assumed. However, it would be interesting to know if this restriction can be dropped.
Problems arising from non left-linear rewriting are directly transposed to left-linear conditional rewriting. The semi-closure condition is sufficient to avoid this, and it seems to provide the counterpart of left-linearity for unconditional rewriting. However, two remarks have to be made about this restriction. First, it would be interesting to know if it is a necessary condition and besides, to characterize a class of non semi-closed systems that can be translated into equivalent semi-closed ones. Second, semi-closed terminating join systems behave like normal systems. But normal systems can be easily translated into equivalent non-conditional systems. Moreover such a translation preserves good properties such as left-linearity and non ambiguity. As many practical uses of rewriting rely on terminating systems, semi-closed join systems may be in practice essentially an intuitive way to design rewrite systems that can be then efficiently implemented by non-conditional rewriting.
by adding or removing several type constructions, like tensor prod- uct, polymorphism and type fixpoints. Our work follows in spirit the same approach focusing on the encoding of (co)algebras.
Gaboardi et al. [ 17 ] have studied the expressivity of the different light logics by designing embeddings from the light logics to Lin- ear Logic by Level [ 4 ], another logic providing a characterization of polynomial time but based on more general principles. Interest- ingly, in Linear Logic by Level the § modality commutes with all the other type constructions. It would be interesting to study what is the expressivity of this logic with respect to the encoding of alge- bras and coalgebras. Baillot et al. [ 6 ] have approached the problem of improving the expressivity of LAL by designed a programming language with recursion and pattern matching around it. We take inspiration by their work but instead of adding extra constructions we focus on the constructions that can be defined in LAL itself. 9. Conclusion and future works
Remarks : 1) With these rules, the rst proof above is not directly a proof in the restriction: the axiom rule of LJ has to be encoded in the restriction by an axiom rule followed by a contraction rule.
2)This calculus appears also in Danos et al  with a slight dierence in the treatment of structural rules. Like its classical version LKT, it has been considered by Danos et al for its good behaviour w.r.t. embedding into linear logic. Thecalculus LJT appears also as a fragment of ILU, the intuitionistic neutral fragment of unied logic described by Girard in . Thecalculus ILU is itself a form of LJ constrained with a stoup, for which Girard pointed out that \the formula [in the stoup] (if there is one) is the analogue of the familiar
ii) with <ν − 1, i> if ν > the depth of this occurrence ; and after that, we replace this occurrence of <ν − 1, i> withthe closed term given by the environment e.
Now, we obtain (λ n u)[e] (which is closed) by the substitution (ii) on the free occurrences of <ν, i> in u such that ν > the depth of this occurrence in u. Indeed, they are exactly the free occurrences in λ n u. Then, one step of weak head reduction on (λ n u)[e] ¯ φ 1 . . . ¯ φ n performs
19th April 2007
LJQ is a focused sequent calculus for intuitionistic logic, with a simple re- striction on the first premiss of the usual left introduction rule for implication. In a previous paper we discussed its history (going back to about 1950, or be- yond) and presented its basic theory and some applications; here we discuss in detail its relation to call-by-value reduction in lambdacalculus, establishing a connection between LJQ and the CBV calculus λ C of Moggi. In particular,
Expr ession also needs an in terpreted attribute. For DTD-technical reasons, only the two most important values are specified for the val attribute (similarly, only two ord values are given). The DTD also does not enforce context-dependent attribute values such as <Equal oriented="no"> being normally used in conditions. Moreover, while the DTD does not prevent Lambda formulas to occur on the lhs of (both kinds of) equations, a static analyzer should confine them to the rhs of oriented equations. A more precise XSD is part of the emerging Functional RuleML 0.9 [http://www.ruleml.org/fun] .
In differential proof nets the 0-ary tensor and the 0-ary par can be added freely in the sense that we still have a natural interpretation in MRel and M ∞ . These
operations can be translated in our calculus as an exception mechanism. With on one side a τ (Q) that “raises” the exception (or test) Q by burning its ap- plicative context (whenever these applications do not have any linear compo- nent, otherwise it diverges). And with on the other side a ¯ τ(M ) that “catch” the exceptions in M by burning the abstraction context of M (whenever this abstraction is dummy). The main difference with a usual exception system is the divergence of the catch if no exception are raised.
