On the mechanisation of the logic of partial functions

PhD ThesisIt is well known that partial functions arise frequently in formal reasoning about programs. A partial function may not yield a value for every member of its domain. Terms that apply partial functions thus may not denote, and coping with such terms is problematic in two-valued classical logic. A question is raised: how can reasoning about logical formulae that can contain references to terms that may fail to denote (partial terms) be conducted formally? Over the years a number of approaches to coping with partial terms have been documented. Some of these approaches attempt to stay within the realm of two-valued classical logic, while others are based on non-classical logics. However, as yet there is no consensus on which approach is the best one to use. A comparison of numerous approaches to coping with partial terms is presented based upon formal semantic definitions. One approach to coping with partial terms that has received attention over the years is the Logic of Partial Functions (LPF), which is the logic underlying the Vienna Development Method. LPF is a non-classical three-valued logic designed to cope with partial terms, where both terms and propositions may fail to denote. As opposed to using concrete undfined values, undefinedness is treated as a \gap", that is, the absence of a defined value. LPF is based upon Strong Kleene logic, where the interpretations of the logical operators are extended to cope with truth value \gaps". Over the years a large body of research and engineering has gone into the development of proof based tool support for two-valued classical logic. This has created a major obstacle that affects the adoption of LPF, since such proof support cannot be carried over directly to LPF. Presently, there is a lack of direct proof support for LPF. An aim of this work is to investigate the applicability of mechanised (automated) proof support for reasoning about logical formulae that can contain references to partial terms in LPF. The focus of the investigation is on the basic but fundamental two-valued classical logic proof procedure: resolution and the associated technique proof by contradiction. Advanced proof techniques are built on the foundation that is provided by these basic fundamental proof techniques. Looking at the impact of these basic fundamental proof techniques in LPF is thus the essential and obvious starting point for investigating proof support for LPF. The work highlights the issues that arise when applying these basic techniques in LPF, and investigates the extent of the modifications needed to carry them over to LPF. This work provides the essential foundation on which to facilitate research into the modification of advanced proof techniques for LPF.EPSR

Extending a first order predicate calculus with partially defined iota terms

We extend the classical first order logic with partially defined iota terms in order to model the way partial functions are treated in common mathematical practice

On Modal Logics of Partial Recursive Functions

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established

Semantics and correctness proofs for programs with partial functions

This paper presents a portion of the work on specification, design, and implementation of safety-critical systems such as reactor control systems. A natural approach to this problem, once all the requirements are captured, would be to state the requirements formally and then either to prove (preferably via automated tools) that the system conforms to spec (program verification), or to try to simultaneously generate the system and a mathematical proof that the requirements are being met (program derivation). An obstacle to this is frequent presence of partially defined operations within the software and its specifications. Indeed, the usual proofs via first order logic presuppose everywhere defined operations. Recognizing this problem, David Gries, in ``The Science of Programming,`` 1981, introduced the concept of partial functions into the mainstream of program correctness and gave hints how his treatment of partial functions could be formalized. Still, however, existing theorem provers and software verifiers have difficulties in checking software with partial functions, because of absence of uniform first order treatment of partial functions within classical 2-valued logic. Several rigorous mechanisms that took partiality into account were introduced [Wirsing 1990, Breu 1991, VDM 1986, 1990, etc.]. However, they either did not discuss correctness proofs or departed from first order logic. To fill this gap, the authors provide a semantics for software correctness proofs with partial functions within classical 2-valued 1st order logic. They formalize the Gries treatment of partial functions and also cover computations of functions whose argument lists may be only partially available. An example is nuclear reactor control relying on sensors which may fail to deliver sense data. This approach is sufficiently general to cover correctness proofs in various implementation languages

Identifiers in Registers - Describing Network Algorithms with Logic

We propose a formal model of distributed computing based on register automata that captures a broad class of synchronous network algorithms. The local memory of each process is represented by a finite-state controller and a fixed number of registers, each of which can store the unique identifier of some process in the network. To underline the naturalness of our model, we show that it has the same expressive power as a certain extension of first-order logic on graphs whose nodes are equipped with a total order. Said extension lets us define new functions on the set of nodes by means of a so-called partial fixpoint operator. In spirit, our result bears close resemblance to a classical theorem of descriptive complexity theory that characterizes the complexity class PSPACE in terms of partial fixpoint logic (a proper superclass of the logic we consider here).Comment: 17 pages (+ 17 pages of appendices), 1 figure (+ 1 figure in the appendix

The notion of problem, intuitionism and partiality

Problems are defined as abstract procedures. An explication of procedures as used in Transparent Intensional Logic (TIL) and called constructions is presented and the subclass of constructions called concepts is defined. Concepts as closed constructions modulo α- and η-conversion can be associated with meaningful expressions of a natural or professional language in harmony with Church’s conception. Thus every meaningful expression expresses a concept. Since every problem can be unambiguously determined by a concept we can state that every problem is a concept and every concept can be viewed as a problem.Kolmogorov’s idea of a connection between problems and Heyting’s calculus is examined and the non-classical features of the latter are shown to be compatible with realistic logic using partial functions

A Dempster-Shafer theory inspired logic.

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic