We investigate a novel intuitionistic modal logic, called Propositional Lax Logic, with promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness `up to' behavioural constraints --- a central notion in hardware verification --- as a logical modality. The resulting logic is unorthodox in several respects. As a modal logic it is special since it features a single modal operator fl that has a flavour both of possibility and of necessity. As for hardware verification it is special since it is an intuitionistic rather than classical logic which so far has been the basis of the great majority of approaches. Finally, its models are unusual since they feature worlds with inconsistent information and furthermore the only frame condition is that the fl -frame be a subrelation of the oe-frame. In the paper we will provide the motivation for Propositional Lax Logic and present several technical results. We will investigate...

We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

In this report we introduce a new proof-theoretic approach to the semantics of Constraint Logic Programming, based on an intuitionistic first-order modal logic, called QLL. The distinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual extensional aspects of Logic Programming, i.e. "which queries are successful, " but also some of the intensional aspects, i.e. "what is the answer constraint and how is it constructed." It provides for a direct link between the model-theoretic and the operational semantics following a formulas-as-programs and proofs-as-constraints principle. This approach makes use of logic in a different way than other approaches based on logic calculi. On the one side it is to be distinguished from the well-known provability semantics which is concerned merely with what is derivable as opposed to how it is derivable, paying attention to the fact that it is the how that determines the answer constraint. ...

this paper. This calculus satises cut-elimination, as for instance shown (in a more complicated form) in [Wij90]. This calculus is dierent from what is usually taken as the basic constructive system K, as we do not assume the distribution of possibility (3) over disjunctions neither in its binary form 3(A _ B) ! (3A _ 3B) nor in its nullary form 3? ! ? The sequent calculus above corresponds to an axiomatic formulation given by axioms for intuitionistic logic, plus axioms: 2(A ! B) ! (2A ! 2B) 2(A ! B) ! (3A ! 3B) 2A3B ! 3(A B) together with rules for Modus Ponens and Necessitation: ` A ! B ` A ` B MP ` A ` 2A Nec Wijesekera proved a Craig interpolation theorem, one of the usual consequences of syntactic cut-elimination and produced Kripke, algebraic and topological semantics for a calculus very similar to the one above. The only dierence is that he does assume 3? ! ?. From our \wish list" for logical systems only a natural deduction formulation and a categorical semantics are missing. These we proceed to discuss

We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency design, etc. Both systems have so far been studied mainly from a type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

Up until now classical logic has been the logic of choice in formal hardware verification. This report advances the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes at the same time. 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 [4], in which validity is validity up to stabilization. We show that this semantics is equivalently characterized in terms of stabilization bounds so that implication oe comes out as "boundedly gives rise to." An int...

This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic

In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1 , where the calculus for PLL 1 is duplication--free while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in relation to the problem of Formal Hardware Verification. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Namely, we enlarge the language containing the usual sings T and F with the new sign F c . PLL and PLL 1 logics are characterized by a Kripke--semantics which is a "weak" version of the semantics for ordinary intuitionistic modal logics. In this paper we establish the soundness and completeness theorems for these calculi.

In this paper we relate the lax modality O to Intuitionistic Propositional Logic (IPL) and give a complete characterisation of inhabitation in Computational Type Theory (CTT) as a logic of constraint contexts. This solves a problem open since the 1940's, when Curry was the first to suggest a formal syntactic interpretation of O in terms of contexts.

. We develop within higher order logic (HOL) a general and exible method of abstraction and renement, which specically addresses the problem of handling constraints. The method is based on an interpretation of rst-order Lax Logic in HOL, which can be seen as a modal extension of deliverables. It provides a new technique for automating reasoning about behavioural constraints. We show how the method can be applied in several dierent tasks, for example to achieve a formal separation of the logical and timing aspects of hardware design, and to generate systematically timing constraints for a simple sequential device from a formal proof of its abstract behaviour. The method and all proofs in the paper have been implemented in Isabelle as a denitional extension of the HOL logic. We assume the reader is familiar with higher order logic but do not assume detailed knowledge of circuit design. 1