Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have been proposed. However, little is known about the complexity theoretic aspects of circular proofs, which exhibit sophisticated loop structures atypical of more common 'recursion schemes'. This paper attempts to bridge the gap between circular proofs and implicit computational complexity. Namely we introduce a circular proof system based on Bellantoni and Cook's famous safe-normal function algebra, and we identify suitable proof theoretical constraints to characterise the polynomial-time and elementary computable functions.

In chapters 2.5–7 of the Prior Analytics Aristotle is concerned with what he calls circular proof. He gives an account of circular proofs within the framework of his syllogistic theory, and discusses how they come about in the three figures of categorical syllogisms. The results of this discussion are summarized at the end of chapter 2.7, at 59a32–41. The summary contains several statements to the effect that certain circular proofs come about in the third figure. Some of these statements are problematic because the circular proofs in question are actually not in the third figure of categorical syllogisms; in fact, these circular proofs are not categorical syllogisms at all, but what Theophrastus called prosleptic syllogisms. Hence, the statements are incorrect if they are understood to refer to the third figure of Aristotle’s categorical syllogisms. Since it seems natural to understand them in this way, Ross and others conclude that the passage at 59a32–41 is spurious and should be excised, although it is found in all MSS. 1 By contrast, this paper aims to show that the passage is not spurious. Following Pacius, I argue that the problematic statements in it refer not to the third figure of categorical syllogisms, but to the third figure of prosleptic syllogisms. On this interpretation, the statements are correct and can be regarded as genuine. Given that they are genuine, they show that Aristotle was aware of a classification of prosleptic syllogisms into three figures, even though such a classification does not occur elsewhere in his writings. Thus, the passage at 59a32–41 appears to be the earliest evidence we have of figures of prosleptic syllogisms. I begin with an overview of Aristotle’s treatment of circular proofs in Prior Analytics 2.5–7, focussing on his use of prosleptic syllogisms (§1). Readers familiar with the contents of Prior Analytics 2.5–7 may wish to skip this overview. Next we consider the problematic statements in 59a32–41 (§2). I will argue that these statements refer to the third figure of prosleptic syllogisms, and that there is no reason to doubt Aristotle’s authorship of the passage (§3).

One of the authors introduced in [Santocanale, FoSSaCS, 2002] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [Santocanale, FoSSaCS, 2002] is cut-free; even if sound and complete for provability, it lacked an important property for the semantics of proofs, namely fullness w.r.t. the class of intended categorical models (called mu-bicomplete categories in [Santocanale, ITA, 2002]). In this paper we fix this problem by adding the cut rule to the calculus and by modifying accordingly the syntactical constraint ensuring soundness of proofs. The enhanced proof system fully represents arrows of the canonical model (a free mu-bicomplete category). We also describe a cut-elimination procedure as a a model of computation arising from the above mentioned categorical operations. The procedure constructs a cut-free proof-tree with possibly infinite branches out of a finite circular proof with cuts.

One of the authors introduced in [1] a calculus ofcircular proofs for studying the computability arising from thefollowing categorical operations: finite products and coproducts,initial algebras, final coalgebras. The calculus of[1] is cut-free; yet, even if sound and complete forprovability, it lacks an important property for the semantics ofproofs, namely fullness w.r.t. the class of natural categorical modelscalled μ-bicomplete category in [2].We fix, with this work, this problem by adding the cut rule to thecalculus. To this goal, we need to modifying the syntacticalconstraints on the cycles of proofs so to ensure soundness of thecalculus and at same time local termination of cut-elimination. Theenhanced proof system fully represents arrows of the intended model, afree μ-bicomplete category. We also describe a cut-eliminationprocedure as a model of computation arising from the above mentionedcategorical operations. The procedure constructs a cut-freeproof-tree with infinite branches out of a finite circular proof withcuts.[1] Luigi Santocanale. A calculus of circular proofs and its categorical semantics. In Mogens Nielsen and Uffe Engberg, editors, FoSSaCS, volume 2303 of Lecture Notes in Computer Science, pages 357–371. Springer, 2002.[2] Luigi Santocanale. μ-bicomplete categories and parity games. Theoretical Informatics and Applications, 36:195–227, September 2002.

We introduce a new way of composing proofs in rule-based proof systems that generalizes tree-like and dag-like proofs. In the new definition, proofs are directed graphs of derived formulas, in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs circular. We show that, for all sets of standard inference rules, circular proofs are sound. We first focus on the circular version of Resolution, and see that it is stronger than Resolution since, as we show, the pigeonhole principle has circular Resolution proofs of polynomial size. Surprisingly, as proof systems for deriving clauses from clauses, Circular Resolution turns out to be equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: (1) polynomial-time (LP-based) algorithms that find circular Resolution proofs of constant width, (2) examples that separate circular from dag-like Resolution, such as the pigeonhole principle and its variants, and (3) exponentially hard cases for circular Resolution. Contrary to the case of circular resolution, for Frege we show that circular proofs can be converted into tree-like ones with at most polynomial overhead.

Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, as directed acyclic graphs of derived formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type of structural restriction for proofs in rule-based proof systems. In this definition, proofs are directed graphs of derived formulas in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs “circular”. We show that, for all sets of standard inference rules with single or multiple conclusions, circular proofs are sound. We start the study of the proof complexity of circular proofs at Circular Resolution, the circular version of Resolution. We immediately see that Circular Resolution is stronger than dag-like Resolution since, as we show, the propositional encoding of the pigeonhole principle has circular Resolution proofs of polynomial size. Furthermore, for derivations of clauses from clauses, we show that Circular Resolution is, surprisingly, equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: (1) polynomial-time (LP-based) algorithms that find Circular Resolution proofs of constant width, (2) examples that separate Circular from dag-like Resolution, such as the pigeonhole principle and its variants, and (3) exponentially hard cases for Circular Resolution. Contrary to the case of Circular Resolution, for Frege we show that circular proofs can be converted into tree-like proofs with at most polynomial overhead.

Logics based on the \(\mu \)-calculus are used to model inductive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. The traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads.

Implicit characterizations of FPTIME and NC revisited

Proof theory provides a foundation for studying and reasoning about programming languages, most directly based on the well-known Curry-Howard isomorphism between intuitionistic logic and the typed lambda-calculus. More recently, a correspondence between intuitionistic linear logic and the session-typed pi-calculus has been discovered. In this paper, we establish an extension of the latter correspondence for a fragment of substructural logic with least and greatest fixed points. We describe the computational interpretation of the resulting infinitary proof system as session-typed processes, and provide an effectively decidable local criterion to recognize mutually recursive processes corresponding to valid circular proofs as introduced by Fortier and Santocanale. We show that our algorithm imposes a stricter requirement than Fortier and Santocanale's guard condition, but is local and compositional and therefore more suitable as the basis for a programming language.

We present a sequent-style proof system for provability logic GL that admits so-called circular proofs. For these proofs, the graph underlying a proof is not a finite tree but is allowed to contain cycles. As an application, we establish Lindon interpolation for GL syntactically.

The thesis applies the ICC tecniques to the probabilistic polinomial complexity classes in order to get an implicit characterization of them. The main contribution lays on the implicit characterization of PP (which stands for Probabilistic Polynomial Time) class, showing a syntactical characterisation of PP and a static complexity analyser able to recognise if an imperative program computes in Probabilistic Polynomial Time. The thesis is divided in two parts. The first part focuses on solving the problem by creating a prototype of functional language (a probabilistic variation of lambda calculus with bounded recursion) that is sound and complete respect to Probabilistic Prolynomial Time. The second part, instead, reverses the problem and develops a feasible way to verify if a program, written with a prototype of imperative programming language, is running in Probabilistic polynomial time or not. This thesis would characterise itself as one of the first step for Implicit Computational Complexity over probabilistic classes. There are still open hard problem to investigate and try to solve. There are a lot of theoretical aspects strongly connected with these topics and I expect that in the future there will be wide attention to ICC and probabilistic classes.

We give a recursion-theoretic characterization of the complexity classes NCkfor ki¾? 1. In the spirit of implicit computational complexity, it uses no explicit bounds in the recursion and also no separation of variables is needed. It is based on three recursion schemes, one corresponds to time (time iteration), one to space allocation (explicit structural recursion) and one to internal computations (mutual in place recursion). This is, to our knowledge, the first exact characterization of NCkby function algebra over infinite domains in implicit complexity.

The Recursion Scheme from the Cofree Recursive Comonad

A recurring theme in theory of computation is that of a function algebra 1 i.e., a smallest class of functions containing certain initial functions and closed under certain operations (especially substitution and primitive recursion).2 In 1904, G.H. Hardy [Har04] used related concepts to define sets of real numbers of cardinality ℵ1. In 1923, Th. Skolem [Sko23] introduced the primitive recursive functions, and in 1925, as a technical tool in his claimed sketch proof of the continuum hypothesis, D. Hilbert [Hil25] defined classes of higher type functionals by recursion. In 1928, W. Ackermann [Ack28] furnished a proof that the diagonal function φ a (a, a) of Hilbert [Hi125], a variant of the Ackermann function, is not primitive recursive. In 1931, K. Godel [God31] defined the primitive recursive functions, there calling them “rekursive Funktionen” , and used them to arithmetize logical syntax via Godel numbers for his incompleteness theorem. Generalizing Ackermann’s work, in 1936 R. Peter [Pet36] defined and studied the k-fold recursive functions. The same year saw the introduction of the fundamental concepts of Turing machine (A.M. Turing [Tur37]), λ-calculus (A. Church [Chu36]) and μ-recursive functions (S.C. Kleene [Kle36a]). By restricting the scheme of primitive recursion to allow only limited summations and limited products, the elementary functions were introduced in 1943 by L. Kalmar [Kal43].

We obtain symmetric joint eigenfunctions for the commuting partial differential operators associated to the hyperbolic Calogero-Moser |$N$|-particle system. The eigenfunctions are constructed via a recursion scheme, which leads to representations by multidimensional integrals whose integrands are elementary functions. We also tie in these eigenfunctions with the Heckman–Opdam hypergeometric function for the root system |$A_{N-1}$|⁠.