We consider synthesis problems from logical specifications over infinite data domains, expressed in the logic constraint LTL (CLTL), which extends LTL with predicates over an infinite set of data values. We consider register-bounded synthesis, where the goal is to automatically generate, if it exists, a transducer with r registers that realizes a given CLTL formula, where r is also given as input. We prove that CLTL register-bounded synthesis is 2ExpTime-c for various data domains such as any infinite set with equality, (Q,<), and (N,<). For the latter domain, this contrasts with known undecidability results of (unbounded) register CLTL synthesis, by Bhaskar and Praveen. Lastly, we consider synthesis in a partial observation setting by extending CLTL with invisible variables. 2012 ACM Subject Classification Theory of computation -> Automata over infinite objects; Theory of computation -> Modal and temporal logics
Probabilistic models of computation have been studied for over three decades now, from foundational, logical, coalgebraic, categorical, as well as more practical verification-motivated point of view. In my work, and in this talk, we focus on the foundational, semantics side of probabilistic automata and transition systems, and in particular the relevant monads, and their algebras. The interplay of probability and non-determinism has been particularly challenging from a semantics point of view for some decades, as it does not just yield a monad. There are by now well-studied solutions to this and several monads for probability and non-determinism: some enriching the structure with convexity to obtain a monad, albeit a non-commutative one, others deliberately simplifying it, by imposing commutativity. It is well known that monads have two faces: a computational one - we think here of the powerset monad modelling non-determinism, or the probability distribution monad modelling probabilistic choices, and a universal-algebraic one - where we think of the algebraic presentation of the monads, like semilattices for the powerset monad and (variants of) convex algebras for (variants of) the probability distribution monad. Combining non-determinism and probability yields other combined monads, among which probably the most studied is the convex-subsets-of-distributions monad. This monad is presented by so-called convex semilattices, algebraic structures that are both a semilattice and a convex algebra, with suitable distributivity connecting the operations. From a semantics point of view, the algebraic presentations give us a nice way to define (and sometimes compute) language (aka trace) equivalence of the corresponding automata. Moreover, the presentations are useful for axiomatizations of language equivalence. From an algebraic point of view, these algebras are interesting and many questions about them are still open. We will discuss language semantics, its axiomatization, as well as some obtained and open algebraic problems for convex algebras and convex semilattices - and their computational consequences. In particular, we have a full characterization of congruences of (variants of) convex algebras, which for example yields decidability of distribution semantics; other results on congruences, e.g. a result on a congruence being finitely generated as a subalgebra, surprisingly accellerated the proof of completeness of infinite trace semanitcs, etc. We will review some existing results: all congruences are described and fp = fg for convex algebras, termination is a black hole, useful functors are proper on convex algebras, cancellativity of convex semi-lattices; as well as mention ongoing work on: mid-point-cancellativity, subalgebras, and homomorphisms for convex semi-lattices, as well as topological convex semilattices. This talk is based on previous joint works with Filippo Bonchi, Alexandra Silva, Valeria Vignudelli, and Harald Woracek, as well as on ongoing work and discussions with Matteo Mio, Alex Simpson, and Harald Woracek. 2012 ACM Subject Classification Theory of computation -> Probabilistic computation; Theory of computation -> Categorical semantics; Theory of computation -> Logic and verification; Theory of computation -> Program verification; Theory of computation -> Randomness, geometry and discrete structures
We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We prove that the proof may fail for adjunctions between wild categories. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, called 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda. 2012 ACM Subject Classification Theory of computation -> Type theory
Temporal logics are a powerful tool to specify properties of computational systems. For concurrent programs, Higher Dimensional Automata (HDA) are a very expressive model of non-interleaving concurrency. HDA recognize languages of partially ordered multisets, or pomsets. Recent work has shown that Monadic Second Order (MSO) logic is as expressive as HDA for pomset languages. In the case of words, Kamp's theorem states that First Order (FO) logic is as expressive as Linear Temporal Logic (LTL). In this paper, we extend this result to pomsets. To do so, we first investigate the class of pomset languages that are definable in FO. As expected, this is a strict subclass of MSO-definable languages. Then, we define a Linear Temporal Logic for pomsets (LTLPoms), and show that it is equivalent to FO. 2012 ACM Subject Classification Theory of computation. Modal and temporal logics; Theory of computation -> Concurrency
