Papers by Aleksy Schubert
Curry-Howard isomorphism makes it possible to obtain functional programs from proofs in logic. We... more Curry-Howard isomorphism makes it possible to obtain functional programs from proofs in logic. We analyse the problem of program synthesis for ML programs with algebraic types and relate it to the proof search problems in appropriate logics. The problem of synthesis for closed programs is easily equivalent to the proof construction in intuitionistic propositional logic and thus fits in the class of PSPACE-complete problems. We focus further attention on the synthesis problem relative to a given external library of functions. It turns out that the problem is undecidable for unbounded instantiation in ML. However its restriction to instantiations with atomic types only results in a case equivalent to proof search in a restricted fragment of intuitionistic first-order logic, being the core of Σ 1 level of the logic in the Mints hierarchy. This results in EXPSPACEcompleteness for this special case of the ML program synthesis problem.
One of the biggest obstacles in the formalisation of the Java bytecode is that the language consi... more One of the biggest obstacles in the formalisation of the Java bytecode is that the language consists of around 200 instructions. However, a rigorous handling of metatheoretic properties of a programming language requires a formalism which is compact in size. Therefore, the actual Java bytecode instruction set is never used in the context. Instead, the existing formalisations usually cover a 'representative' set of instructions. This paper describes a design of formalisation that provides a concise set of abstract, generic instructions that can be specialised to obtain any particular bytecode instruction. In this way one can work with a manageable set of instructions to prove general facts about the Java bytecode, but at the same time all the bytecode instructions are available to enable direct verification of actual bytecode programs. A considerable part of the design has been realised in Coq.
Journal of Logic and Computation, Nov 16, 2021
Predicate intuitionistic logic is a well-established fragment of dependent types. Proof construct... more Predicate intuitionistic logic is a well-established fragment of dependent types. Proof construction in this logic, as the Curry–Howard isomorphism states, is the process of program synthesis. We present automata that can handle proof construction and program synthesis in full intuitionistic first-order logic. Given a formula, we can construct an automaton such that the formula is provable if and only if the automaton has an accepting run. As further research, this construction makes it possible to discuss formal languages of proofs or programs, the closure properties of the automata and their connections with the traditional logical connectives.
Lecture Notes in Computer Science, 2005
We consider terms of simply typed lambda calculus in which copy of a subterm may not be inserted ... more We consider terms of simply typed lambda calculus in which copy of a subterm may not be inserted to the argument of itself during the reduction. The terms form wide class which includes linear terms. We show that corresponding variant of the dual interpolation problem i.e. problem in which all expressions can be restricted to terms of this kind. Thus the model for this kind of expressions can admit fully abstract semantics.
Automata theoretical techniques are developed that handle inhabitant search in the simply typed l... more Automata theoretical techniques are developed that handle inhabitant search in the simply typed lambda calculus. The automata-theoretic model for inhabitant search, which can be viewed as proof search by the Curry-Howard isomorphism, is proven to be adequate by reduction of the inhabitant existence problem to the emptiness problem for the automata. To strengthen the claim, it is demonstrated that the latter has the same complexity as the former. We also discuss the basic closure properties of the automata.
Different sequences of proof steps may result in different experiences of the overall formal proo... more Different sequences of proof steps may result in different experiences of the overall formal proof clarity. There is a conjecture that sequence in which more steps refer to the content of the preceding statement (1) is more comprehensible than the one in which references span longer distances (2). We studied the claim in experimental setting where subjects indicated their reading preference with two versions of the same proof. The difference between their reports indicates that contrary to the conjecture, proofs of the kind (2) can give cognitive advantage in a statistically significant way.
The presentation will contrast the complexity results for proving assertions in classical and int... more The presentation will contrast the complexity results for proving assertions in classical and intuitionistic logic. The comparison will be built upon the known results for propositional logic and predicate one. The predicate case will be based upon the Mints hierarchy in intuitionistic logic which will be contrasted with its counterpart i.e. the prenex hierarchy in classical logic. The di erences in complexity will be illustrated with examples of particular proving mechanisms that are responsible for the divergence, which should facilitate the understanding of where the mathematics is done constructively.
Lecture Notes in Computer Science, 2000
We introduce several structures between Church-style and Curry-style based on partially typed ter... more We introduce several structures between Church-style and Curry-style based on partially typed terms formalism. In the uniform framework, we study the static properties of the λ-terms between the two styles. It is proved that type checking, typability, and type inference for domain-free λ2 are in general undecidable. A simple instance of the second-order unification problem is reduced to the problem of type inference for domain-free λ2. The typability problem is undecidable even for a predicative fragment of domain-free λ2, called the rank 2 fragment. It is also found that making polymorphic domains free and the use of type-holes [ ] are independently responsible for the undecidability of the partial polymorphic type reconstruction problem.
Positive Logic Is 2-Exptime Hard
One of the biggest obstacles in the formalisation of the Java bytecode is that the language consi... more One of the biggest obstacles in the formalisation of the Java bytecode is that the language consists of around 200 instructions. However, a rigorous handling of metatheoretic properties of a programming language requires a formalism which is compact in size. Therefore, the actual Java bytecode instruction set is never used in the context. Instead, the existing formalisations usually cover a 'representative' set of instructions. This paper describes a design of formalisation that provides a concise set of abstract, generic instructions that can be specialised to obtain any particular bytecode instruction. In this way one can work with a manageable set of instructions to prove general facts about the Java bytecode, but at the same time all the bytecode instructions are available to enable direct verification of actual bytecode programs. A considerable part of the design has been realised in Coq.
