Free Function Symbols Research Articles

Interpolation Results for Arrays with Length and MaxDiff

In this paper, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. As is well-known, some diff operation (i.e., some kind of difference function showing where two unequal array differ) is needed to keep interpolants quantifier-free in array theories; our maxdiff operation returns the max index where two arrays differ and so it has a univocally determined semantics. The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays. Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory and we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols. In this paper, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. It is known from the literature that a diff operation is required in order for the theory of arrays 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): instead, our maxdiff operation returns the max index where two arrays differ and so it is univocally determined at the semantic level. The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays (which are defined in their specified allocation memory). Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory and we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols. The array theory in the paper has been modified so as to model real arrays used in common programming languages (we now require them to be ‘contiguous’, i.e., not undefined in any of their allocation entries); moreover strong amalgamation and interpolation with free function symbols are proved. The interpolation algorithm avoids full instantiation routines and unbounded loops, thus achieving the above mentioned polynomial reduction complexity. The present paper is a substantially revised version of a previous conference paper presented at FoSSaCS 2021.

Cardinality constraints for arrays (decidability results and applications)

Enriching logic formalisms with counting capabilities is an important task in view of the needs of many application areas, ranging from database theory to formal verification. In this paper, we consider a very expressive language obtained by enriching linear integer arithmetic with free function symbols and cardinality constraints for interpreted sets. We obtain positive results for a flat fragment via a reduction to decidability of Presburger arithmetic with unary counting quantifiers (Schweikhart in Arithmetic, first-order logic, and counting quantifiers, ACM TOCL, New York, 2004). We isolate also an easier simple flat subfragment, whose satisfiability is in NP, and we show that this subfragment is adequate to formalize problems arising in the area of the verification of fault-tolerant distributed algorithms. We finally discuss our first implementation, the related experimental results, as well as further algorithmic problems suggested by model-checking applications.

Tractable and intractable instances of combination problems for unification and disunification

The algorithms for combining decision procedures for (dis)unification in the union of arbitrary disjoint equational theories known from the literature generate an exponential number of 'pure' output problems over the component theories. Despite several optimization techniques, no general results are known that describe in which cases a polynomial-time combination procedure is (im)possible. In the first part of this paper we characterize a large class of equational theories E where E-unification with free function symbols is NP-hard and where a polynomial-time optimization of the combination algorithm is impossible (unless P = NP). This supports the conjecture that for the theories in this class there is no polynomial-time algorithm for combining E-unification with constants with free unification. In the second part of the paper we introduce another class of equational theories where (dis)unification algorithms can be combined deterministically and in polynomial time. Under reasonable assumptions on the form of the given unification procedure, each unitary, regular and collapse-free equational theory with a polynomial-time unification algorithm belongs to this class.

Any ground associative-commutative theory has a finite canonical system

It is shown that theories presented by a set of ground equations with several associative-commutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering that is AC-compatible and total on the set of congruence classes generated by the associativity and commutativity axioms. As far as we know, this is the first ordering with such properties when several AC-function symbols and free-function symbols are allowed. Such an ordering is also a fundamental tool for deriving a complete theorem proving strategies with built-in associative commutative unification.

Decidability and Expressiveness for First-Order Logics of Probability

We consider decidability and expressiveness issues for two first-order logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is Π21 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, Π1∞ hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is Π21 complete with as little as one unary predicate in the language, even without equality. With equality, we get Π1∞ hardness with only a constant symbol. We then turn our attention to an analysis of what causes this overwhelming complexity. For example, we show that if we require rational probabilities then we drop from Π21 to Π11. In many contexts it suffices to restrict attention to domains of bounded size; fortunately, the logics are decidable in this case. Finally, we show that, although the two logics capture quite different intuitions about probability, there is a precise sense in which they are equi-expressive.

Competing for theAC-Unification Race

We describe our implementation of the unification algorithm for terms involving some associative-commutative operators plus free function symbols described by Boudetet al. The first goal of this implementation is efficiency, more precisely competing for theAC Unification Race. Although our implementation has been designed for good performance when applied to non-elementaryAC-unification problems, it is also very efficient on elementary problems. Our implementation, written in C and running on Sun workstations, is to be compared with the implementations in LISP, on Symbolics LIPS machines.

Unification in a combination of arbitrary disjoint equational theories

The unification problem in a disjoint combination of equational theories, E 1 +...+E n , is reduced to a combination of two kinds of problems in the E j 's : the pure unification, problem with free (Uninterpreted) constants and the constant-elimination problem. The constant-elimination problem is to find, given terms t i , 1≤i≤m and free constants c i , 1≤i≤m , all substitutions σ such that for all i with 1≤i≤m , σt i is equal to some t′ i that does not contain c i . The soundness and completeness of the method shows, that a disjoint combination of theories is finitary, provided every theory is finitary and constant-elimination problems in every theory are finitary solvable. In particular, any combination of finitary unifying regular theories, of Boolean rings, of Abelian groups or of theories with terminating basic narrowing, is of unification-type finitary. As a special case, unification in a combination of a free Boolean ring with free function symbols is decidable and finitary; the same holds for Abelian groups. This paper extends the known results on unification in a combination of regular and collapse-free theories in two aspects: i) Arbitrary theories are admissable. ii) The described general procedure can be used for enumerating a complete set of unifiers, even if some unification procedure for a paticular theory produces an infinite complete set of unifiers. It is demonstrated that unifiability of unification problems is decidable in the general case E 1 +...+E n if for every i∈{1,…,n} , there is a method to decide unification problems in a combination of E i with free function symbols.