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Focusing is traditionally seen as a means of reducing inessential nondeterminism in backward-reasoning strategies such as uniform proof-search or tableaux systems. In this paper we construct a form of focused derivations for propositional linear logic that is appropriate for forward reasoning in the inverse method. We show that the focused inverse method conservatively generalizes the classical hyperresolution strategy for Horn-theories, and demonstrate through a practical implementation that the focused inverse method is considerably faster than the non-focused version.

Abstract. The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this "bureaucracy" from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative canonicity directly in the cut-free sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multifocused proofs. We validate this definition by proving a bijection to the well-known proof-nets for the unit-free multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments.

OverviewTLAPS, the TLA + proof system, is a platform for the development and mechanical verification of TLA + proofs. The TLA + proof language is declarative, and understanding proofs requires little background beyond elementary mathematics. The language supports hierarchical and non-linear proof construction and verification, and it is independent of any verification tool or strategy. Proofs are written in the same language as specifications; engineers do not have to translate their high-level designs into the language of a particular verification tool. A proof manager interprets a TLA + proof as a collection of proof obligations to be verified, which it sends to backend verifiers that include theorem provers, proof assistants, SMT solvers, and decision procedures.The first public release of TLAPS is available from [1], distributed with a BSD-like license. It handles almost all the non-temporal part of TLA + as well as the temporal reasoning needed to prove standard safety properties, in particular invariance and step simulation, but not liveness properties. Intuitively, a safety property asserts what is permitted to happen; a liveness property asserts what must happen; for a more formal overview, see [3,10]. FoundationsTLA + is a formal language based on TLA (the Temporal Logic of Actions) [12]. It was designed for specifying the high-level behavior of concurrent and distributed systems, but it can be used to specify safety and liveness properties of any discrete system or algorithm. A behavior is a sequence of states, where a state is an assignment of values to state variables. Safety properties are expressed by describing the allowed steps (state transitions) in terms of actions, which are first-order formulas involving two copies v and v of each state variable, where v denotes the value of the variable at the current state and v its value at the next state. These properties are proved by reasoning about actions, using a small and restricted amount of temporal reasoning. Proving liveness properties requires propositional linear-time temporal logic reasoning plus a few TLA proof rules.It has always been possible to assert correctness properties of systems in TLA + , but not to write their proofs. We have added proof constructs based on a hierarchical style for writing informal proofs [11]. The current version of the language is essentially the 1

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The inverse method is a generalization of resolution that can be applied to non-classical logics. We have recently shown how Andreoli's focusing strategy can be adapted for the inverse method in linear logic. In this paper we introduce the notion of focusing bias for atoms and show that it gives rise to forward and backward chaining, generalizing both hyperresolution (forward) and SLD resolution (backward) on the Horn fragment. A key feature of our characterization is the structural, rather than purely operational, explanation for forward and backward chaining. A search procedure like the inverse method is thus able to perform both operations as appropriate, even simultaneously. We also present experimental results and an evaluation of the practical benefits of biased atoms for a number of examples from different problem domains.

We present the theory and implementation of a theorem prover for first-order intuitionistic linear logic based on the inverse method. The central proof-theoretic insights underlying the prover concern resource management and focused derivations, both of which are traditionally understood in the domain of backward reasoning systems such as logic programming. We illustrate how resource management, focusing, and other intrinsic properties of linear connectives affect the basic forward operations of rule application, contraction, and forward subsumption. We also present some preliminary experimental results obtained with our implementation.

Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016International audienceFocusing is a general technique for transforming a sequent proof system into one with a syntactic separation of non-deterministic choices without sacrificing completeness. This not only improves proof search, but also has the representational benefit of distilling sequent proofs into synthetic normal forms. We show how to apply the focusing technique to nested sequent calculi, a generalization of ordinary sequent calculi to tree-like instead of list-like structures. We thus improve the reach of focusing to the most commonly studied modal logics, the logics of the modal S5 cube. Among our key contributions is a focused cut-elimination theorem for focused nested sequents