Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, as opposed to the more traditional approaches that deal with tree-like objects such as formulas or sequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, is just a special, conservative fragment of this more general and, in a sense, more basic logic -- the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof complexities offered by cirquent calculus, and more. Among the main purposes of this paper is to provide an introductory-style starting point for what, as the author wishes to hope, might have a chance to become a new line of research in proof theory -- a proof theory based on circuits instead of formulas.
Deep Dive into Cirquent calculus deepened.
Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, as opposed to the more traditional approaches that deal with tree-like objects such as formulas or sequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, is just a special, conservative fragment of this more general and, in a sense, more basic logic – the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof comp
Among the main objectives of the introductory section of a well-written paper should be to help the reader determine whether he or she is willing to invest time into reading the rest of it. The following rhetorical question1 may contribute to making such a determination in the present case:
What is the most natural representation of Boolean functions, formulas or circuits? Those who are not quite clear about the meaning of the word “natural”, may try to replace it with “direct”, “reasonable” or “efficient”, and think about where the computer industry would be at present if, for some strange reason, computer engineers had insisted on tree-rather than graph-style cicuitries. Or ask why one does not hear theoretical computer scientists speak about formula complexity nearly as often as about circuit complexity.
Should then proof-theoreticians continue sticking to formulas, especially now that logic is increasingly CS-oriented, and efficiency is of much greater concern than it was in the days of Frege, Hilbert and Gentzen? The author believes that there are no good reasons for such conservatism other than habit and tradition, if not laziness. And this paper is for those who might feel potentially ready to accept the same view, or be curious enough to be willing to take a look at what happens when one gives the idea a try. It is devoted to (re)introducing and advancing the foundations of cirquent calculus, the circuit-based proof theory.
Unlike the more traditional syntactic approaches that manipulate tree-or forest-like objects such as formulas or sequents and where proofs are often also trees, cirquent calculus deals with circuit-style constructs called cirquents, in which children may be shared between different parent nodes. Furthermore, being intrinsically a deep inference (see later) approach, it makes possible combining, within a single cirquent, what would otherwise be different parallel nodes (formulas, sequents) of a proof tree, meaning that eventually not only subcirquents, but also subtransformations are amenable to being shared. Sharing thus allows us to achieve higher efficiency, whether it be the compactness of representations of Boolean functions or other objects of study, or the numbers of steps in derivations and proofs. Indeed, in natural situations, specifically ones arising in the world of computing, prohibitively long formulas typically owe their sizes to reoccurring subformulas, and explosively large proof trees often emerge as a result of the necessity to perform identical or similar steps over and over again.
The possibility of compressing formulas or proofs is not the only -in fact, not even the primary -appeal of cirquent calculus. Generality, flexibility and expressiveness are other, more fundamental, advantages to point out. Cirquent calculus is more general than the calculus of structures (Guglielmi et al. [3,4,9,10]); the latter is more general than hypersequent calculus (Avron [1], Pottinger [17]); and the latter, in turn, is more general than sequent calculus (Gentzen). Each framework in this hierarchy permits to successfully axiomatize certain logics that the predecessor frameworks fail to tame. Cirquent calculus itself was originally introduced as a deductive system for the resource-conscious computability logic [12,13,15] after it had become evident that neither sequent calculus nor the more flexible and promising calculus of structures were sufficient to axiomatize it.
While in classical logic circuits do not offer any additional expressive power, they -more precisely, cirquents that are more general than circuits -turn out to be properly more expressive than formulas when it comes to finer semantical approaches such as resource logics, with computability logic ( [12,13,15,16]) and abstract resource semantics ( [14]) being two examples. Efficiency considerations totally aside, it was exactly this expressive power that in [14] made a difference between axiomatizability and unaxiomatizability for computability logic or abstract resource semantics: even if one is only trying to set up a deductive system that proves all (and only) valid formulas, intermediate steps in proofs of such formulas still inherently require using objects (cirquents) that cannot be written as formulas.
Switching from formulas to cirquents indeed becomes imperative -not only syntactically but also semantically -if one wants to systematically develop resource logics. Fine-level resource-semantical approaches intrinsically require the ability to account for the possibility of resource sharing, the ability that linear logic or other formula-or sequent-based approaches do not and cannot possess. The following naive example may provide some insights.
We are talking about a vending machine that has slots for 25-cent (25c) coins, with each slot taking a single coin. Coins can be authentic or counterfeited. Let us instead use the more generic terms true and false here, as there are various particular situati
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
