We study the computational expressivity of proof systems with fixed point operators, within the 'proofs-as-programs' paradigm. We start with a calculus muLJ (due to Clairambault) that extends intuitionistic logic by least and greatest positive fixed points. Based in the sequent calculus, muLJ admits a standard extension to a 'circular' calculus CmuLJ. Our main result is that, perhaps surprisingly, both muLJ and CmuLJ represent the same first-order functions: those provably total in $Π^1_2$-$\mathsf{CA}_0$, a subsystem of second-order arithmetic beyond the 'big five' of reverse mathematics and one of the strongest theories for which we have an ordinal analysis (due to Rathjen). This solves various questions in the literature on the computational strength of (circular) proof systems with fixed points. For the lower bound we give a realisability interpretation from an extension of Peano Arithmetic by fixed points that has been shown to be arithmetically equivalent to $Π^1_2$-$\mathsf{CA}_0$ (due to Möllerfeld). For the upper bound we construct a novel computability model in order to give a totality argument for circular proofs with fixed points. In fact we formalise this argument itself within $Π^1_2$-$\mathsf{CA}_0$ in order to obtain the tight bounds we are after. Along the way we develop some novel reverse mathematics for the Knaster-Tarski fixed point theorem.

Abstract A bilateralist take on proof-theoretic semantics can be understood as demanding of a proof system to display not only rules giving the connectives’ provability conditions but also their refutability conditions. On such a view, then, a system with two derivability relations is obtained, which can be quite naturally expressed in a proof system of natural deduction but which faces obstacles in a sequent calculus representation. Since in a sequent calculus there are two derivability relations inherent, one expressed by the sequent sign and one by the horizontal lines holding between sequents, in a truly bilateral calculus both need to be dualized. While dualizing the sequent sign is rather straightforwardly corresponding to dualizing the horizontal lines in natural deduction, dualizing the horizontal lines in sequent calculus, uncovers problems that, as will be argued in this paper, shed light on deeper conceptual issues concerning an imbalance between the notions of proof vs. refutation. The roots of this problem will be further analyzed and possible solutions on how to retain a bilaterally desired balance in our system are presented.

Given the high cost of formal verification, a large system may include differently analyzed components: a few are fully verified, and the rest are tested. Currently, there is no reasoning system that can soundly compose these heterogeneous analyses and derive the overall formal guarantees of the entire system. The traditional compositional reasoning technique—rely-guarantee reasoning—is effective for verified components, which undergo over-approximated reasoning, but not for those components that undergo under-approximated reasoning, e.g., using testing or other program analysis techniques. The goal of this paper is to develop a formal, logical foundation for composing heterogeneous analysis, deploying both over-approximated (verification) and under-approximated (testing) reasoning. We focus on systems that can be modeled as a collection of communicating processes. Each process owns its internal resources and a set of channels through which it communicates with other processes. The key idea is to quantify the guarantees obtained about the behavior of a process as a test level, which captures the constraints under which this guarantee is analyzed to be true. We design a novel proof system LabelBI based on the logic of bunched implications that enables rely-guarantee reasoning principles for a system of differently analyzed components. We develop trace semantics for this logic, against which we prove our logic is sound. We also prove cut elimination of our sequent calculus. We demonstrate the expressiveness of our logic via a case study.

Abstract This paper presents a comprehensive proof-theoretic analysis of Jaśkowski’s discussive (or discursive) logic, working with a set of connectives including classical negation and disjunction, as well as so-called (right-)discussive conjunction and discussive implication. By employing established techniques two labelled frameworks are introduced: sequent and natural deduction systems. The paper explores the ability of the proposed calculi to accurately represent Jaśkowski’s discussive logic, particularly in light of its paraconsistent nature, and establishes cut-admissibility and normalization theorems. Additionally, the introduced sequent calculus – shown to allow terminating proof search – is employed to prove the embedding of discussive logic within modal logic $$\textbf{S5}$$ S 5 . Finally, it is proved that the natural deduction calculus translates into the corresponding sequent system, with soundness and completeness established for both calculi. Concluding remarks highlight the potential for expanding this study and suggest directions for future research.

Abstract The quest of smoothly combining logics so that connectives from different logics can co-exist in peace has been a fascinating topic of research. In 2015, Dag Prawitz introduced a natural deduction system for an ecumenical first-order logic, unifying classical and intuitionistic logics within a shared language. Building upon this foundation, we introduced, in a series of works, sequent systems for ecumenical logics and modal extensions. In this work we propose a new pure sequent calculus version for Prawitz’s original system, where each rule features precisely one logical operator. This is achieved by extending sequents with an additional context, called stoup, and establishing the ecumenical concept of polarities. We smoothly extend these ideas for handling modalities, presenting a new pure labelled system for ecumenical modal logics. Finally, we show how this allows for naturally retrieving the ecumenical modal nested system proposed in a previous work.

