We study propositional logic in detail, defining its formal syntax in OCaml together with parsing and printing support. We discuss some of the key propositional algorithms and prove the compactness theorem, as well as indicating the surprisingly rich applications of propositional theorem proving.

The syntax of propositional logic

Propositional logic is a modern version of Boole's algebra of propositions as presented in Section 1.4. It involves expressions called formulas that are intended to represent propositions, i.e. assertions that may be considered true or false. These formulas can be built from constants ‘true’ and ‘false’ and some basic atomic propositions (atoms) using various logical connectives (‘not’, ‘and’, ‘or’, etc.). The atomic propositions are like variables in ordinary algebra, and we sometimes refer to them as propositional variables or Boolean variables. As the word ‘atomic’ suggests, we do not analyze their internal structure; that will be considered when we treat first-order logic in the next chapter.

Representation in OCaml

We represent propositional formulas using an OCaml datatype by analogy with the type of expressions in Section 1.6. We allow the ‘constant’ propositions False and True and atomic formulas Atom p, and can build up formulas from them using the unary operator Not and the binary connectives And, Or, Imp (‘implies’) and Iff (‘if and only if’). We defer a discussion of the exact meanings of these connectives, and deal first with immediate practicalities.

The underlying set of atomic propositions is largely arbitrary, although for some purposes it's important that it be infinite, to avoid a limit on the complexity of formulas we can consider. In abstract treatments it's common just to index the primitive propositions by number.

In this chapter we introduce logical reasoning and the idea of mechanizing it, touching briefly on important historical developments. We lay the groundwork for what follows by discussing some of the most fundamental ideas in logic as well as illustrating how symbolic methods can be implemented on a computer.

What is logical reasoning?

There are many reasons for believing that something is true. It may seem obvious or at least immediately plausible, we may have been told it by our parents, or it may be strikingly consistent with the outcome of relevant scientific experiments. Though often reliable, such methods of judgement are not infallible, having been used, respectively, to persuade people that the Earth is flat, that Santa Claus exists, and that atoms cannot be subdivided into smaller particles.

What distinguishes logical reasoning is that it attempts to avoid any unjustified assumptions and confine itself to inferences that are infallible and beyond reasonable dispute. To avoid making any unwarranted assumptions, logical reasoning cannot rely on any special properties of the objects or concepts being reasoned about. This means that logical reasoning must abstract away from all such special features and be equally valid when applied in other domains. Arguments are accepted as logical based on their conformance to a general form rather than because of the specific content they treat. For instance, compare this traditional example:

All men are mortal

Socrates is a man

Therefore Socrates is mortal

with the following reasoning drawn from mathematics:

All positive integers are the sum of four integer squares

15 is a positive integer

Therefore 15 is the sum of four integer squares

These two arguments are both correct, and both share a common pattern:

By their very nature, sensor network applications are often platform-specific: the application uses a particular set of sensors on a mote-specific sensor board, to measure application-specific conditions. The hardware, and hence the application, is typically not directly reusable on another mote platform. Furthermore, some applications may want to push a mote platform to its limits, e.g. to maximize sampling rate, or minimize the latency in reacting to an external event. Getting to these limits normally requires extensive platform-specific tuning, including platform-specific code (possibly even written in assembly language).

Conversely, large portions of sensor network applications are portable: multi-hop network protocols, radio stacks for commonly available radio chips, signal processing, etc. need little or no change for a new platform. Thus, while a sensor network application is not typically directly portable, TinyOS should make it easy to port applications to new platforms by minimizing the extent of the necessary changes.

Portability and the hardware abstraction architecture

TinyOS's main tool to maximize portability while maintaining easy access to platform-specific features is a multi-level hardware abstraction architecture (HAA), shown in Figure 12.1. The device driver components that give access to a mote's various hardware resources are divided into three categories:

• The hardware interface layer (HIL): a device driver is part of the HIL if it provides access to a device (radio, storage, timers, etc.) in a platform-independent way, using only hardware independent interfaces.

