Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way. We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic [15], in which validity is validity up to stabilisation, and implication oe comes out as "boundedly gives rise to." We show that this semantics is equivalently characterised by a notion of realisability with stabilisation bounds as realisers...

The paper presents a new logical specification language, called Propositional Stabilisation Theory (PST), to capture the stabilisation behaviour of combinational input-output systems. PST is an intuitionistic propositional modal logic interpreted over sets of waveforms. The language is more economic than conventional specification formalisms such as timed Boolean functions, temporal logic, or predicate logic in that it separates function from time and only introduces as much syntax as is necessary to deal with stabilisation behaviour. It is a purely propositional system but has secondorder expressiveness. One and the same Boolean function can be represented in various ways as a PST formula, giving rise to different timing models which associate different stabilisation delays with different parts of the functionality and adjust the granularity of the data-dependency of delays within wide margins. We show how several standard timing analyses can be characterised as algorithms computing c...

1 An introduction to inquisitive semantics 1 1.1 Information states and the classical update perspective...... 1 1.2 Limitations of the classical picture................. 2 1.3 The inquisitive programme: propositions as proposals...... 3

2.1 Syntax and semantics of intermediate logics.......... 4 2.2 Operations on Kripke frames.................. 5

Sono allora un soldatino di piombo appeso a un filo sul mare aperto I miei pensieri passano come velieri Leonardo Martellini, Velieri To my schoolteacher Mauro, who taught me how to doubt i Contents Acknowledgements............................. Overview of the thesis........................... iv v

Abstract. We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property. Dedicated to Petr Hájek, on the occasion of his 70th Birthday 1. Preface The three authors of this paper have enjoyed Petr Hájek’s acquaintance since the late eighties, when a lively community interested in the metamathematics of arithmetic shared ideas and traveled among the beautiful cities of Prague, Moscow, Amsterdam, Utrecht, Siena, Oxford and Manchester. At that time, Petr Hájek and Pavel Pudlák were writing their landmark book Metamathematics of First-Order Arithmetic [HP91], which Petr Hájek tried out on a small group of eager graduate students in Siena in the months of February and March 1989. Since then, Petr Hájek has been a role model to us in many ways. First of all, we have always been impressed by Petr’s meticulous and clear use of correct notation, witness all his different types of dots and corners, for example in the Tarskian ‘snowing’-snowing lemmas [HP91]. But also as a human being, Petr has been a role model by his example of living in truth, even in averse circumstances [Hav89]. The tragic story of the Logic Colloquium 1980, which was planned to be held in Prague and of which Petr Hájek was the driving force, springs to mind [DvDLS82]. Finally, we were moved by Petr’s open-mindedness when coming to terms with a situation that turned out to look disconcertingly unlike the ‘standard model ’ 1. Therefore, in this paper, we would like to pay homage to Petr Hájek. Unfortunately, we cannot hope to emulate his correct use of dots and corners. Instead, we do our best to provide some pleasing non-standard models and non-classical arithmetics. 2.

the intermediate logic of