Abstract The equational theory of the class of lattices with a pair of unary residuated operations is shown to be decidable in $$O(n^5)$$ O ( n 5 ) time. The same complexity holds in the bounded case. The equational theory of the class of lattices, as well as the class of bounded lattices, with a unary operator is shown to be decidable in $$O(n^3)$$ O ( n 3 ) time. Explicit algorithms are given for deciding the above equational theories. These algorithms use a dynamic programming approach and are based on a sequent calculus that extends Whitman’s sequent calculus for lattices.

Abstract This paper presents a novel proof-theoretic approach to a logic specifically designed to handle exceptions and typicality. Our method extends the classical first-order sequent calculus by incorporating a specialized framework to manage both negative extra-logical information (explicit exceptions) and positive information (background assumptions). We prove that the resulting sequent calculi satisfy the cut-elimination theorem, thereby ensuring strong analytical properties. Furthermore, we show how this framework effectively models traditional reasoning patterns involving conflicting information and typicality. Finally, we establish a natural correspondence between our approach and the Kraus-Lehmann-Magidor postulates, further grounding our work within established theoretical foundations.

Abstract Alongside the sequent calculus for Classical Core Logic $${\mathbb {C}}^+$$ C + we set forth some new sequent calculi that we call $${\mathbb {T}}$$ T , $${\mathbb {C}}^{++}$$ C + + , and $${\mathbb {K}}$$ K . $${\mathbb {T}}$$ T encodes truth-tabular reasoning; $${\mathbb {C}}^{++}$$ C + + classicizes Core Logic $${\mathbb {C}}$$ C by having multiple succedents; and $${\mathbb {K}}$$ K is a cut-free sequent calculus inspired by insights of Ketonen. Our aim is to establish that $${\mathbb {C}}^{++}$$ C + + is weakly complete, and to do so in a way that makes no use of standard semantic notions or the rule of Cut. All the concepts defined and applied in the course of this study will be proof-theoretic, turning on rules of inference as meaning-constitutive. It is intended to be a contribution to the program of proof-theoretic semantics, from the distinctive vantage point of the Core logician.

In this paper a framework to distinguish in a Fregean manner between sense and denotation of \(\lambda\)-term-annotated derivations will be applied to a bilateralist sequent calculus displaying two derivability relations, one for proving and one for refuting. Therefore, a two-sorted typed \(\lambda\)-calculus will be used to annotate this calculus and a Dualization Theorem will be given, stating that for any derivable sequent expressing a proof, there is also a derivable sequent expressing a refutation and vice versa. By having joint \(\lambda\)-term annotations for proof systems in natural deduction and sequent calculus style, a comparison with respect to sense and denotation between derivations in those systems will be feasible, since the annotations elucidate the structural correspondences of the respective derivations. Thus, we will have a basis for determining in which cases, firstly, derivations expressing a proof vs. derivations expressing a refutation and, secondly, derivations in natural deduction vs. in sequent calculus can be identified and on which level.

Minimal Sequent Calculus for Teaching First-Order Logic: Lessons Learned

There is a quite intentional resemblance between the Cut Rule and Aristotle’s Syllogism. In this paper some deep connections between Sequent Calculus and Syllogistics will be investigated. Taking into consideration Alvarez & Correia’s ´axiomatization of Syllogistics, currently the most complete available in the literature, I will show how this ancient logical system can be put into correspondence with the structural features of a special Sequent Calculus system, SS. On the grounds of this discovery I will present some improvements of the expressive power of Alvarez & Correia’s system. As for the philosophical consequences of ´the correspondence, I will give answers to several concerns Manuel Correia had on his system. A somewhat new philosophical relevance of the Cut-Elimination Theorem will be highlighted in the end.

Abstract Recent work by Afshari et al. introduces a notion of Herbrand schemes for first-order logic by associating a higher-order recursion scheme to a sequent calculus proof. Calculating the language of associated Herbrand schemes directly yields Herbrand disjunctions. As such, these schemes can be seen as programs extracted from proofs. The present article generalizes this computational interpretation by removing the restriction of acyclicity from Herbrand schemes which amounts to admitting recursively defined programs. It is shown that the notion of proof corresponding to these generalized Herbrand schemes is cyclic proofs, considered here in the context of classical theories of inductively defined predicates. In particular, for cyclic proofs of generalized $\varSigma _{1}$-sequents, Herbrand schemes extract sets of witnessing terms via a simulation of non-wellfounded cut elimination.

Compiler intermediate representations have to strike a balance between being high-level enough to allow for easy translation of surface languages into them and being low-level enough to make code generation easy. An intermediate representation based on a logical system typically has the former property and additionally satisfies several meta-theoretical properties which are valuable when optimizing code. Recently, classical sequent calculus, which is widely recognized and impactful within the fields of logic and proof theory, has been proposed as a natural candidate in this regard, due to its symmetric treatment of data and control flow. For such a language to be useful, however, it must eventually be compiled to machine code. In this paper, we introduce an intermediate representation that is based on classical sequent calculus and demonstrate that this language can be directly translated to conventional hardware. We provide both a formal description and an implementation. Preliminary performance evaluations indicate that our approach is viable.

Global condition check strategy for a cyclic sequent calculus of temporal logic

Cut-free sequent calculi are handy tools for backward proof-search of logical formulas or sequents. In the present paper, we introduce a Gentzen-type sequent calculus for the logic of common knowledge. To maintain a deterministic backward proof-search process, we do not include cut or cut-like rules in the introduced calculus. Also, derivation loops are used to define provable sequents and to establish termination of backward proof-search. Using this sound and complete finitary loop-type sequent calculus we construct a decision procedure for the logic of common knowledge. The procedure allows to efficiently determine whether an arbitrary formula or sequent is valid in the logic.

Abstract We consider a Gentzen-type cut-free sequent calculus GS5 for the modal logic S5 with a restriction on backward applications of modal rule $(\Box \Rightarrow )$. Using Schütte’s method of reduction trees, we prove that the calculus is complete for S5. We also prove that all rules are invertible and the cut rule is admissible in the calculus. We show that any backward proof search terminates, obtaining a decision procedure for S5 using the introduced calculus GS5.

We investigate the proof complexity of systems based on positive branching programs, i.e. non-deterministic branching programs (NBPs) where, for any 0-transition between two nodes, there is also a 1-transition. Positive NBPs compute monotone Boolean functions, just like negation-free circuits or formulas, but constitute a positive version of (non-uniform) NL, rather than P or NC1, respectively. The proof complexity of NBPs was investigated in previous work by Buss, Das and Knop, using extension variables to represent the dag-structure, over a language of (non-deterministic) decision trees, yielding the system eLNDT. Our system eLNDT+ is obtained by restricting their systems to a positive syntax, similarly to how the 'monotone sequent calculus' MLK is obtained from the usual sequent calculus LK by restricting to negation-free formulas. Our main result is that eLNDT+ polynomially simulates eLNDT over positive sequents. Our proof method is inspired by a similar result for MLK by Atserias, Galesi and Pudl\'ak, that was recently improved to a bona fide polynomial simulation via works of Je\v{r}\'abek and Buss, Kabanets, Kolokolova and Kouck\'y. Along the way we formalise several properties of counting functions within eLNDT+ by polynomial-size proofs and, as a case study, give explicit polynomial-size poofs of the propositional pigeonhole principle.

Abstract We say that a Kripke model is a GL-model (Gödel and Löb model) if the accessibility relation $\prec $ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots $ , and $t_\omega $ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_\omega $ . A non-normal modal logic $\mathbf {D}$ , which was studied by Beklemishev [3], is characterized as follows. A formula $\varphi $ is a theorem of $\mathbf {D}$ if and only if $\varphi $ is true at $t_\omega $ in any D-model. $\mathbf {D}$ is an intermediate logic between the provability logics $\mathbf {GL}$ and $\mathbf {S}$ . A Hilbert-style proof system for $\mathbf {D}$ is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for $\mathbf {D}$ , and show the cut-elimination theorem. We also introduce new Hilbert-style systems for $\mathbf {D}$ by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $\omega $ . Finally, we show a general result as follows. Let X and $X^+$ be arbitrary modal logics. If the relationship between semantics of X and semantics of $X^+$ is equal to that of $\mathbf {GL}$ and $\mathbf {D}$ , then $X^+$ can be axiomatized based on X in the same way as the new axiomatization of $\mathbf {D}$ based on $\mathbf {GL}$ .

Abstract Fractional semantics provides a multi-valued interpretation of a variety of logics, governed by purely proof-theoretic principles. This approach employs a method of systematic decomposition of formulas through a well-disciplined sequent calculus, assigning a fractional value that measures the “quantity of identity” (intuitively, “quantity of truth”) within a sequent. A key consequence of this framework is the breakdown of the traditional symmetry between truth and contradiction. In this paper, we explore the ramifications of this novel perspective on classical logic. Specifically, we (i) introduce an alternative paraconsistent consequence relation, and (ii) show how the gradual character of contradictions induces a corresponding characterization of tautologies, thereby obtaining a full-fledged informational refinement of classical logic.

Some of the more prominent contributions to the last fifty years of scholarship on Aristotle’s syllogistic suggest a conceptual framework under which the syllogistic is a logic, a system of inferential reasoning, only if it is not a theory, a system concerned with ontology or general facts. I argue that this a misleading interpretative framework. I begin by noting that the syllogistic exhibits one mark of contemporary logics: syllogisms are inferences and not implications. The debate on this question has focused on the interpretation of indirect proof. But I argue that this evidence is neutral on the question, and offer new considerations in favor of the interpretation of syllogisms as inferences. I next note that the syllogistic exhibits one mark of theories: it employs a distinct underlying logic so to derive derivative structures from primitive structures. So the syllogistic is something sui generis: by our lights, it is arguably neither clearly a logic, nor clearly a theory. I also discuss whether the syllogistic is better represented as a natural deduction system, a sequent calculus or an axiomatic system, and conclude with a few remarks on the use of modern systems to represent historical logics.