However, one of the shortcomings of L tac is that tactics used to construct proofs are unsafe 7 . This language can be improved by means of types for tactics as is suggested by
Efficiency in theorem provers can also be gained by improving the machinery to guide and verify proofs, by means of improvements in the interaction withthe kernel and the tactic engine. Improvements for system development have many contributions: libraries for specialised theories (see the list of users’ contributions 8 ), tactic languages to enhance the mechanisation of proofs [ 36 , 118 ], improvements to the user back-end framework for example the asynchronous edition of proofs [ 17 ], interaction with other theorem provers to cooperate in a large developments, etc. All of this turns C OQ into a sophisticated programming language where the program development arises naturally, for instance the R USSELL extension [ 101 ] to develop programs with dependent types.
In this chapter, we are interested in the conﬂuence of rewriting systems. In par- ticular, we study a problem arising from the combination of rewrite rules with β- reduction. Remember that conﬂuence is a highly desirable property of the λΠ- Calculus Modulo for several reasons. First, conﬂuence is the most direct way to prove the product compatibility property (Theorem 2.6.11). Second, as soon as the rewrite relation is also strongly normalizing, conﬂuence entails the decidability of the congruence: two terms are convertible if and only if they have the same normal form. Third, conﬂuence has also been used in the previous chapter for proving that weakly-well-formed rewrite rules are permanently well-typed. More generally, any property based on uniﬁcation will require conﬂuence. Lastly, conﬂuence is used to prove strong normalization when there are type-level rewrite rules [Bla05b].
Although the original λµe µ calculus of [ 2 ] has a system of simple types based on the sequent calculus, the untyped version is a Turing-complete language for computation with explicit repre- sentation of control, as well as code. In this work we try to give a meaning to untyped λµe µ calculus and understand its behaviour. We interpret its variant closed under call-by-name reduction in the category of negated domains, and the variant closed under call-by-value reduction in the Kleisli category. As far as we know, this is the first interpretation of untyped λµe µ calculus. We also prove the confluence of both versions.
5.2 Tailoring LambdaCalculus for an ASM
Let F be the family of interpretations of all static symbols in the initial state. The adequate Lambdacalculus to encode the ASM is Λ F .
Let us argue that this is not an unfair trick. An algorithm does decompose a task in elementary ones. But “elementary” does not mean “trivial” nor “atomic”, it just means that we do not detail how they are performed: they are like oracles. There is no absolute notion of elementary task. It depends on what big task is under investigation. For an algorithm about matrix product, multiplication of integers can be seen as elementary. Thus, algorithms go with oracles.
Kozen (1979) gives a semantics of imperative probabilistic pro- grams as partial measurable functions from infinite random traces to final states, which serves as the model for our trace semantics. Kozen also proves this semantics equivalent to a domain-theoretic one. Park et al. (2008) give an operational version of Kozen’s trace- based semantics for a λ-calculuswith recursion, but “do not inves- tigate measure-theoretic properties”. Cousot and Monerau (2012) generalise Kozen’s trace-based semantics to consider probabilistic programs as measurable functions from a probability space into a semantics domain, and study abstract interpretation in this setting.
hievements and Perspe
Using various proof te
hniques, we have established that the ~ x-
ulus is strongly normalizing. F or that purpose, we have formalized a proof te
hnique of SN via PSN. Let us mention that we have su
essfully applied this te
hnique, with some adjustments, to prove SN of the -
ed in [3℄) for the rst time, as far as we know. We also used it to establish that PSN implies SN for the -
ulus [1℄, for whi
h PSN is known to fail [10℄, showing that, for this
ulus, the only problem of SN is in PSN.