Useful evaluation is an optimised evaluation mechanism for functional programming languages. It relies on representing terms with sharing and imposing a restricted notion of useful substitutions, that intuitively disallows copying subterms that do not contribute to the progress of the computation. In particular, useful call-by-value evaluation optimises the standard call-by-value strategy by preserving its original semantics. This preservation result has been shown by means of syntactical rewriting techniques, difficult to adapt to alternative variants of the calculi at play. In this work, we present the first semantic model of useful call-by-value evaluation through the non-idempotent intersection type system u. Our first contribution is a characterisation of termination for useful call-by-value evaluation via system u. That is, a term is typable in system u if and only if it terminates in the useful call-by-value strategy. As a second contribution, we show that system u provides a quantitative interpretation for useful call-by-value evaluation, offering exact step-count information for program evaluation. Our third contribution is that termination in call-by-value and useful call-by-value are equivalent. This ensures in particular that call-by-value, which is (potentially) erasing, and useful call-by-value, which is non-erasing, are observationally equivalent. Even though the specification of the operational semantics of useful evaluation is highly complex, system u is notably simple. As far as we know, system u is one of the scarce quantitative type systems capturing exactly the substitution step-count for variables and abstractions in an open call-by-value strategy. 2012 ACM Subject Classification Theory of computation -> Lambda calculus; Theory of computation -> Type theory; Theory of computation -> Operational semantics
We study small witnesses for the inequivalence of two regular languages. A natural witness is a distinguishing word, e.g. a word in exactly one of the two languages. We propose using more succinct witnesses in the form of witnessing DFAs. A witnessing DFA recognizes a subset of one of the languages and contains at least one distinguishing word. In this way the DFA expresses behaviour contained in the first language but not the second. We show witnessing DFAs can be used to present more concise witnesses for the inequivalence of two regular languages. We show that the decision problem for the existence of a witnessing DFA of certain size is NP-complete in general, and in P in the special case of unary DFAs. Besides these computational aspects, we study structural properties of witnessing DFAs. Not all languages can be a minimal witness. It turns out that minimal witnesses are exactly the languages that are not decomposable in the union of languages with smaller state-complexity, the so-called prime languages as studied earlier by Kupferman and Mosheiff. 2012 ACM Subject Classification Theory of computation -> Regular languages
Many computational problems can be modelled as the class of all finite structures A that satisfy a fixed first-order sentence. hereditarily, i.e., we require that every (induced) substructure of A satisfies phi. We call the corresponding computational problem the hereditary model checking problem for phi, and denote it by Her(phi). We present a complete description of the quantifier prefixes for. such that Her(phi) is in P; we show that for every other quantifier prefix there exists a formula. with this prefix such that Her(phi) is coNP-complete. Specifically, we show that if Q is of the form.for all*there exists for all* or of the form for all*there exists*, then Her(phi) can be solved in polynomial time whenever the quantifier prefix of phi is Q. Otherwise, Q contains there exists there exists for all or there exists for all there exists as a subword, and in this case, there is a first-order formula phi whose quantifier prefix is Q and Her(phi) is coNP-complete. Moreover, we show that there is no algorithm that decides for a given first-order formula. whether Her(phi) is in P (unless P=NP). 2012 ACM Subject Classification Theory of computation -> Complexity theory and logic
We extend the existing HOL Light Library for Modal Systems (HOLMS) to support a modular implementation of modal reasoning within the HOL Light proof assistant. We deeply embed axiomatic calculi and relational semantics for seven normal modal logics (K, T, B, K4, S4, S5, GL) and formalise modal adequacy theorems for these systems. We then leverage those formalisations to implement a mechanism for automated reasoning via proof-search in the associated labelled sequent calculi, which we shallowly embed in HOL Light's goal-stack mechanism. This way, we equip the general-purpose proof assistant with (semi)decision procedures for these logics that, in case of failure to construct a proof for the input formula, return a certified countermodel within the appropriate class for the logic under consideration. On the methodological side, we propose a precise measure of the modularity of our approach by systematically adopting Christopher Strachey's distinction between ad hoc and parametric polymorphism throughout the library. 2012 ACM Subject Classification Theory of computation -> Modal and temporal logics; Theory of computation -> Higher order logic; Theory of computation -> Automated reasoning; Theory of computation -> Type theory; Theory of computation -> Proof theory; Theory of computation -> Logic and verification
The interaction between a system and the components modeling its environment is traditionally modeled by a multi-player game played on a finite graph. In zero-sum games, the players have conflicting objectives, and it is clear that increasing the memory of the environment players can only make it harder for the system to win. In non-zero-sum games, the objectives of the players may overlap. There, typical questions concern the stability of the game and the equilibria the players may reach. In particular, in rational synthesis (RS), the goal is to find an equilibrium that satisfies the objective of the system. We study how the memory of the environment players may affect the existence of an RS solution. As we show, the picture is diverse, even when the objectives of all players are memoryless. On the one hand, when stability amounts to a Nash equilibrium (NE), then increasing the memory of the environment may only help the system to suggest an RS solution. On the other hand, when the notion of stability involves deviations by coalitions of environment players, for example in a strong Nash equilibrium (SNE), then increasing their memory may sometimes enable and sometimes prevent the existence of an RS solution. We study memory bounds for the players, showing that the memory required may be polynomial in an NE-RS solution and exponential in an SNE-RS solution. We also solve the SNE-RS problem, show that it is PSPACE-complete, and relate the differences between NE and SNE with the differences between cooperative and non-cooperative RS. 2012 ACM Subject Classification Theory of computation -> Formal languages and automata theory; Theory of computation -> Semantics and reasoning
The 0-imax-successor algebra, where imax: N x N -> N is the function defined by imax( n, 0) = 0 and imax( n, S(m)) = max( n, S(m)), is used to represent universe levels in impredicative type theory, in particular with universe polymorphism which introduces level variables, so it is present in proof systems such as Rocq and Lean. In particular, we need to know when two elements of this algebra are equivalent, and we may also want to decide the inequality. In this article, we introduce a canonical form for the terms of this algebra, and we provide a canonization algorithm. It permits deciding level equivalence by checking the canonical form equality, and also permits easily checking if a level is smaller than another one. 2012 ACM Subject Classification Theory of computation -> Type theory; Theory of computation -> Equational logic and rewriting
Semiring provenance was originally introduced in database theory with the aim of explaining why certain tuples are (not) contained in the answer of a query. To this end, logical statements are not just evaluated to true or false but to values in a commutative semiring. Depending on the underlying semiring, this allows us to track descriptions of the atomic facts that are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. Recently, this approach has been expanded to a systematic study of semiring semantics for first-order logic and other logical systems. This raises the question to what extent model-theoretic results can be generalised to semiring semantics and how this relates to the algebraic properties of the underlying semiring. Here we investigate the availability of compactness in semiring semantics. The appropriate setting for this is based on absorptive semirings with well-defined infinitary products. Compactness can be stated either in terms of satisfiability or in terms of entailment, and these two variants are trivially equivalent in Boolean semantics. However, this is no longer the case in semiring semantics. Compactness in terms of satisfiability, defined as the existence of non-zero valuations, indeed generalises to every infinitary absorptive semiring. For compactness in terms of entailment the situation is different. The entailment relation naturally extends to semiring semantics (via the natural order on the semiring) but this yields a stronger variant of compactness, which fails for certain important semirings, including the tropical semiring and the Lukasiewicz semiring. Our main positive results show that strong compactness does indeed hold for all finite semirings and all lattice semirings. 2012 ACM Subject Classification Theory of computation -> Logic
Interface theories, notably interface automata, serve as expressive frameworks for component-based design, specifying component behavior and interaction in concurrent systems. Traditional interface formalisms specify assumptions that a component's environment must satisfy and the guarantees that each component provides. This qualitative view of component interaction based on imposing strict assumptions and Boolean guarantees may, however, not be expressive enough to capture the system's allowed or desired behaviors under different environments. In this paper, we introduce reward interfaces to support component-based design while accommodating multi-valued correctness requirements and adaptive best-effort satisfaction of component's guarantees. Building upon interface automata, our framework enables modeling a rich class of quantitative component specifications. We propose formal notions of implementation, refinement and compatibility for reward interfaces. We study a class of reward interfaces with automata-based representations, for which we provide algorithms for checking compatibility and refinement, and existence of best-effort implementations. Our framework offers a comprehensive approach to reward interface specification and design. 2012 ACM Subject Classification Theory of computation -> Formal languages and automata theory
The relationship between classical and constructive logics has long been illuminated by a series of conservation results, beginning with Kolmogorov's negative translation and Glivenko's double negation theorem, and later extended by Kuroda and Segerberg to first-order and minimal logics respectively. These results reveal how certain classical principles can be interpreted or recovered within weaker constructive frameworks, either via translations or through minimal extensions that satisfy specific logical properties. In this paper, we propose a unifying generalisation of these conservation theorems, that consolidates and expands the abstract methods introduced in earlier studies, offering a unified perspective on the interplay between classical provability and constructive reasoning.
