Model checking is the best-known and most successful approach to formally verifying that systems satisfy specifications, expressed as temporal logic formulae. In this article, we develop the theory of equilibrium checking, a related but distinct problem. Equilibrium checking is relevant for multi-agent systems in which system components (agents) are assumed to be acting rationally in pursuit of delegated goals, and is concerned with understanding what temporal properties hold of such systems under the assumption that agents select strategies in equilibrium. The formal framework we use to study this problem assumes agents are modelled using REACTIVE MODULES, a system modelling language that is used in a range of practical model checking systems. Each agent (or player) in a REACTIVE MODULES game is specified as a nondeterministic guarded command program, and each player's goal is specified with a temporal logic formula that the player desires to see satisfied. A strategy for a player in a REACTIVE MODULES game defines how that player selects enabled guarded commands for execution over successive rounds of the game. For this general setting, we investigate games in which players have goals specified in Linear Temporal Logic (in which case it is assumed that players choose deterministic strategies) and in Computation Tree Logic (in which case players select nondeterministic strategies). For each of these cases, after formally defining the game setting, we characterise the complexity of a range of problems relating to Nash equilibria (e.g., the computation or the verification of existence of a Nash equilibrium or checking whether a given temporal formula is satisfied on some Nash equilibrium). We then go on to show how the model we present can be used to encode, for example, games in which the choices available to players are specified using STRIPS planning operators.

Possibility and necessity measures are commonly defined over Boolean algebras. This work consider a generalization of these kinds of measures over MV-algebras as a possibilistic counterpart of the (probabilistic) notion of state on MV-algebras. Two classes of possibilistic states over MV-algebras of functions are characterized in terms of (generalized) Sugeno integrals. For reasoning about these representable classes of possibilistic states, we introduce many-valued modal logics based on the Rational Lukasiewicz Logic, that are be shown to be complete with respect to corresponding classes of Kripke models equipped with those states.

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.

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