Building on previous work by Mummert et al. (2015, The modal logic of Reverse Mathematics. Archive for Mathematical54 (3–4) 425–437), we study the logic underlying the web of implications and non-implications which constitute the so called reverse mathematics zoo. We introduce a tableaux system for this logic and natural deduction systems for important fragments of the language.

It is well known that satisfiability is decidable for Horn clauses of the class . Since arbitrary Horn clauses can naturally be approximated by -clauses, can be used for realizing any program analysis which can be specified by means of Horn clauses. Recently, we have shown that decidability for Horn clauses from is retained if the clauses are either extended with tests for disequality between subterms identified by paths or for disequality between homomorphic images of terms. These two results refer to orthogonal extensions of -clauses. Here, we provide a generalization of both results. For that, we introduce hom-path disequalities and show that for each finite set of -clauses extended with such tests an equivalent tree automaton with hom-path disequalities can be constructed. Since emptiness for that class of automata has been shown decidable by Godoy et al. in 2010, we conclude that satisfiability is decidable for -clauses with hom-path disequalities.

Coinductive predicates express persisting ‘safety’ specifications of transition systems. Previous observations by Hermida and Jacobs identify coinductive predicates as suitable final coalgebras in a fibration – a categorical abstraction of predicate logic. In this paper, we follow the spirit of a seminal work by Worrell and study final sequences in a fibration. Our main contribution is to identify some categorical ‘size restriction’ axioms that guarantee stabilization of final sequences after ω steps. In its course, we develop a relevant categorical infrastructure that relates fibrations and locally presentable categories, a combination that does not seem to be studied a lot. The genericity of our fibrational framework can be exploited for binary relations (i.e. the logic of ‘binary predicates’) for which a coinductive predicate is bisimilarity, constructive logics (where interests are growing in coinductive predicates) and logics for name-passing processes.

A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This algebraic structure is shown to provide faithful interpretations of finitary process algebras, for an extension of the standard notion of testing semantics, leading to a model that is both denotational (in the sense that the internal workings of processes are ignored) and non-interleaving. Constructions on algebras and their subspaces enjoy a good structure that make them (nearly) a model of differential linear logic, showing that the underlying approach to the representation of non-determinism as linear combinations is the same.

We consider term rewriting under sharing in the form of compression by singleton tree grammars (STG), which is more general than the term dags. Algorithms for the subtasks of rewriting are analysed: finding a redex for rewriting by locating a position for a match, performing a rewrite step by constructing the compressed result and executing a sequence of rewrite steps. The first main result is that locating a match of a linear term s in another term t can be performed in polynomial time if s, t are both STG-compressed. This generalizes results on matching of STG-compressed terms, matching of straight-line-program-compressed strings with character-variables, where every variable occurs at most once, and on fully compressed matching of strings. Also, for the case where s is directed-acyclic-graph (DAG)-compressed, it is shown that submatching can be performed in polynomial time. The general case of compressed submatching can be computed in non-deterministic polynomial time, and an algorithm is described that may be exponential in the worst case, its complexity is nO(k), where k is the number of variables with double occurrences in s and n is the size of the input. The second main result is that in case there is an oracle for the redex position, a sequence of m parallel or single-step rewriting steps under STG-compression can be performed in polynomial time. This generalizes results on DAG-compressed rewriting sequences. Combining these results implies that for an STG-compressed term rewrite system with left-linear rules, m parallel or single-step term rewrite steps can be performed in polynomial time in the input size n and m.

In the article below, Theorem 3.4 requires the additional assumption that A is Kan as well. Indeed, the inductive proof as given only shows that if W(f)<α is a Kan complex, then W(f)<α+1 → A is a Kan fibration.

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environment Beluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs in Beluga relies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage of Beluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploit Beluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate that Beluga provides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments.

The aim of this paper is to provide mathematical foundations of a graph transformation language, called UnCAL, using categorical semantics of type theory and fixed points. About 20 years ago, Buneman et al. developed a graph database query language UnQL on the top of a functional meta-language UnCAL for describing and manipulating graphs. Recently, the functional programming community has shown renewed interest in UnCAL, because it provides an efficient graph transformation language which is useful for various applications, such as bidirectional computation.