We illustrate these notions in Figure 1, for the example λx.λy. ((λz.z)y)x. The scope of the abstraction λx is the entire subterm, λy. ((λz.z)y)x (which may or may not be taken to include λx itself). Note that with explicit substitution, the scope may grow or shrink by lifting explicit substitutions in or out. The skeleton is the term λx.λy. (wy)x where the subterm λz.z is lifted out as an (ex- plicit) substitution [λz.z/w]. The spine of a term, indicated in the second image, cannot naturally be expressed with explicit substitution, though one can get an impression with capturing substitutions: it would be λx.λy.wx, withthe sub- term (λz.z)y extracted by a capturing substitution [(λz.z)y/w]. Observe that the skeleton can be described as the iterated spine: it is the smallest subgraph of the syntax tree closed under taking the spine of each abstraction, i.e. that contains the spine of every abstraction it contains.
These collapse results require innocence – or at least a substitute ensuring that composition is deadlock-free. But beyond the sequential deterministic case, there was for a long time no adequate notion of innocence [ Harmer and McCusker 1999 ]. This changed only a few years ago, with two notions of non-deterministic innocent strategies (using concurrent games [ Castellan et al. 2014 ] and sheaves [ Tsukada and Ong 2015 ]). These two models depart from traditional game semantics in ways that are technically very different, but conceptually similar: they both record more intensional behavioural information. This change of perspective recently allowed a quantitative extension of the relational collapse [ Castellan et al. 2018 ] for a probabilistic language, using concurrent games. Concurrent games are a family of game semantics initiated in [ Abramsky and Melliès 1999 ], with intense activity in the past decade prompted by a new non-deterministic generalization based on event structures [ Rideau and Winskel 2011 ]. Building on notions from concurrency theory, they are a natural fit for the semantics of concurrent programs [ Castellan and Clairambault 2016 ; Castellan and Yoshida 2019 ]. It is perhaps more surprising that their adoption has a strong impact even when studying sequential programs such as the quantum λ-calculus: they offer a fine-grained causal presentation of the behaviour of programs that contrasts withthe temporal presentation of traditional games models. This has far-reaching consequences. For the present paper, both our collapse theorem and the congruence of the observational quotient required for full abstraction rely on a visibility condition, a substitute for innocence ensuring a deadlock-free composition – visibility bans certain impure causal patterns, leveraging the expressiveness of concurrent games. Thus, our constructions rely heavily on the fact that the model of [ Clairambault et al. 2019 ] was developed within concurrent games. Our collapse theorem follows in the footsteps of the probabilistic collapse [ Castellan et al. 2018 ], which we generalize to the quantum case.
We have introduced the structural λj-calculus, a concise but expressive λ-calculuswith jumps. No prior knowledge of Linear Logic is necessary to understand λj, despite their strong connection. We have established many different sanity prop- erties for λj such as confluence and PSN. We have used λj as an operational framework to elaborate new characterisations of the well-known notions of full developments and L-developments, and to obtain the new, more powerful notion of XL-development. Finally, we have modularly added commutation of indepen- dent jumps, σ-equivalence and two kinds of propagations of jumps, while showing that PSN still holds.
Thus, in order to characterize normalization in λ µ , we will resort to intersection and union type. But moreover, we want to do that in a quantitative way, as it was announced in the introduction of Part II. Indeed, the non-idempotent approach provides very simple combinatorial arguments, only based on a decreasing measure, to characterize head or strongly normalizing terms by means of typability (recall Sec. 3.4 for HN and Sec. 5.2 for SN). We show that for every typable term t with type derivation Π, if t reduces to t 0 , then t 0 is typable with a type derivation Π 0 such that the measure of Π is strictly greater than that of Π 0 . In the well-known case of the λ-calculus, this measure is simply based on the structure of type tree derivations given by the number of its nodes (Definition 3.2), which strictly decreases along reduction. However, in the λ µ -calculus, the creation of nested applications during µ-reduction may increase the number of nodes of the corresponding type derivations, so that such a naive definition of measure is not decreasing anymore. We then take into account not only the structure of derivations, but also the structure (multiplicity and size) of certain types appearing in the derivations, thus ensuring an overall decreasing of the measure during reduction.