• […]

We've considered various algorithms (tableaux, resolution, etc.) for verifying that a first-order formula is logically valid, if indeed it is. But these will not in general tell us when a formula is not valid. We'll see in Chapter 7 that there is no systematic procedure for doing so. However, there are procedures that work for certain special classes of formulas, or for validity in certain special (classes of) models, and we discuss some of the more important ones in this chapter. Often these naturally generalize common decision problems in mathematics and universal algebra such as equation-solving or the ‘word problem’.

The decision problem

There are three natural and closely connected problems for first-order logic for which we might want an algorithmic solution. By negating the formula, we can according to taste present them in terms of validity or unsatisfiability.

1. Confirm that a logically valid (or unsatisfiable) formula is indeed valid (resp. unsatisfiable), and never confirm an invalid (satisfiable) one.

2. Confirm that a logically invalid (or satisfiable) formula is indeed invalid (resp. satisfiable), and never confirm a valid (unsatisfiable) one.

3. Test whether a formula is valid or invalid (or whether it is satisfiable or unsatisfiable).

Evidently (3) encompasses both (1) and (2). Conversely, solutions to both (1) and (2) could be used together to solve (3): just run the verification procedures for validity and invalidity (or satisfiability and unsatisfiability) in parallel. Now, we have presented explicit solutions to (1), such as tableaux or resolution. But these do not solve (3). Given a satisfiable formula, these algorithms, while at least not incorrectly claiming they are unsatisfiable, will not always terminate.

In this appendix we collect together some useful mathematical background. Readers may prefer to read the main text and refer to this appendix only if they get stuck. We do not give much in the way of proofs and the style is terse and rather dull, so this is not a substitute for standard texts. For example, Forster (2003) discusses in detail almost all the topics here, as well as much relevant material in logic and computability and some more advanced topics in set theory.

Mathematical notation and terminology

We use ‘iff’ as a shorthand for ‘if and only if’ and ‘w.r.t.’ for ‘with respect to’. We write x | y, read ‘x divides y’, to mean that y is an integer multiple of x, e.g. 3 | 6, 1 | x and x | 0. We use the usual arithmetic operations (‘+’ etc.) on numbers; we generally write xy for the product of x and y, but sometimes write x y to emphasize that there is an operation involved and make the syntax more regular. An operation such as addition for which the order of the two arguments is irrelevant (x + y = y + x) is called commutative, and an operation where the association does not matter (x + (y + z) = (x + y) + z) is said to be associative. We also use the conventional equality and inequality relations (‘=’, ‘≤’ etc.) on numbers, and sometimes emphasize that an equation is the definition of a concept by decorating the equality sign with def, e.g. tan(x) =def sin(x)/cos(x).

This chapter presents TinyOS 's execution model, which is based on split-phase operations, run-to-completion tasks and interrupt handlers. Chapter 3 introduced components and modules, Chapter 4 introduced how to connect components together through wiring. This chapter goes into how these components execute, and how you can manage the concurrency between them in order to keep a system responsive. This chapter focuses on tasks, the basic concurrency mechanism in nesC and TinyOS. We defer discussion of concurrency issues relating to interrupt handlers and resource sharing to Chapter 11, as these typically only arise in very high-performance applications and low-level drivers.

Overview

As we saw in Section 3.4, all TinyOS I/O (and long-running) operations are split-phase, avoiding the need for threads and allowing TinyOS programs to execute on a single stack. In place of threads, all code in a TinyOS program is executed either by a task or an interrupt handler. A task is in effect a lightweight deferred procedure call: a task can be posted at anytime and posted tasks are executed later, one-at-a-time, by the TinyOS scheduler. Interrupts, in contrast can occur at any time, interrupting tasks or other interrupt handlers (except when interrupts are disabled).

While a task or interrupt handler is declared within a particular module, its execution may cross component boundaries when it calls a command or signals an event (Figure 5.1).As a result, it isn't always immediately clear whether a piece of code is only executed by tasks or if it can also be executed from an interrupt handler.

