Ori Lahav

From Frame Properties to Hypersequent Rules in Modal Logics

We provide a general method for generating cutfree and/or analytic hypersequent Gentzen-type calculi for a variety of normal modal logics. The method applies to all modal logics characterized by Kripke frames, transitive Kripke frames, or symmetric Kripke frames satisfying some properties, given by first-order formulas of a certain simple form. This includes the logics KT, KD, S4, S5, K4D, K4.2, K4.3, KBD, KBT, and other modal logics, for some of which no Gentzen calculi was presented before. Cut-admissibility (or analyticity in the case of symmetric Kripke frames) is proved semantically in a uniform way for all constructed calculi. The decidability of each modal logic in this class immediately follows.

Taming release-acquire consistency

We introduce a strengthening of the release-acquire fragment of the C11 memory model that (i) forbids dubious behaviors that are not observed in any implementation; (ii) supports fence instructions that restore sequential consistency; and (iii) admits an equivalent intuitive operational semantics based on point-to-point communication. This strengthening has no additional implementation cost: it allows the same local optimizations as C11 release and acquire accesses, and has exactly the same compilation schemes to the x86-TSO and Power architectures. In fact, the compilation to Power is complete with respect to a recent axiomatic model of Power; that is, the compiled program exhibits exactly the same behaviors as the source one. Moreover, we provide criteria for placing enough fence instructions to ensure sequential consistency, and apply them to an efficient RCU implementation.

Owicki-Gries Reasoning for Weak Memory Models

We show that even in the absence of auxiliary variables, the well-known Owicki-Gries method for verifying concurrent programs is unsound for weak memory models. By strengthening its non-interference check, however, we obtain OGRA, a program logic that is sound for reasoning about programs in the release-acquire fragment of the C11 memory model. We demonstrate the usefulness of this logic by applying it to several challenging examples, ranging from small litmus tests to an implementation of the RCU synchronization primitives.

A unified semantic framework for fully structural propositional sequent systems

We identify a large family of fully structural propositional sequent systems, which we call basic systems. We present a general uniform method for providing (potentially, nondeterministic) strongly sound and complete Kripke-style semantics, which is applicable for every system of this family. In addition, this method can also be applied when: (i) some formulas are not allowed to appear in derivations, (ii) some formulas are not allowed to serve as cut formulas, and (iii) some instances of the identity axiom are not allowed to be used. This naturally leads to new semantic characterizations of analyticity (global subformula property), cut admissibility and axiom expansion in basic systems. We provide a large variety of examples showing that many soundness and completeness theorems for different sequent systems, as well as analyticity, cut admissibility, and axiom expansion results, easily follow using the general method of this article.

Modular reasoning about heap paths via effectively propositional formulas

First order logic with transitive closure, and separation logic enable elegant interactive verification of heap-manipulating programs. However, undecidabilty results and high asymptotic complexity of checking validity preclude complete automatic verification of such programs, even when loop invariants and procedure contracts are specified as formulas in these logics. This paper tackles the problem of procedure-modular verification of reachability properties of heap-manipulating programs using efficient decision procedures that are complete: that is, a SAT solver must generate a counterexample whenever a program does not satisfy its specification. By (a) requiring each procedure modifies a fixed set of heap partitions and creates a bounded amount of heap sharing, and (b) restricting program contracts and loop invariants to use only deterministic paths in the heap, we show that heap reachability updates can be described in a simple manner. The restrictions force program specifications and verification conditions to lie within a fragment of first-order logic with transitive closure that is reducible to effectively propositional logic, and hence facilitate sound, complete and efficient verification. We implemented a tool atop Z3 and report on preliminary experiments that establish the correctness of several programs that manipulate linked data structures.

Kripke semantics for basic sequent systems

We present a general method for providing Kripke semantics for the family of fully-structural multiple-conclusion propositional sequent systems. In particular, many well-known Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obtain semantic characterizations of analytic sequent systems of this type, as well as of those admitting cut-admissibility. These characterizations serve as a uniform basis for semantic proofs of analyticity and cut-admissibility in such systems.

SAT-Based Decision Procedure for Analytic Pure Sequent Calculi

We identify a wide family of analytic sequent calculi for propositional non-classical logics whose derivability problem can be uniformly reduced to SAT. The proposed reduction is based on interpreting these calculi using non-deterministic semantics. Its time complexity is polynomial, and, in fact, linear for a useful subfamily. We further study an extension of such calculi with Next operators, and show that this extension preserves analyticity and is subject to a similar reduction to SAT. A particular interesting instance of these results is a HORNSAT-based linear-time decision procedure for Gurevich and Neeman’s primal infon logic and several natural extensions of it.

Automated Support for the Investigation of Paraconsistent and Other Logics

We automate the construction of analytic sequent calculi and effective semantics for a large class of logics formulated as Hilbert calculi. Our method applies to infinitely many logics, which include the family of paraconsistent C-systems, as well as to other logics for which neither analytic calculi nor suitable semantics have so far been available.

Finite-valued Semantics for Canonical Labelled Calculi

We define a general family of canonical labelled calculi, of which many previously studied sequent and labelled calculi are particular instances. We then provide a uniform and modular method to obtain finite-valued semantics for every canonical labelled calculus by introducing the notion of partial non-deterministic matrices. The semantics is applied to provide simple decidable semantic criteria for two crucial syntactic properties of these calculi: (strong) analyticity and cut-admissibility. Finally, we demonstrate an application of this framework for a large family of paraconsistent logics.

On Constructive Connectives and Systems

Canonical inference rules and canonical systems are defined in the framework of non-strict single-conclusion sequent systems, in which the succeedents of sequents can be empty. Important properties of this framework are investigated, and a general non-deterministic Kripke-style semantics is provided. This general semantics is then used to provide a constructive (and very natural), sufficient and necessary coherence criterion for the validity of the strong cut-elimination theorem in such a system. These results suggest new syntactic and semantic characterizations of basic constructive connectives.