Modal Tableaux with Propagation Rules and Structural Rules

We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automaton-labelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cut-free and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar logics using our calculus.KeywordsDescription LogicKripke ModelKripke FramePropositional Dynamic LogicNegation Normal FormThese 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.

First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the true semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation

Analytic tableau systems for the family of non-deterministic semantics are introduced. These are based on tableaux for many-valued logics using sets-as-signs DNF representations. Karnaugh maps illustrate the construction of tableau rules. In contrast to classical many-valued tableaux, we add a rule called sign intersection. Soundness and completeness are shown. As an example demonstrates, some tableau systems would be incomplete without sign intersection. There is a correspondence to well-studied canonical calculi based on sequent systems: Tableau systems can be translated into canonical calculi, but not vice-versa (structural rules are missing on the tableau side).

We investigate the systematic development of refined tableau systems for a subset of the logics of confluence, which are modal logics comprising of instances of the Scott-Lemmon axioms. In particular, we look at rule refinements aiming to decrease branching, perform fewer inferences and reduce the application of rules which create new labels in the tableau. Propagation rules are common forms of refined rules, that construct smaller pre-models sufficient to determine satisfiability, without needing to construct full concrete models satisfying the correspondence properties which would require a lot more inference steps. These rules have already been developed for the confluence logics that are part of the modal logic cube, but are lacking for some instances outside the cube. Such instances can be awkward, as the nature of their correspondence properties makes the development of propagation rules particularly challenging. These are the logics K G0111, K G and K De for which we propose refined tableau systems. We also present refined tableau systems for the combined logics K alt1De, K BG0111 and K DDe. Soundness and completeness results for all the systems are established.

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.

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.

CPAP--a gene mutated in primary microcephaly--is required for procentriole formation. Here we show that CPAP degradation and function is controlled by the poly(ADP-ribose) polymerase tankyrase 1. CPAP is PARsylated by tankyrase 1 in vitro and in vivo. Overexpression of tankyrase 1 leads to CPAP proteasomal degradation, preventing centriole duplication, whereas depletion of tankyrase 1 stabilizes CPAP in G1, generating elongated procentrioles and multipolarity. Tankyrase 1 localizes to centrosomes exclusively in G1, coinciding with CPAP degradation. Hence, tankyrase 1-mediated PARsylation regulates CPAP levels during the cell cycle to limit centriole elongation and ensure normal centrosome function.

MetTeL is a generic tableau prover for various modal, intuitionistic, hybrid, description and metric logics. The core component of MetTeL is a logic-independent tableau inference engine. A novel feature is that users have the ability to flexibly specify the set of tableau rules to be used in derivations. Termination can be achieved via a generalisation of a standard loop checking mechanism or unrestricted blocking.

This paper explores proof-theoretic aspects of hybrid type-logical grammars, a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present both a sequent and a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited to a non-associative version and a multimodal version incorporating structural rules and unary modes.

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503

In this paper, we present a general strategy that enables the translation of tableau proofs using different Skolemization rules into machine-checkable proofs. It is part of a framework that enables (i) instantiation of the strategy into algorithms for different sets of tableau rules (e.g., different logics) and (ii) easy soundness proof which relies on the local extensibility of user-defined rules. Furthermore, we propose an instantiation of this strategy for first-order tableaux that handles notably pre-inner Skolemization rules, which is, as far as the authors know, the first one in the literature. This deskolemization strategy has been implemented in the Goéland [17] automated theorem prover, enabling an export of its proofs to Coq [8] and Lambdapi [2]. Finally, we have evaluated the algorithm performances for inner and pre-inner Skolemization rules through the certification of proofs from some categories of the TPTP [39] library.

In the paper we propose a new approach to formalization of cognitive logics. By cognitive logics we understand supraclassical, but non-trivial consequence operations, defined in a propositional language. We extend some paradigm of tableau methods, in which classical consequence Cn is defined, to stronger logics - monotonic, as well as non-monotonic ones - by specific use of non-classical tableau rules. So far, in that context tableaus have been treated as a way of formalizing other approaches to supraclassical logics, but we use them autonomically to generate various consequence operations. It requires a description of the hierarchy of non-classical tableau rules that result in different supraclassical consequence operations, so we give it.

Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK\(^{-}\), cKN4, scBK\(^{-}\), scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK\(^{-}\) and cKN4 are connexive variants of the FDE-based modal logics BK\(^{-}\) and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK\(^{-}\), scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics \(\mathbf {cBK}^-\) and \(\mathbf {cMBL}\) are shown to be algebraizable. The algebraizability of \(\mathbf {cMBL}\) is derived from proving \(\mathbf {cMBL}\) to be definitionally equivalent to \(\mathbf {MBL}\). All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics.

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this article. Modal logics could be very helpful in modelling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently the Connectionist Modal Logics has been presented, which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness and the fact that the system itself is a decision procedure are proved in this article. The main advantages of this approach are: first, the system is deterministic, i.e. it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula (if such exists); and third, the language of deduction and the language of a logic coincide. Some of these advantages are compared with other classical approaches.

The initial contractile event in muscle is the binding of Ca2+ ions to troponin C of the troponin complex, leading to a series of conformational changes in the members of the thin and thick filaments. Knowledge of the crystal structure of turkey skeletal muscle troponin C has provided a structural basis for the modeling of the first stage of this process in atomic detail. This crystal structure probably represents the molecule in the relaxed state of muscle, with two of the maximum of 4 Ca2+ ions bound. The basis for the model presented here is that upon binding of the additional two Ca2+ ions, the regulatory domain of the molecule undergoes a conformational transition to become closely similar in structure to the domain which always binds Ca2+ or Mg2+ under physiological conditions. The root mean square discrepancy in atomic coordinates between the apo and the modeled Ca2+-bound states of the regulatory domain is 4.8 A, with some shifts as large as 10-15 A in the region near the linker between the two Ca2+ binding sites. It is demonstrated that this Ca2+-bound conformation of the regulatory domain conforms to accepted protein structure rules and that the change in conformation can be accomplished without encountering any barriers too high to be surmounted on the physiological time scale.