Intuitionistic G\"odel-L\"ob logic, \`a la Simpson: labelled systems and birelational semantics

We derive an intuitionistic version of G\"odel-L\"ob modal logic ($\sf{GL}$) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, $\sf{\ell IGL}$, by restricting a non-wellfounded labelled system for $\sf{GL}$ to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that $\sf{GL}$'s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of $\sf{GL}$ are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that $\sf{\ell IGL}$ coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic $\sf{IGL}$. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of $\sf{IGL}$.Comment: 25 pages including 8 pages appendix, 4 figure

Cyclic proof systems for modal fixpoint logics

This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one

Sequent calculus proof systems for inductive definitions

Inductive definitions are the most natural means by which to represent many families of structures occurring in mathematics and computer science, and their corresponding induction / recursion principles provide the fundamental proof techniques by which to reason about such families. This thesis studies formal proof systems for inductive definitions, as needed, e.g., for inductive proof support in automated theorem proving tools. The systems are formulated as sequent calculi for classical first-order logic extended with a framework for (mutual) inductive definitions. The default approach to reasoning with inductive definitions is to formulate the induction principles of the inductively defined relations as suitable inference rules or axioms, which are incorporated into the reasoning framework of choice. Our first system LKID adopts this direct approach to inductive proof, with the induction rules formulated as rules for introducing atomic formulas involving inductively defined predicates on the left of sequents. We show this system to be sound and cut-free complete with respect to a natural class of Henkin models. As a corollary, we obtain cut-admissibility for LKID. The well-known method of infinite descent `a la Fermat, which exploits the fact that there are no infinite descending chains of elements of well-ordered sets, provides an alternative approach to reasoning with inductively defined relations. Our second proof system LKIDw formalises this approach. In this system, the left-introduction rules for formulas involving inductively defined predicates are not induction rules but simple case distinction rules, and an infinitary, global soundness condition on proof trees — formulated in terms of “traces” on infinite paths in the tree — is required to ensure soundness. This condition essentially ensures that, for every infinite branch in the proof, there is an inductive definition that is unfolded infinitely often along the branch. By an infinite descent argument based upon the well-foundedness of inductive definitions, the infinite branches of the proof can thus be disregarded, whence the remaining portion of proof is well-founded and hence sound. We show this system to be cutfree complete with respect to standard models, and again infer the admissibility of cut. The infinitary system LKIDw is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic” proof system, CLKIDw, is suitable for formal reasoning since proofs have finite representations and the soundness condition on proofs is thus decidable. We show how the formulation of our systems LKIDw and CLKIDw can be generalised to obtain soundness conditions for a general class of infinite proof systems and their corresponding cyclic restrictions. We provide machinery for manipulating and analysing the structure of proofs in these essentially arbitrary cyclic systems, based primarily on viewing them as generating regular infinite trees, and we show that any proof can be converted into an equivalent proof with a restricted cycle structure. For proofs in this “cycle normal form”, a finitary, localised soundness condition exists that is strictly stronger than the general, infinitary soundness condition, but provides more explicit information about the proof. Finally, returning to the specific setting of our systems for inductive definitions, we show that any LKID proof can be transformed into a CLKIDw proof (that, in fact, satisfies the finitary soundness condition). We conjecture that the two systems are in fact equivalent, i.e. that proof by induction is equivalent to regular proof by infinite descent

Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs

A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2005 paper by Brotherston showed that the provability of CLKID-omega includes the provability of LKID, first order classical logic with inductive definitions in Martin-L\"of's style, and conjectured the equivalence. The equivalence has been left an open question since 2011. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions