Applying Program Transformations for Deductive Verification of the List Reverse Program

The program transformation methods to simplify the deductive verification of programs with recursive data types are investigated. The list reversion program is considered as an example. A source program in the C language is translated to the cP functional language which includes no pointers. The resulting program is translated further to the WhyML language to perform deductive verification of the program. The cP language includes the same constructs of the C language except pointers. In the C program, all actions that include pointers are replaced by the equivalent fragments without pointers. These replacement are performed by the special transformations using the results of the program dataflow analysis. Three variants of deductive verification of the transformed list reverse program in the Why3 verification platform with SMT solvers (Z3 4.8.6, CVC3 2.4.1, CVC4 1.7) are performed. First, the recursive WhyML program supplied with specifications was automatically verified successfully using only SMT solvers. Second, the recursive program was translated to the P predicate language. Correctness formulae were constructed for the P program and translated further to the why3 specification language. The formulae proving correctness were easy like the first variant. But correctness formulae for the first and second variants were different. Third, the "imperative" WhyML program that included while loop with additional invariant specifications was verified. The proving was easy but not automatic. So, for deductive verification, recursive program variant appears to be more preferable against imperative program variant.

On the complexity of index sets for finite predicate logic programs which allow function symbols

We study the recognition problem in the metaprogramming of finite normal predicate logic programs. That is, let $\mathcal{L}$ be a computable first-order predicate language with infinitely many constant symbols and infinitely many $n$-ary predicate symbols and $n$-ary functions symbols for all $n \geq 1$. Then we can effectively list all the finite normal predicate logic programs $Q_0,Q_1,\ldots $ over $\mathcal{L}$. Given some property $\mathcal{P}$ of finite normal predicate logic programs over $\mathcal{L}$, we define the index set $I_{\mathcal{P}}$ to be the set of indices $e$ such that $Q_e$ has property $\mathcal{P}$. We classify the complexity of the index set $I_{\mathcal{P}}$ within the arithmetic hierarchy for various natural properties of finite predicate logic programs. For example, we determine the complexity of the index sets relative to all finite predicate logic programs and relative to certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having no recursive stable models, (iii) having at least one stable model, (iv) having at least one recursive stable model, (v) having exactly $c$ stable models for any given positive integer $c$, (vi) having exactly $c$ recursive stable models for any given positive integer $c$, (vii) having only finitely many stable models, (viii) having only finitely many recursive stable models, (ix) having infinitely many stable models and (x) having infinitely many recursive stable models.

Praedicaturi supponimus. Is Gilbert of Poitiers’ approach to the problem of linguistic reference a pragmatic one?

The article investigates how the problem of (linguistic) reference is treated in Gilbert of Poitiers’ Commentaries on Boethius’ Opuscula sacra. In this text the terms supponere, suppositus,-a,-um, and suppositio mainly concern the act of a speaker (or of the author of a written text) that consists of referring—by choosing a name as subject term in a proposition—to one or more subsistent things as what the speech act (or the written text) is about. Supposition is for Gilbert an action performed by a speaker, not a property of terms, and his ‘contextual approach’ has a pragmatic touch: “we do not predicate in order to supposit as much as we supposit in order to predicate”. Language is considered by Gilbert as a system for communication between human beings, key notions are the ‘sense in the author’s mind’ (sensus mentis eius qui loquitur) and the ‘interpreter’s understanding’ (intelligentia lectoris). The phenomenon of ‘disciplinal’ discourse (“man is a species of individuals”) is treated by means of these hermeneutic notions and not by means of a special kind of supposition.

The logic of demand in Haskell

Haskell is a functional programming language whose evaluation is lazy by default. However, Haskell also provides pattern matching facilities which add a modicum of eagerness to its otherwise lazy default evaluation. This mixed or “non-strict” semantics can be quite difficult to reason with. This paper introduces a programming logic, P-logic, which neatly formalizes the mixed evaluation in Haskell pattern-matching as a logic, thereby simplifying the task of specifying and verifying Haskell programs. In p-logic, aspects of demand are reflected or represented within both the predicate language and its model theory, allowing for expressive and comprehensible program verification.

An undecidable two sorted predicate calculus

Let L be a first order predicate language with two sorts of variables and a single dyadic predicate letter whose first place is to be filled by variables of one sort and whose second place is to be filled by variables of the other sort. In answer to a question of M. H. Löb we show that there is no decision procedure for determining whether or not a sentence of L is universally valid.