Disproving in First-Order Logic with Definitions, Arithmetic and Finite Domains

This thesis explores several methods which enable a first-order reasoner to conclude satisfiability of a formula modulo an arithmetic theory. The most general method requires restricting certain quantifiers to range over finite sets; such assumptions are common in the software verification setting. In addition, the use of first-order reasoning allows for an implicit representation of those finite sets, which can avoid scalability problems that affect other quantified reasoning methods. These new techniques form a useful complement to existing methods that are primarily aimed at proving validity. The Superposition calculus for hierarchic theory combinations provides a basis for reasoning modulo theories in a first-order setting. The recent account of ‘weak abstraction’ and related improvements make an mplementation of the calculus practical. Also, for several logical theories of interest Superposition is an effective decision procedure for the quantifier free fragment. The first contribution is an implementation of that calculus (Beagle), including an optimized implementation of Cooper’s algorithm for quantifier elimination in the theory of linear integer arithmetic. This includes a novel means of extracting values for quantified variables in satisfiable integer problems. Beagle won an efficiency award at CADE Automated theorem prover System Competition (CASC)-J7, and won the arithmetic non-theorem category at CASC-25. This implementation is the start point for solving the ‘disproving with theories’ problem. Some hypotheses can be disproved by showing that, together with axioms the hypothesis is unsatisfiable. Often this is relative to other axioms that enrich a base theory by defining new functions. In that case, the disproof is contingent on the satisfiability of the enrichment. Satisfiability in this context is undecidable. Instead, general characterizations of definition formulas, which do not alter the satisfiability status of the main axioms, are given. These general criteria apply to recursive definitions, definitions over lists, and to arrays. This allows proving some non-theorems which are otherwise intractable, and justifies similar disproofs of non-linear arithmetic formulas. When the hypothesis is contingently true, disproof requires proving existence of a model. If the Superposition calculus saturates a clause set, then a model exists, but only when the clause set satisfies a completeness criterion. This requires each instance of an uninterpreted, theory-sorted term to have a definition in terms of theory symbols. The second contribution is a procedure that creates such definitions, given that a subset of quantifiers range over finite sets. Definitions are produced in a counter-example driven way via a sequence of over and under approximations to the clause set. Two descriptions of the method are given: the first uses the component solver modularly, but has an inefficient counter-example heuristic. The second is more general, correcting many of the inefficiencies of the first, yet it requires tracking clauses through a proof. This latter method is shown to apply also to lists and to problems with unbounded quantifiers. Together, these tools give new ways for applying successful first-order reasoning methods to problems involving interpreted theories

Zilber's notion of logically perfect structure: Universal Covers

We sketch recent interactions between model theory and a roughly 150-year old study of analytic functions involving complex analysis, algebraic topology, and number theory, centered in canonicity of universal covers. Towards this goal we discuss in a systematic and unified way several examples indicating the main ideas of the proofs and the necessary changes in method for different situations: exponential covers, modular and Shimura curves, Shimura and abelian varieties, and coherent families of smooth covers.Comment: 1 figur

Some Concerns Regarding Ternary-relation Semantics and Truth-theoretic Semantics in General

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel