Multi types-aka non-idempotent intersection types-have been used to obtain quantitative bounds on higherorder programs, as pioneered by de Carvalho. Notably, they bound at the same time the number of evaluation steps and the size of the result. Recent results show that the number of steps can be taken as a reasonable time complexity measure. At the same time, however, these results suggest that multi types provide quite lax complexity bounds, because the size of the result can be exponentially bigger than the number of steps.Starting from this observation, we refine and generalise a technique introduced by Bernadet & Graham-Lengrand to provide exact bounds for the maximal strategy. Our typing judgements carry two counters, one measuring evaluation lengths and the other measuring result sizes. In order to emphasise the modularity of the approach, we provide exact bounds for four evaluation strategies, both in the λ-calculus (head, leftmostoutermost, and maximal evaluation) and in the linear substitution calculus (linear head evaluation).Our work aims at both capturing the results in the literature and extending them with new outcomes. Concerning the literature, it unifies de Carvalho and Bernadet & Graham-Lengrand via a uniform technique and a complexity-based perspective. The two main novelties are exact split bounds for the leftmost strategythe only known strategy that evaluates terms to full normal forms and provides a reasonable complexity measure-and the observation that the computing device hidden behind multi types is the notion of substitution at a distance, as implemented by the linear substitution calculus.
International audienceThe CDCL procedure for SAT is the archetype of conflict-driven procedures for satisfiability of quantifier-free problems in a single theory. In this paper we lift CDCL to CDSAT (Conflict-Driven Satisfia-bility), a system for conflict-driven reasoning in combinations of disjoint theories. CDSAT combines theory modules that interact through a global trail representing a candidate model by Boolean and first-order assignments. CDSAT generalizes to generic theory combinations the model-constructing satisfiability calculus (MCSAT) introduced by de Moura and Jovanovi´cJovanovi´c. Furthermore, CDSAT generalizes the equality sharing (Nelson-Oppen) approach to theory combination, by allowing theories to share equality information both explicitly through equalities and dis-equalities, and implicitly through assignments. We identify sufficient conditions for the soundness, completeness, and termination of CDSAT
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