This chapter describes components, the building blocks of nesC programs. Every component has a signature, which describes the functions it needs to call as well as the functions that others can call on it. A component declares its signature with interfaces, which are sets of functions for a complete service or abstraction. Modules are components that implement and call functions in C-like code. Configurations connect components into larger abstractions. This chapter focuses on modules, and covers configurations only well enough to modify and extend existing applications: Chapter 4 covers writing new configurations from scratch.

Component signatures

A nesC program is a collection of components. Every component is in its own source file, and there is a one-to-one mapping between component and source file names. For example, the file LedsC.nc contains the nesC code for the component LedsC, while the component PowerupC can be found in the file PowerupC.nc. Components in nesC reside in a global namespace: there is only one PowerupC definition, and so the nesC compiler loads only one file named PowerupC.nc.

There are two kinds of components: modules and configurations. Modules and configurations can be used interchangeably when combining components into larger services or abstractions. The two types of components differ in their implementation sections. Module implementation sections consist of nesC code that looks like C. Module code declares variables and functions, calls functions, and compiles to assembly code. Configuration implementation sections consist of nesC wiring code, which connects components together. Configurations are the major difference between nesC and C (and other C derivatives).

In this chapter, we look at the design and implementation of SoundLocalizer, a somewhat more complex sensor network application. SoundLocalizer implements a coordinated event detection system where a group of motes detect a particular event – a loud sound – and then communicate amongst themselves to figure out which mote detected the event first and is therefore presumed closest to where the event occurs. To ensure timely event detection, and accurately compare event detection times, this application needs to use some low-level interfaces from the platform's hardware abstraction and hardware presentation layers (HAL, HPL, as described in the previous chapter). As a result, this application is not directly portable – we implement it here for micaz motes with an mts300 sensor board. In the design and implementation descriptions below, we discuss how the application and code are designed to simplify portability and briefly describe what would be involved in porting this application to another platform.

The HAL and HPL components used by SoundLocalizer offer lower-level interfaces (interrupt-driven, controlled by a Resource interface, etc.) than the high-level HIL components we used to build the AntiTheft application of Chapter 6. As a result, SoundLocalizer's implementation must use atomic statements and arbitration to prevent concurrency-induced problems, as we saw in Chapter 11.

The complete code for SoundLocalizer is available from TinyOS's contributed code directory (under “TinyOS Programming”).

Sound Localizer design

Figure 13.1 shows a typical setup for the SoundLocalizer application. A number of detector motes are placed on a surface a couple of feet apart. When the single coordinator mote is switched on, it sends a series of radio packets that let the detector motes synchronize their clocks.

This book is about writing TinyOS systems and applications in the nesC language. This chapter gives a brief overview of TinyOS and its intended uses. TinyOS is an open-source project which a large number of research universities and companies contribute to. The main TinyOS website, www.tinyos.net, has instructions for downloading and installing the TinyOS programming environment. The website has a great deal of useful information which this book doesn't cover, such as common hardware platforms and how to install code on a node.

Networked, embedded sensors

TinyOS is designed to run on small, wireless sensors. Networks of these sensors have the potential to revolutionize a wide range of disciplines, fields, and technologies. Recent example uses of these devices include:

Golden Gate Bridge safety High-speed accelerometers collect synchonizedd ata on the movement of and oscillations within the structure of San Francisco's Golden Gate Bridge. This data allows the maintainers of the bridge to easily observe the structural health of the bridge in response to events such as high winds or traffic, as well as quickly assess possible damage after an earthquake [10]. Being wireless avoids the need for installing and maintaining miles of wires.

Volcanic monitoring Accelerometers and microphones observe seismic events on the Reventador and Tungurahua volcanoes in Ecuador. Nodes locally compare when they observe events to determine their location, and report aggregate data to a camp several kilometers away using a long-range wireless link. Small, wireless nodes allow geologists and geophysicists to install dense, remote scientific instruments [30], obtaining data that answers other questions about unapproachable environments.