Lecture Notes in Computer Science
Lecture Notes in Computer Science, 2009
It is known that the first-order theory with a single predicate → that denotes one-step rewriting... more It is known that the first-order theory with a single predicate → that denotes one-step rewriting reduction on terms is undecidable already for formulae with ∃∀ prefix. Several decidability results exist for the fragment of the theory in which the formulae start with the ∃ prefix only. This paper considers a similar fragment for a predicate → p which denotes the parallel one-step rewriting reduction. We show that the theory is related to the type entailment problem and prove that the first-order theory of → p is undecidable already for formulae with ∃ prefix.
Lecture Notes in Computer Science, 2005
We consider terms of simply typed lambda calculus in which copy of a subterm may not be inserted ... more We consider terms of simply typed lambda calculus in which copy of a subterm may not be inserted to the argument of itself during the reduction. The terms form wide class which includes linear terms. We show that corresponding variant of the dual interpolation problem i.e. problem in which all expressions can be restricted to terms of this kind. Thus the model for this kind of expressions can admit fully abstract semantics.
Theoretical Computer Science, 2014
We study type checking, typability, and type inference problems for type-free style and Curry sty... more We study type checking, typability, and type inference problems for type-free style and Curry style second-order existential systems where the type-free style differs from the Curry style in that the terms of the former contain information on where the existential quantifier elimination and introduction take place but omit the information on which types are involved. We show that all the problems are undecidable employing reduction of second-order unification in case of the typefree system and semiunification in case of the Curry style system. This provides a fine border between problems yielding to a reduction of second-order unification problem and the semiunification problem. In addition, we investigate the subject reduction property of the system in the Curry-style. existential type system λ ∃ [Fuj05] can be regarded as a subsystem of λ2. On the 1998 ACM Subject Classification: F.4.1.
A note on subject reduction in -Curry with respect to complete developments
Information Processing Letters, 2014
ABSTRACT We prove subject reduction of the Curry style existential system (→,∃) with regard to co... more ABSTRACT We prove subject reduction of the Curry style existential system (→,∃) with regard to complete developments. We use here a remote adaptation of the technique of Barbanera et al. used before for systems with union types.
Information and Computation, 2012
We consider here a number of variations on System F that are predicative second-order systems who... more We consider here a number of variations on System F that are predicative second-order systems whose terms are intermediate between the Curry style and the Church style. As in the Church style, the terms we deal with here contain the information on where universal quantifier elimination and introduction in the type inference process must take place. However, they omit the information on what types are involved in the rules, which is similar to Curry forms. This can be viewed as a version of the partial type reconstruction problem considered by Boehm and Pfenning in which type erasure is done in a systematic way. In this paper we prove the undecidability of the type checking, type inference, and typability problems for the system. This demonstrates that the reason for undecidability is not the absence of the information where the second-order rules should be applied but the actual shape of the polymorphic types to be used in the derivation. Moreover, the proof works for the predicative version of the system with finitely stratified polymorphic types. The result includes bounds on the Leivant levels of types used in the instances leading to undecidability.
The essay presents an analysis of "Jabberwocky" by Lewis Carroll. The analysis is based... more The essay presents an analysis of "Jabberwocky" by Lewis Carroll. The analysis is based on such sources as Carroll’s explanations, Martin Gardner’s suggestions in "The Annotated Alice", and dictionary research. Each sense clarification leads to a solution in Polish, which is then used in a new translation of Carroll’s poem
We show that a restricted variant of constructive predicate logic with positive (covariant) quant... more We show that a restricted variant of constructive predicate logic with positive (covariant) quantification is of super-elementary complexity. The restriction is to limit the number of eigenvariables used in quantifier introductions rules to a reasonably usable level. This construction suggests that the known non-elementary decision algorithms for positive logic may actually be best possible. 1998 ACM Subject Classification F.4.1 Mathematical Logic, I.2.3 Deduction and Theorem Proving
Theory and Practice of Logic Programming, 2018
We propose an interpretation of the first-order answer set programming (FOASP) in terms of intuit... more We propose an interpretation of the first-order answer set programming (FOASP) in terms of intuitionistic proof theory. It is obtained by two polynomial translations between FOASP and the bounded-arity fragment of the Σ1 level of the Mints hierarchy in first-order intuitionistic logic. It follows that Σ1 formulas using predicates of fixed arity (in particular unary) is of the same strength as FOASP. Our construction reveals a close similarity between constructive provability and stable entailment, or equivalently, between the construction of an answer set and an intuitionistic refutation. This paper is under consideration for publication in Theory and Practice of Logic Programming
ACM Transactions on Computational Logic, 2016
We show that the constructive predicate logic with positive (covariant) quantification is hard fo... more We show that the constructive predicate logic with positive (covariant) quantification is hard for doubly exponential universal time, that is, for the class co- 2-N exptime . Our approach is to represent proof-search as computation of an alternating automaton. The memory of the automaton is structured in a way that strictly corresponds to scopes of the binders used in the constructed proof. This provides an application of automata-theoretic techniques in proof theory.
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Papers by Aleksy Schubert