Analytic Tableaux for Non-deterministic Semantics

In this paper we generalize the existing tableau methods for modal logics. First of all, while usual modal tableaux are based on trees, our basic structures are rooted directed acyclic graphs (RDAG). This allows natural tableau rules for some modal logics that are difficult to capture in the usual way (such as those having an accessibility relation that is dense or confluent). Second, tableau rules rewrite patterns, which are (schemas of) parts of a RDAG. A particular case of these rules are the single-step rules recently proposed by Massacci. This allows in particular tableau rule presentations for K5, KD5, K45, KD45, and S5 that respect the subformula property. Third, we divide modal tableau rules into propagation rules and structural rules. Structural rules construct new edges and nodes (without adding formulas to nodes), while propagation rules add formulas to nodes. This distinction allows to prove completeness in a modular way.

I show two limitations of multiplicative (or context-independent) sequent calculi: contraction can neither be restricted to atoms nor to the bottom of a proof tree. There is a a set of rules for classical propositional logic, system SKS [1], with two interesting properties: 1. Contraction can be restricted to atoms. As a consequence, no rule in system SKS requires the duplication of formulae of unbounded size. 2. Contraction can be restricted to the bottom of a proof. As a consequence, a proof in SKS can be separated into two phases (seen bottom-up): In the first phase atoms are duplicated and in the second phase all other rules are applied and the size of the frontier of the proof (in terms of number of atoms) does not increase. System SKS is not a sequent calculus but is presented in a more general formalism, the calculus of structures [2]. While the sequent calculus restricts the application of rules to the main connective of a formula, the calculus of structures is more expressive by admitting rules that can be applied anywhere deep inside formulae. This leads to the question whether the extra expressive power of the calculus of structures is needed to observe these properties, or whether they can be observed in sequent systems as well. This question seems quickly answered by sequent system G3cp [3] in which contraction is admissible and can thus trivially be restricted to atoms or to the bottom of a proof. However, the R∧-rule in G3cp is additive (or context-sharing), so none of the above mentioned interesting consequences for system SKS follow for G3cp. Even though contraction is admissible, formulae of unbounded size still have to be duplicated in the context of an additive R∧. The separation of a proof into two phases as in system SKS also is not observable in G3cp, simply because there is no rule that just duplicates formulae. To answer the question whether similar properties to those of system SKS can be achieved in sequent systems, I thus consider systems with a multiplicative R∧-rule. For the specific case of system GS1p [3], a Gentzen-Schutte formulation of classical logic shown in Fig. 1, I now prove by counterexample that it does not admit the properties of SKS. Atoms can be either positive or negative and are denoted by a or b. The negation of an atom a is denoted by ā and is again an atom. In a proof, contraction is considered to be at the bottom if there is no application of a rule different from contraction such that one of its premises is the conclusion of a contraction. An application of the contraction rule is considered atomic if its principal formula is an atom. Theorem 1. There is a sequent that is valid but for which there is no cut-free proof in GS1p with multiplicative context treatment in which all contractions are at the bottom. Ax A, Ā Γ,A ∆, Ā Cut Γ,∆ Γ,A,B R∨ Γ,A ∨B Γ,A ∆,B R∧ Γ,∆,A ∧B Γ,A,A RC Γ,A Γ RW Γ,A Fig. 1. GS1p with multiplicative context treatment Proof. Consider the following sequent: a ∧ a, ā ∧ ā . It suffices to show that a ∧ a, . . . , a ∧ a, ā ∧ ā, . . . , ā ∧ ā (for any number of occurrences of the formulae a ∧ a and ā ∧ ā) is not provable in GS1p without contraction and cut. The connective ∨ does not occur in the conclusion. Thus, the only rules that can appear in the proof are RW ,Ax and R∧. A proof has to have all branches closed with axioms A, Ā. Since no rule that may occur introduces new formulae, the only formula that can take the place of A in an axiom is the atom a. There can be no such proof since there is always one branch in the derivation tree in which there is at most one single atom, as is shown by induction on the derivation tree: Base Case There is only one branch in a ∧ a, . . . , a ∧ a, ā ∧ ā, . . . , ā ∧ ā and it contains at most one (that is, no) single atom. Inductive Case Weakening does not increase the number of single atoms. When a R∧ rule is applied, the context is split, so the only single atom that may occur in the conclusion goes to one branch. Choose the other branch, which has at most one (that is, exactly one) atom. Theorem 2. There is a sequent that is valid but for which there is no cut-free proof in GS1p with multiplicative context treatment in which all contractions are atomic.

We present an amusing semantic tableau system for propositional S5 which is actually quite efficient in practice. We assume the reader is familiar with the propositional tableau system using unsigned formulas as presented in [4], Ch. II. We continue the a, β classification of formulas, and add two new categories, necessaries (v) and possibles (π). These, together with their respective components v0 and π0 are defined by the following tables: v v0 Ή π0

We analyze different features related to the mechanization of von Wright's logic of elsewhere e. First, we give a new proof of the NP-completeness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although e and the well-known propositional modal S5 share numerous common features we show that e is strictly more expressive than S5 (in a sense to be specified). Second, we present a prefixed tableau system for e and we prove both its soundness and completeness. Two extensions of this system are also defined, one related to the logical consequence relation and the other related to the addition of modal operators (without increasing the expressive power). An example of tableau proof is also presented. Different continuations of this work are proposed, one of them being to implement the defined tableau system, another one being to extend this system to richer logics that can be found in the literature.

The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general structure of proofs in the original calculus in a way ensuring preservation of the weak cut elimination theorem under the transformation. The described transformation metod is illustrated on several concrete examples of many-valued logics, including a new application to information sources logics.

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

This paper presents an application of automated theorem proving techniques to natural language processing, particularly to Categorial Grammar (CG) parsing. It describes the system LLKE, a labelled analytic tableau system which supports a framework for specification and proof for several calculi, starting with the Lambek calculus L, covering its substructural extensions such as LP, LPE, LPE, NL etc, and allowing for the implementation of modalities. The tableau expansion algorithms are presented and heuristics to improve their performance based on domain-specific knowledge are discussed along with comparisons of our system with other systems that implement categorial deduction.

This paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Löwenheim-Skolem theorem.The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

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.

In Ilić and Boričić (2014, Log. J. IGPL, 22, 673–695), the right-handed cut-free sequent calculus $GRW$ for the contraction-less relevant logic $RW$ is defined. In this paper, we show that the enlargement of the system $GRW$ with the structural rule of intensional contraction (WI) presents the sequent system for the principal relevant logic $R$ but the rule of cut cannot be eliminated in $GRW+$(WI).

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.

Tableau-based proof systems can be elegantly speci ed and directly executed by a tabled Logic Programming (LP) system. Our experience with the XMC model checker shows that such an encoding can be used to search for the existence of a proof very eAEciently. However, the users of a tableau system are often interested in getting suAEcient evidence (in terms of the tableau proof rules) on why a proof does or does not exist. In this paper, we address the problem of constructing such an evidence without introducing any additional computational overhead to the proof search. A tabled LP system maintains a memo table of \lemmas that were tried and possibly proved during query evaluation. We propose the concept of justi er for extracting suAEcient evidence for the truth or falsehood of literals in a logic program, by post-processing the memo tables created during query evaluation. Based on this logic program justi er, we show how to construct evidence for the presence/absence of tableau in a tableau-based proof system. We provide experimental results showing the e ectiveness of the justi er in constructing succinct evidence of the evaluation performed by the XMC model checker. Finally we discuss the role of the justi er as a programming abstraction for encoding efcient algorithms as tabled logic programs.

PA (Process algebra) is the name that has become common use to denote the algebra with a sequential and parallel operator (without communication), plus recursion. PA-processes are a superset of both Basic Parallel Processes (BPP) [Chr93] and context-free processes (BPA).We study three problems for PA-processes: The reachability problem, the partial deadlock reachability problem (“Is it possible to reach a state where certain actions are not enabled ?”) and the partial livelock reach-ability problem (“Is it possible to reach a state where certain actions are disabled forever ?”). We present sound and complete tableau systems for these problems and compare them to non-tableau algorithms.Keywordstableau systemstemporal logicprocess algebrasPA-processes

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.