Intuitionistic computability logic

Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and "truth" is understood as algorithmic solvability, CL potentially offers a comprehensive logical basis for constructive applied theories and computing systems inherently requiring constructive and computationally meaningful underlying logics. Among the best known constructivistic logics is Heyting's intuitionistic calculus INT, whose language can be seen as a special fragment of that of CL. The constructivistic philosophy of INT, however, has never really found an intuitively convincing and mathematically strict semantical justification. CL has good claims to provide such a justification and hence a materialization of Kolmogorov's known thesis "INT = logic of problems". The present paper contains a soundness proof for INT with respect to the CL semantics. A comprehensive online source on CL is available at http://www.cis.upenn.edu/~giorgi/cl.htm

Glue TAG semantics for binary branching syntactic structures

This thesis presents Gl-TAG, a new semantics for a fragment of natural language including simple in/transitive sentences with quantifiers. Gl-TAG utilises glue semantics, a proof-theoretic semantics based on linear logic, and TAG, a tree-based syntactic theory. We demonstrate that Gl-TAG is compositional, and bears interesting similarities to other approaches to the semantics of quantifiers. Chapter 1, rather than discussing the arguments of the thesis as a whole, outlines the global picture of language and semantic theory we adopt, introducing different semantics for quantification, so that Gl-TAG is understood in the proper context. Chapter 2, the heart of the thesis, introduces Gl-TAG, illustrating its application to quantifier scope ambiguity (Qscope ambiguity) and binding. Ways of constricting quantifier scope where necessary are suggested, but their full development is a topic of future research. Chapter 3 demonstrates that our semantics is compositional in certain formal senses there distinguished. Our account of quantification bears striking similarities to that proposed in Heim and Kratzer (1998), and also to Cooper storage (Cooper ((1983))); in fact, we can set up a form of Cooper storage within Gl-TAG. We suggest in conclusion that the features in common between frameworks highlight the possible formal similarities between the approaches. One philosophically interesting aspect of our semantics left aside is that it depends on proof theoretic methods; glue semantics combines semantic values both by harnessing the inferential power of linear logic and by exploiting the Curry-Howard isomorphism (CHI) familiar from proof theory (see chapter 2 for a brief explanation of the CHI). The semantic value of a proposition is thus a proof, as some proof theorists have desired (see Martin-Lof (1996). This raises a question for future research; namely, whether Gl-TAG is an inferential semantics in the sense that some philosophers have discussed (Murzi and Steinberger (2015))

Proof search issues in some non-classical logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed.This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin’s cutfree LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1–1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called ‘permutation-free’ calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a ‘permutation-free’ calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to besound and complete with respect to a proof-theoretic semantics and (weak) cutelimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter

Proof Search Issues in Some Non-Classical Logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli (citeandreoli-92) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut-free LJT (citeherb-95, here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1--1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called `permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding emphet al (citeheu-sey-zim-96). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a `permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut-elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4

Programming with Bunched Implications

Submitted for the degree of Doctor of PhilosophyWe give an operational semantics for the logic programming language BLP, based on the hereditary Harrop fragment of the logic of bunched implications, BI. We introduce BI, explaining the account of the sharing of resources built into its semantics, and indicate how it may be used to give a logic programming language. We explain that the basic input/output model of operational semantics, used in linear logic programming, will not work for bunched logic. We show how to obtain a complete, goal-directed proof theory for hereditary Harrop BI and how to reformulate the operational model to account for the interaction between multiplicative and additive structure. We give examples of how the resulting programming language handles sharing and non-sharing use of resources purely logically and contrast them with Prolog. We describe the use of modules and their applications and discuss the possibilities offered in this context by multiplicative quanti ers. We provide a denotational semantics based on the construction of a least xed point of Herbrand interpretations. Finally we provide an annotated implementation of the operational semantics using the continuation-passing style (CPS)

Glue TAG semantics for binary branching syntactic structures

This thesis presents Gl-TAG, a new semantics for a fragment of natural language including simple in/transitive sentences with quantifiers. Gl-TAG utilises glue semantics, a proof-theoretic semantics based on linear logic, and TAG, a tree-based syntactic theory. We demonstrate that Gl-TAG is compositional, and bears interesting similarities to other approaches to the semantics of quantifiers. Chapter 1, rather than discussing the arguments of the thesis as a whole, outlines the global picture of language and semantic theory we adopt, introducing different semantics for quantification, so that Gl-TAG is understood in the proper context. Chapter 2, the heart of the thesis, introduces Gl-TAG, illustrating its application to quantifier scope ambiguity (Qscope ambiguity) and binding. Ways of constricting quantifier scope where necessary are suggested, but their full development is a topic of future research. Chapter 3 demonstrates that our semantics is compositional in certain formal senses there distinguished. Our account of quantification bears striking similarities to that proposed in Heim and Kratzer (1998), and also to Cooper storage (Cooper ((1983))); in fact, we can set up a form of Cooper storage within Gl-TAG. We suggest in conclusion that the features in common between frameworks highlight the possible formal similarities between the approaches. One philosophically interesting aspect of our semantics left aside is that it depends on proof theoretic methods; glue semantics combines semantic values both by harnessing the inferential power of linear logic and by exploiting the Curry-Howard isomorphism (CHI) familiar from proof theory (see chapter 2 for a brief explanation of the CHI). The semantic value of a proposition is thus a proof, as some proof theorists have desired (see Martin-Lof (1996). This raises a question for future research; namely, whether Gl-TAG is an inferential semantics in the sense that some philosophers have discussed (Murzi and Steinberger (2015))