Complete Sequent Calculi for Induction and Infinite Descent

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the meta-theory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programming-style search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semi-automated proof of cut-elimination for Girard’s Logic of Unicity (LU).

First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when capacity limits are put on the cognitive resources. Finally, we investigate the correlation between a number of mathematical complexity measures defined on graphs and sentences and some psychological complexity measures that were recorded in the experiment.

Tableau-based proof systems have been designed for many logics extending classical first-order logic. This paper proposes a sound tableau calculus for temporal logics of the first-order CTL-family. Until now, a tableau calculus has only been presented for the propositional version of CTL. The calculus considered operates with prefixed formulas and may be regarded as an instance of a labelled deductive system. The prefixes allow an explicit partial description of states and paths of a potential Kripke counter model in the tableau. It is possible in particular to represent path segments of finite but arbitrary length which are needed to process reachability formulas. Furthermore, we show that by using prefixed formulas and explicit representation of paths it becomes possible to express and process fairness properties without having to resort to full CTL. The approach is suitable for use in interactive proof-systems.KeywordsTemporal LogicFunction SymbolLinear Temporal LogicVariable AssignmentState FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present Zenon, an automated theorem prover for first order classical logic (with equality), based on the tableau method. Zenon is intended to be the dedicated prover of the Focal environment, an object-oriented algebraic specification and proof system, which is able to produce OCaml code for execution and Coq code for certification. Zenon can directly generate Coq proofs (proof scripts or proof terms), which can be reinserted in the Coq specifications produced by Focal. Zenon can also be extended, which makes specific (and possibly local) automation possible in Focal.

We present sound and complete sequent calculi for the modal mu-calculus with converse modalities, aka two-way modal mu-calculus. Notably, we introduce a cyclic proof system wherein proofs can be represented as finite trees with back-edges, i.e., finite graphs. The sequent calculi incorporate ordinal annotations and structural rules for managing them. Soundness is proved with relative ease as is the case for the modal mu-calculus with explicit ordinals. The main ingredients in the proof of completeness are isolating a class of non-wellfounded proofs with sequents of bounded size, called slim proofs, and a counter-model construction that shows slimness suffices to capture all validities. Slim proofs are further transformed into cyclic proofs by means of re-assigning ordinal annotations.

Formal reasoning about inductively defined relations and structures is widely recognized not only for its mathematical interest but also for its importance in computer science, and has applications in verifying properties of programs and algorithms. Recently, several proof systems of inductively defined predicates based on sequent calculus including the cyclic proof system CLKID-omega and the infinite-descent proof system LKID-omega have attracted much attention. Although the relation among their provabilities has been clarified so far, the logical complexity of these systems has not been much studied. The infinite-descent proof system LKID-omega is an infinite proof system for inductive definitions and allows infinite paths in proof figures. It serves as a basis for the cyclic proof system. This paper shows that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete. To show this, first it is shown that the validity for inductive definitions in standard models is equivalent to the validity for inductive definitions in standard term models. Next, using this equivalence, this paper extends the truth predicate of omega-languages, as given in Girard's textbook, to inductive definitions by employing arithmetical coding of inductive definitions. This shows that the validity of inductive definitions in standard models is a (Pi-1-1) relation. Then, using the completeness of LKID-omega for standard models, it is shown that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete.

This book constitutes the proceedings of the 28th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, TABLEAUX 2019, held in London, UK, in September 2019, colocated with the 12th International Symposium on Frontiers on Combining Systems, FroCoS 2019. The 25 full papers presented were carefully reviewed and selected from 43 submissions.They present research on all aspects of the mechanization of tableaux-based reasoning and related methods, including theoretical foundations, implementation techniques, systems development and applications. The papers are organized in the following topical sections: tableau calculi, sequent calculi, semantics and combinatorial proofs, non-wellfounded proof systems, automated theorem provers, and logics for program or system verification.

This paper presents a sound and complete logical system whose atomic sentences are the equalities of recursive terms involving sets. There are two interpretations of this language: one makes use of non-wellfounded sets with finite transitive closure, and the other uses pointed finite graphs modulo bisimulation. Our logical system is a sequent-style deduction system. The main axioms and inference rules come from the $\mbox{{\it FLR}$_0$}$ -proof system from [6], including the Recursion Inference Rule (but an additional axiom is needed), and also axioms corresponding to the extensionality axiom of set theory.

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

Pure functional programming languages have been proposed as a vehicle to describe, simulate and manipulate circuit specifications. We propose an extension to Haskell to solve a standard problem when manipulating data types representing circuits in a lazy functional language. The problem is that circuits are finite graphs - but viewing them as an algebraic (lazy) datatype makes them indistinguishable from potentially infinite regular trees. However, implementations of Haskell do indeed represent cyclic structures by graphs. The problem is that the sharing of nodes that creates such cycles is not observable by any function which traverses such a structure. In this paper we propose an extension to call-by-need languages which makes graph sharing observable. The extension is based on non updatable reference cells and an equality test (sharing detection) on this type. We show that this simple and practical extension has well-behaved semantic properties, which means that many typical source-to-source program transformations, such as might be performed by a compiler, are still valid in the presence of this extension.

A proof system for the first-order relational calculus

We introduce an extended tableau calculus for answer set programming (ASP). The proof system is based on the ASP tableaux defined in [Gebser&Schaub, ICLP 2006], with an added extension rule. We investigate the power of Extended ASP Tableaux both theoretically and empirically. We study the relationship of Extended ASP Tableaux with the Extended Resolution proof system defined by Tseitin for clause sets, and separate Extended ASP Tableaux from ASP Tableaux by giving a polynomial length proof of a family of normal logic programs {Π n } for which ASP Tableaux has exponential length minimal proofs with respect to n. Additionally, Extended ASP Tableaux imply interesting insight into the effect of program simplification on the length of proofs in ASP. Closely related to Extended ASP Tableaux, we empirically investigate the effect of redundant rules on the efficiency of ASP solving.KeywordsLogic ProgramStable ModelProof SystemDeduction RuleEmpty ClauseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

We present a tableau based proof system for νTrTL, a trace based temporal logic with fixpoints. The proof system generalises similar systems for standard interleaving temporal logics with fixpoints. In our case special attention has to be given to the modal rule: First we give a system with an interleaving style modal rule, later we use a technique similar to the sleep set method (known from finite state model checking) to obtain a more efficient proof rule. We briefly highlight the relation of the improved rule with recent advances in tableau systems for classical propositional logic, the tamed cut of the system KE.The treatment of fixpoints leads to possibly infinite tableaux, which however can be finitely represented, yielding treelike structures with back loops: we show this using an automata construction. Indirectly we obtain a (known) decidability result.KeywordsModel CheckTemporal LogicModal RuleProof SystemAtomic PropositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.