Separation Logic (SL) is a well-established framework for reasoning about programs that manipulate dynamic memory. To express and verify properties of custom recursive data structures, SL is extended with spatial predicates defined by user-specified inductive rules. Many verification problems reduce to deciding entailments between formulas involving these predicates. While the general entailment problem is undecidable, a broad class of inductive rules - known as PCE (Progressing, Connected, and Established) - has been identified for which entailment is decidable. In this work, we extend the study of the entailment problem to Dynamic Separation Logic (DSL), an extension of SL that includes dynamic modalities for reasoning about actions on the heap and store. We show that entailment in DSL remains decidable for PCE rules by proving that dynamic modalities can be automatically eliminated. 2012 ACM Subject Classification Theory of computation -> Logic and verification; Theory of computation -> Automated reasoning
Quantitative semantics are those denotational semantics that inherit from linear logic [24] a sensitivity to the multiplicity of resources involved in computation. Those include the relational model [24] and its numerous variations (such as finiteness spaces [18], weighted relational models [29] and their extensions [20, 17], generalized species of structure [22], span models [32, 12], etc), as well as related syntactic methods such as non-idempotent intersection types [15] and Taylor expansion of lambda-terms [21]. Interactive semantics are usually also quantitative, but in addition they present the interactive behaviour of proofs and programs, generally organized chronologically - those include the many variants of game semantics (starting with [27, 2]), and other frameworks such as Geometry of Interaction [25] or ludics [26]. Both families are cornerstones of modern denotational semantics, and both have associated Alonzo Church awards: game semantics in 2017, and quantitative semantics (in particular, differential linear logic and the differential.-calculus) in 2024. It has more or less always been clear to the experts that the two, sharing an origin in linear logic, are conceptually related. Yet there are differences, which seem fundamental: in particular, while quantitative models compose relationally, the composition of strategies follows an intricate "parallel interaction plus hiding" process inspired from concurrency theory [1]. The two families of models have also historically targeted different kinds of languages: whereas quantitative semantics focused on theoretical calculi (and the lambda-calculus in particular), game semantics is known for fully abstract models for languages with elaborate combinations of effects including local state [3], control operators [28], and concurrent primitives [23]. Early on, researchers have explored the relationship between the two [16, 4], and investigations on this question have spanned decades [5, 6, 35, 33]. In particular, Mellies' work on asynchronous games [31, 30] made significant conceptual contributions, showing that the issue was enlightened by adopting a positional formulation of game semantics, where points in the relational model simply arise as certain positions. This talk surveys recent developments in this line of work, shedding light on the connection between those two families. Our work is set in so-called "thin concurrent games" [8, 9], an extension with symmetry of Rideau and Winskel's concurrent games on event structures [34]. Event structures being one of the main "truly concurrent" models of concurrency [36], it is perhaps expected that thin concurrent games can model concurrent languages: they provide a truly concurrent refinement of Ghica and Murawski's fully abstract model of Idealized Concurrent Algol [7, 9]. But beyond the semantics of concurrency, thin concurrent games are also a deep reworking on game semantics built from causal principles, inheriting from asynchronous games a positional flavour. In thin concurrent games, strategies have a dual nature: an event-based nature where they appear as certain event structures composed via parallel interaction plus hiding; or a positional nature where they appear as certain spans of groupoids, composed by pullback (modulo a technical condition on strategies called visibility) - they can be regarded both as a games and a relational model! Leveraging this dual nature, in a sequence of papers with Castellan, de Visme, Olimpieri and Paquet, we have been able to link the single framework of thin concurrent games with numerous other models. This includes various traditional alternating or non-alternating games models [7, 9], the weighted relational model [14], the quantum relational model [11], generalized species of structure [13], and - going beyond quantitative semantics - the linear Scott model [10], a linear decomposition of standard Scott domain semantics [19]. All these distinct models are obtained by projecting away certain aspects of thin concurrent games, giving some support to the claim that thin concurrent games are a Rosetta stone for interactive and quantitative semantics. 2012 ACM Subject Classification Theory of computation -> Denotational semantics