In order to make UnCAL more flexible and fruitful for further extensions and applications, in this paper, we give a more conceptual understanding of UnCAL using categorical semantics. Our general interest of this paper is to clarify what is the algebra of UnCAL. Thus, we give an equational axiomatisation and categorical semantics of UnCAL, both of which are new. We show that the axiomatisation is complete for the original bisimulation semantics of UnCAL. Moreover, we provide a clean characterisation of the computation mechanism of UnCAL called ‘structural recursion on graphs’ using our categorical semantics. We show a concrete model of UnCAL given by the λG-calculus, which shows an interesting connection to lazy functional programming.

This paper defines a new proof- and category-theoretic framework for classical linear logic that separates reasoning into one linear regime and two persistent regimes corresponding to ! and ?. The resulting linear/producer/consumer (LPC) logic puts the three classes of propositions on the same semantic footing, following Benton's linear/non-linear formulation of intuitionistic linear logic. Semantically, LPC corresponds to a system of three categories connected by adjunctions reflecting the LPC structure. The paper's meta-theoretic results include admissibility theorems for the cut and duality rules, and a translation of the LPC logic into category theory. The work also presents several concrete instances of the LPC model.

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle.

A very large amount of work in Natural Language Processing (NLP) use tree structure as the first class citizen mathematical structures to represent linguistic structures, such as parsed sentences or feature structures. However, some linguistic phenomena do not cope properly with trees; for instance, in the sentence ‘Max decides to leave,’ ‘Max’ is the subject of the both predicates ‘to_decide’ and ‘to_leave’. Tree-based linguistic formalisms generally use some encoding to manage sentences like the previous example. In former papers (Bonfante et al. 2011; Guillaume and Perrier 2012), we discussed the interest to use graphs rather than trees to deal with linguistic structures, and we have shown how Graph Rewriting could be used for their processing, for instance in the transformation of the sentence syntax into its semantics. Our experiments have shown that Graph Rewriting applications to NLP do not require the full computational power of the general Graph Rewriting setting. The most important observation is that all graph vertices in the final structures are in some sense ‘predictable’ from the input data, and so we can consider the framework of Non-size increasing Graph Rewriting. In our previous papers, we have formally described the Graph Rewriting calculus we used and our purpose here is to study the theoretical aspect of termination with respect to this calculus. Given that termination is undecidable in general, we define termination criterions based on weight, we prove the termination of weighted rewriting systems, and we give complexity bounds on derivation lengths for these rewriting systems.

The multiset-based relational model of linear logic induces a semantics of the untyped λ-calculus, which corresponds with a non-idempotent intersection type system, System R. We prove that, in System R, the size of type derivations and the size of types are closely related to the execution time of λ-terms in a particular environment machine, Krivine's machine.

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien's ludics nets), we defined a new class of multiplicative/additive polarized proof nets, called J-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by using jumps (that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (called cone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can be partially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.

Rewriting notions like termination, normal forms and confluence can be described in an abstract way referring to rewriting only as a binary relation. Several theorems on rewriting, like Newman's lemma, can be proved in this abstract setting. For investigating possible generalizations of such theorems, it is fruitful to have counterexamples showing that particular generalizations do not hold. In this paper, we develop a technique to find such counterexamples fully automatically, and we describe our tool Carpa that follows this technique. The basic idea is to fix the number of objects of the abstract rewrite system, and to express the conditions and the negation of the conclusion in a satisfiability (SAT) formula, and then call a current SAT solver. In case the formula turns out to be satisfiable, the resulting satisfying assignment yields a counterexample to the encoded property. We give several examples of finite abstract rewrite systems having remarkable properties that are found in this way fully automatically.

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

The exponential modality of linear logic associates to every formula A a commutative comonoid !A which can be duplicated in the course of reasoning. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We apply this general recipe to a series of models of linear logic, typically based on coherence spaces, Conway games and finiteness spaces. This algebraic description unifies for the first time a number of apparently different constructions of the exponential modality in spaces and games. It also sheds light on the duplication policy of linear logic, and its interaction with classical duality and double negation completion.