Interpolation and Amalgamation for Arrays with MaxDiff

Abstract

In this paper, the theory of McCarthy’s extensional arrays enriched with a maxdiff operation (this operation returns the biggest index where two given arrays differ) is proposed. It is known from the literature that a diff operation is required for the theory of arrays in order to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom). Our maxdiff operation significantly increases the level of expressivity; however, obtaining interpolation results for the resulting theory becomes a surprisingly hard task. We obtain such results via a thorough semantic analysis of the models of the theory and of their amalgamation properties. The results are modular with respect to the index theory and it is shown how to convert them into concrete interpolation algorithms via a hierarchical approach.

Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

“…The third variant of the array theory we want to mention is the theory AX maxdi f f , where the axiom ( 8) is strengthened so that diff(a, b) returns the biggest index where a, b differ. This requires adding at least a symbol for a total ordering relation on the sort INDEX (we leave the reader to consult [20] for details). Under suitable mild hypotheses, it is possible to prove that the universal theory AX maxdi f f also has amalgamation and, hence, quantifier-free interpolation (but the proof is surprisingly much more delicate [20]).…”

“…This requires adding at least a symbol for a total ordering relation on the sort INDEX (we leave the reader to consult [20] for details). Under suitable mild hypotheses, it is possible to prove that the universal theory AX maxdi f f also has amalgamation and, hence, quantifier-free interpolation (but the proof is surprisingly much more delicate [20]).…”

“…, a n and b are (not necessarily distinct) variables or constants. A formula is flat iff all literals occurring in it are flat; flat terms are terms occurring in the literals listed above in (20).…”

“…This is crucial because, very often, first-order theories commonly used in verification have a decidable quantifier-free fragment, but are undecidable outside that fragment (this is the case, for instance, of the McCarthy theory of arrays; see Section 3 below); even if general first-order satisfiability remains decidable, the computational cost of a satisfiability test may increase considerably when moving from the quantifier-free fragment to arbitrary first-order formulae (this is the case, for instance, of Presburger arithmetic). This is why some considerable effort has been put into designing theory-specific interpolation algorithms operating at a quantifier-free level [11][12][13][14][15][16][17] and in identifying suitable variants of theories axiomatizing common datatypes enjoying quantifierfree interpolation [18][19][20].…”

General Interpolation and Strong Amalgamation for Contiguous Arrays

“…These are the main reasons why in [17] the theory has been further enriched. There, the semantics of diff, called maxdiff, is uniquely determined in the models of the theory and is more informative: it returns the biggest index where two different arrays differ.…”